After birding on the Farm I started to explore nearby bird-rich locations discovered on the internet. The closest was a small rainforest reserve close by at Booyong a little further along Wilson's River. The Reserve is very dense primary rainforest which once covered this entire area, very little remains. Trails can be walked inside but as is often the case most birds are seen around the edges. This Sulphur-crested Cockatoo sat and watched as I entered the dark forest trail.
Rainforest is my absolute favourite birding habitat. The light is usually terrible, birds are often scarce and hard to get a clear shot at in dense undergrowth, all very challenging but It is therefore very satisfying to get a decent image of birds that most people never see. This little Rufous Fantail for example, one of a pair that followed me around and put on a superb display in the murky light. I was using flash, it was the only way to get anything at all, but I was pleased with the result.
Rustling sounds from the forest floor are the first indication of the presence of this curious bird, the Australian Logrunner. It spends its time in pairs on the ground scraping away the leaf litter in search of insects and grubs, always in dense undergrowth. I was again pleased to get a recognizable shot using flash.
The Eastern Yellow Robin is quite a common little bird, similar to the British Robin in that it is not shy, in fact is one of the easiest birds to photograph. It is not related to the European Robin however, being in a different bird family.
The White-browed Scrub Wren is also quite numerous but much harder to photograph. It's one of those that never stops moving and being an LBJ (little brown job) is quite inconspicuous.
The Golden Whistler on the other hand is much easier. Being larger and mostly bright yellow it is easier to see, plus it advertises its presence with it's constant, whistling song. A very attractive and jolly little chap.
Golden Whistlers are strongly sexually dimorphic, that is the males and females are markedly different. The female (below right) is greeny-brown and nothing like as colourful as the male.
The Grey Fantail is similar to its Rufous cousin in shape and size, just not so colourful. I saw this species around the edges of woodland rather than inside the forest.
There were many more species lurking inside that dark rainforest that I did not manage to photograph. For example the magnificent Regent Bowerbird and I also heard and caught sight of Green Catbird, Rose-crowned Fruit Dove, and possibly Bar-shouldered Dove, although not sure of the latter. Anyway although very small Booyong is an excellent little Reserve with high biodiversity.
While at Booyong I spotted a flock of large, dark and very eerily noisy birds flying overhead. They flew with a very deep, slow and fluid wingbeat so I realised they were something interesting. I headed off in that direction and sure enough found a group of Yellow-tailed Black Cockatoos in a tree plantation, on private land but fortunately not too far from the road.
I had been really keen to see Black Cockatoos. Quite apart from their beauty they have a mystical quality and the Yellow-tailed variety is the largest of Australia's parrot species. Spectacular in sight and sound and beautiful in flight, it was exciting to see and hear them in the wild. This particular species is not one of the rarist but it is in decline due to habitat loss. I hope it does not go the way of the Australian Paradise Parrot which is either extinct or very nearly extinct due to habitat destruction to make way for cattle pasture.
About one hours drive NW of the Farm is the Nightcap Mountain Range National Park, it contains Boulder Creek Dam which provides the region's freshwater. It also happens to be an outstanding reserve for wildlife and birds in particular.
The first interesting birds that I picked up were these stunning Scarlet Honeyeaters feeding on nectar from a large waterside flowering tree.
The Scarlet Honeyeater is the smallest of Australia's Honeyeaters, which is a very large and diverse group. In other parts of the World it is known as the Scarlet Myzomela but I prefer the more descriptive name of Honeyeater. They do in fact occasionally hover in front of a flower to probe for nectar with the long, curved bill in a similar fashion to Hummingbirds and Sunbirds, but they also feed on insects. They were quite difficult to photograph high up in the canopy of a tall flowering tree, with the added complication that they never stop moving for an instant. Perseverence is required.
Every time I visited wooded areas I heard the distinctive and unmistakable sound of the Whipbird. There is no mistaking this noise. Starting with a peeping whistle & culminating in the loud cracking of a whip. The trouble had been in seeing the bird that makes this incredible sound. Well today I got lucky, a pair of Eastern Whipbirds appeared out of the thick undergrowth onto an open patch of grass and I was able to get a shot and actually watch them making their call. The whole body shakes as the whip cracks, it must take a lot of energy to keep producing that sound.
Ravens are quite common in Australia and there are three varieties. They are not easy to distinguish between each other and a couple of other Crow species, they all have white irises and are quite glossy, but I believe the one below is an Australian Raven. It is the only one with long hackles (Loose feathers) below the neck which were prominent when the bird was calling.
This next one however I think is a Forest Raven. Not just because of the thick wooded habitat but I observed it closely and heard its wonderful cawing call, deep, gravelly and far-carrying. A fantastic sound.
A feature of the Reserve here is a superb waterside boarwalk which passes through woodland along the banks of a creek winding its way down to the reservois. I spent some time here watching a startling blue Azure Kingfisher successfully catching fish from its overhanging perch. The light was not very good but I did get some recognizable shots.
I also squeezed off a few shots of a Little Pied Cormorant perched on a log below me. Not easy to get clear shots through the overhanging trees but this was the best of the bunch. I am always pleased to see new birds to add to my life list.
.....but the female is completely different. Grey-brown with a speckled breast, it looks like a thrush.
There were plenty more birds to watch here but many I had already seen elsewhere. In my search for new and interesting species I needed to vary the habitat & seek out new environments.
Birding Down Under. Australia part 1. The Farm.
While Elena was away selling her flat in Russia I flew to Australia to visit my sister Anne & family in Australia. Anne & John live near Ballina/Byron Bay in NSW in an area known as the Northern Rivers. It's beautiful country, not unlike parts of rural England with rolling hills of woodland interspersed with rich cattle pasture and mixed farmland.
Here are a couple of shots of sister Anne & John on their lovely farm.
And here are another pair of residents. This Tawny Frogmouth pair once frequented the veranda but seem now to prefer roosting in a large tree by the chicken house. They are of course active by night but during the day are not at all perturbed by us getting really close, being used to their friendly landlords by now.
The female (I think) below has adopted a typical "I'm just a tree" pose.
I sepent my first full day looking around at some of the bird life on the farm, and there was plenty to see. Laughing Kookaburra's are common in Australia and are easilly seen, often just one bird one its own perched in an exposed position waiting to sight a lizard or similar prey to pounce on. This is typical feeding behaviour of Kingfishers which is what the Kookaburra is, it just doesn't eat fish.
The Superb Fairy Wren is an attractive and very common little bird, overlooked perhaps because it is so tiny but any close examination of any low bushes or tall grass will usually reveal these little gems flitting about after insects, never still so not easy to photograph.
A stretch of Wilson's River runs through the farm and the first bird I saw down there was a stunning White-bellied Sea Eagle, a fairly common raptor in this part of Australia. The white-bellied sea eagle is revered by indiginous people in many parts of Australia, and is the subject of various folk tales throughout its range. They are very susceptible to human disturbance, particularly around nest sites, so numbers are declining quite considerably as a result.
Whilst down at the river I also spotted a pair of roosting White-breasted Woodswallows. These quite small and attractive birds are not related to swallows or hirundines at all, in fact their closest relatives are the crow-like Butcherbirds, Currawongs and Australian Magpies. Very strange.
...and this imposing looking, crow-like bird is a Pied Currawong. Again how different is that from a Woodswallow but the taxonomists assure us they are cousins. Named by indigenous Australians after the sound of its distinctive call the Currawong was very common on the farm. .Anne also fed a Pied Butcherbird daily from a bowl on the kitchen window inside the covered veranda. It was very tame. I could not focus the camera that close so in fact did not get a shot of it at all, but it is almost identical in appearance to the Magpie. Not at all like a Woodswallow!
......as are the Eastern Rosellas which could also be seen from the kitchen veranda foraging on the ground in the grass and occasionally flying up onto a fence post as seen here.
Crested Pigeons were another very attractive bird that could be seen every day hanging around the chicken house, obviously after the grain that John scattered around for his choocks. When startled, the crested pigeon takes to the air with a distinctive whistling 'call', the source of the noise can be attributed to the way the air rushes over a modified primary feather found on the wings. It diverts attention of ariel predators away from birds remaining on the ground.
One of the most common Australian birds is the Willie Wagtail. Not related to the true wagtail family it is named after its habit of waving its long tail up and down while foraging on the ground for insects or seeds. A very familiar sight wherever there are parks or gardens, it has adapted well to human habitation.
...and often numerous Australian White Ibises.
Purple Swamphens and Pacific Black Ducks wandered around the place like they owned it, behaving like domesticated chickens even though they are wild birds, White-headed Pigeons were seen a lot and Noisy Miners were prolific, another of those birds that have benefited from human presence.
Australians are naturally obsessive about invasive alien species, the Indian Mynah being a case in point, but this Noisy Miner is an endemic species that has exploded in numbers since humans have affected the landscape. They are aggressive and greedy territorial gangsters, driving many other species out and almost complelely taking over. Research has found culls of noisy miners can dramatically increase the number of other birds by up to 40 times and the number of species by 10 times in some areas. "There is just no question that if we could control the noisy miner we could have a huge biodiversity impact straight away," said professor Roberts, an eminent ecologist.
I watched this pest in action and noted that where Noisy Miners exist there are very few other birds present, particularly small species. A scourge on biodiversity in Australia.
One little bird species that has not been driven out completely by the wretched Miners is the Fairy Wrens. There are not many but their habitat is grassland and very low thickets, not the haunt of the Miner, so some have survived. The one below is a Red-backed Fairy Wren, a little charmer that I saw in the paddock on the hill above the house.
There are undoubtedly many more but I am not able to identify everything I see. John assures me that he often flushes quail on his morning walks but I wasn't that lucky. It was a great place to be for anyone, let alone a birdwatcher from the other side of the World, so needless to say I thoroughly enjoyed myself.
After a long, hot August with family visitors we finally got back some birding. We joined Derek and Barbara Etherton at their local patch in Zapata and along the Rio Grande, followed by Teba Rock and Fuente de Piedra. Although the latter was very dry we had a surprisingly successful morning.
For example we spotted eight different raptors, Honey Buzzard, Montagu's Harrier, Short-toed and Booted Eagles, Peregrine, Common Buzzard, Kestrel and Griffon Vulture.
The most unusual of these was the Montagu's Harrier, well done to Barbara for spotting it, I thought Elena was a good spotter but Barbara really does have hawk-eyes.
The biggest surprise of the day for me were the nine Black Storks on the Rio Grande. I normally associate this bird with Central & South West of Spain. These nine birds apparently were here from Poland, one did have a ring and I expect they had been identified from that.
In total we picked up 66 species which I thought was quite outstanding given the time of year and how dry it had been recently.
Fuente de Piedra for example has very little water and is quite sad at the moment. We were very pleased though to pick up both Bonellis and Olivaceous Warblers in the shrubs near the Visitor Centre. Very nice sightings.
Waders were very few apart from the ubiquitous Flamingos. The scrape on the left of the entrance driveway is bone dry unfortunately, due to a dispute with a local olive farmer who claims it floods and damages his adjacent olive trees. It seems that farmers always get their own way, often to the detriment of the environment and wildlife. I would suggest the reserve advises him to plant something else which might benefit from free irrigation & continue to keep the scrape wet to attract more birds. That is after all what they are there for..
After such a good morning and having seen 66 different species quite easily I expressed an interest in one of Derek's "Birdathons". That is to try to reach 100 species or more in a single day. So about a week later, Derek, Barbara, Bob Wright and I stood in a moonlit glade up in the Montes de Malaga listening to a noisy Tawny Owl calling from a tree right next to us. As dawn slowly crept up we heard and saw more, including Green Woodpecker, Jay, Blackcap, Hawfinch and Crossbill. However Derek made the observation that we were about ten birds short of what he had hoped for and had picked up on past attempts here.
After a very hearty breakfast we moved on to El Torcal where it was already busy with tourist buses. Again we dipped on a number of expected birds here such as Rock and Cirl Buntings but the unexpected Rock Thrush was a bonus. Well done Barbara again for spotting it.
Other birds here included Blue Rock Thrush, Black Redstart, Griffon Vulture, Black Wheatear, Sardinian and two excellent sightings - Subalpine and Spectacled Warblers.
Our list was building slowly but it became increasingly obvious that we were not getting the numbers we had hoped for. For one thing it was too hot today. In such heat birds are less active, prefering to stay cool and conserve energy. If we were to have any chance of reaching our target we would need to extend our search area. So it was off to Fuente de Piedra to make up for the lack of bird numbers seen.
Flamingo and Lapwing were easy. Olivaceous Warbler took a bit longer but it was still flitting around in the same bush as last week. A few waders were present including Snipe, Green Sandpiper and Black-winged Stilt. Linnets and Serins were seen and finally we started seeing all the Hirundines that had eluded us until now, Barn and Red-Rumped Swallows & House Martin.
From Fuente we drove across country to Teba Rock, picking up along the way a beautiful Common Redstart, probably the best looking bird of the day. I did not have my camera ready unfortunately, this was not a day for photography, it was all about seeing as many birds as we could in 24 hours. We also picked up Northern Wheatear and a couple more Lark species, Raven, Common Buzzard and a Kestrel finally put in appearances and things were looking up.
By the time we reached the Rio Grande however we were still a long way short of our target and it was now the hottest time of the day so all pretty quiet on the birding front and not a Black Stork to be seen. We did pick up a couple of Wagtails and Egrets before moving on to Zapata. Perhaps to make up for the lack of Black Storks a lone White Stork gave us a flypast, and a solitary Black Kite made an appearance. We also added Little Grebe and Turtle Dove, but we were still 28 species short of 100. Our only hope was to proceed on to the Guadalhorce and hope for the best.
As we entered the reserve the first birds we saw were some noisy Monk Parakeets. Another tick. A Cormorant in the riverside trees and both Common and Pallid Swifts and the deficit was coming down. We were all delighted to see a Pied Flycatcher, the first of the year for all of us. From the main hide we picked up some new birds, Kentish Plover, Greenshank, Black-headed and Mediterranean Gulls and best of all a magnificent Osprey eating its fish in a tree on the far side.
We moved on to the Laguna Escondida and found White Headed Duck, Shoveler, Pochard and had excellent views of a Kingfisher.
On the way to Laguna Casillas we spotted a Booted Eagle overhead. This left us needing another 14 birds and Derek identified an overhead Short-toed Lark, I found the first of several Whitethroats and with a Blue-headed Wagtail we were down to a target of 11. At the Wader Hide we picked out a Wood Sandpiper and a Sanderling and finally a small flock of Jackdaws, incredibly the first of the day. Just nine to go and on the "Old River" was a large gull roost within which the telescope revealed a pair of Caspian Terns, a couple of Slender-billed Gulls and at least one Audouins Gull. Only six required now and I called the others to check out a Warbler I had been following in the bushes on the other side of the track, we finally all had excellent views of a lovely Orphean Warbler. Bird of the day for me as it was a "Lifer". Pity I wasn't carrying a camera.
More scoping on the Old River revealed a couple of Little Stints, and then a small flock of Curlew Sandpipers flew in. Just two to go and although we were all very tired at this point we trecked down to the seawatch in the hope of perhaps a Tern, Gannet or Turnstone. As it happened the first bird we saw was a Balearic Shearwater flying quite close in, shortly followed by many more. Then another shearwater appeared but much smaller. The others agreed it must be a Yelkuan Shearwater. I acceded to their superior knowledge gratefully as I had no idea and was delighted to have reached 100 at last. We then retreated to a bar in SanJulien to celebrate with a long, cool drink. As it happened our recorders had forgotten to write down Ringed Plover so our final total was 101 species.
I decided to name this blog The Story of "O"s because we made double O, and three of the best birds of the day were the Olivaceous and Orphean Warblers plus the magnificent Osprey. I have been told The Story of O has other connotations. I wouldn't know.


They are driving impaired. Many hit-and-run drivers are under the influence of drugs or alcohol. Some think they can flee the scene long enough to sober up, while others hope they will never be connected to the accident.
They know they are at fault for the accident. Some drivers break numerous laws at the time of an accident: speeding, reckless driving, distracted driving, street racing, or striking a pedestrian in a crosswalk. These drivers may flee because they know they don't want to take responsibility for their actions.
They are unlicensed drivers. Some hit-and-run drivers do not have a license or are driving under a revoked or suspended license. These drivers also may have a history of similar offenses and know the consequences of being caught will be serious.
They are in the country illegally. In Southern California, a significant number of drivers are illegal immigrants who may fear that stopping at the scene of an accident, even to render life-saving aid, could lead to their deportation.
They are uninsured. Other drivers may not have insurance or are driving cars which are not insured. They flee the scene of an accident to avoid paying for the damage they caused.
They are driving a stolen vehicle. Again, a driver who knows he is breaking a serious law at the time of the crash may weigh his options and decide to risk hit-and-run charges if he is caught.
They are scared. Some drivers simply run because they are scared, even if they haven't done anything wrong necessarily (until fleeing the scene of the crash). These drivers may be young, inexperienced, or simply poor decision makers.
They don't care. Sadly, some hit-and-run drivers simply don't care about the people whom they injured, the damage they caused, or the laws they broke.
They are protecting themselves. Some drivers flee the scene of an accident because they wish to protect their good reputation, avoid exposure, or to keep a secret (such as where they were or where they were going).
If you were injured in a hit-and-run accident and you are in need of legal guidance, we may be able to help. At Russell & Lazarus, our Orange County accident attorneys offer free case evaluations. To schedule a private meeting with one of our lawyers, call us today at (949) 851-0222.


The Premium category does not include polished beads, which have their own category.
Beads which cost more than 20 cents each are included in this category. Thus, this category includes beads which are well made and expensive, but also beads which are merely expensive. I carry many premium-quality beads which are not in this category because they cost less than 20 cents each. My markup on premium beads is no higher than on the less-expensive beads; it is the supplier's price which is high.
I individually inspect about half of the premium beads that I send out. If, in my experience, a batch of premium beads is consistently perfect, I will not inspect them before sending them out. Thus, I only inspect those premium beads that I know to have a condition issue. At 20 cents per bead or higher, you have a right to receive only perfect beads. Thus, if you receive any premium beads that have a defect that makes them unusable, please contact me for a pro-rated refund. Of course, it is always possible that we may disagree as to what constitutes a defect. If you are not satisfied with the condition of the beads, I will pay your return postage cost.


text:
Home Reviews Beats 2019 Netflix – Non-Spoiler
Beats 2019 Netflix – Non-Spoiler
I watch the brand new movie or TV show on Netflix. Today's movie is called Beats. I need you in the comments. Let me know what you thought, or did you expect this film, and let's talk about a young talented man who has suffered a tragic loss in his life, met a disoriented manager trying to regain his career Being the first to be responsible for writing the summaries of the films on IMDB I would like to take a look at that because the wording is not what I just said. I had to spell it out to form a coherent sentence so I could get to this buddy, but okay, we talk about beats that I had no idea what it was about.
I did not even see a trailer that I knew about that he is Anthony Anderson, whom I really like. He knows more about his comedy, but when he retires and takes on a role that's out there, it may be something else, though the films are not running. Not always the best I've ever had. I appreciate what he tells us and I always buy him as an actor when he is deep and emotional in this movie. I bought it and was curious about what he would do here and then the concept of the beats, how that fits in the movie actual story and in the first scenes you see the figure who wears beats headphones. I'm like oh no, this is just a commercial for Beats by Dre.
Do not get me wrong. I love beats for a good couple i had the same couple for Mexico no this is a commercial beginning I did not know which direction this movie would go in and before that scene with the beats I was very excited for the potential because there is a scene at the beginning of this movie that surprised me so much that I did not read the summary.
Beats 2019 Netflix "Non-Spoiler" Movie Review – RapidDrama.com
I would not read the abstract if you had not seen the movie because the summary actually reveals what happened to it at the beginning of this movie, but that shocked me so much it was so well handled and it was emotionally correct and it has this Kid hurt, for that kid that's played on average by Kahlil, now I know him from Cobra Kai, he played a minor role in this show and he was really good, but that's his actual, I think his feature film debut is so cool that he sees it right I like it when a young actor gets the chance to wear a movie, and although he was not the most dynamic actor I've ever seen, he was really good at that role because he did what he did would have.
When he realizes he is the talent has to persecute this child while this girl is being chased by his child, the circle is closing, and I really enjoyed this movie, and all the story points that I've just talked about were entertaining and intriguing emotional, because you buy, how he feels when he finally ventures out of his comfort zone.
You can feel the danger in him. The stress level is very high at that moment and because of what happened, you understand that and Anthony Anderson even though his character does not really know you at what angle he comes because he uses that kid to get his career back in Getting him involved with things, in the end, you bought that relationship, so you're happy, or you're upset about how this movie ended, right, it's a bit cliché, a bit predictable, a bit conventional, It's a movie I've seen elements of before, but it was adorable as well.
It was somehow surprising that it captured the film. In such a situation I have some problems with the film production. It's a bit choppy in terms of the cut. Some of the transitions and the score did not inspire us, even though the score was good, they did not fit into the expected moments to be emotionally resonant, they play that score and somehow overwhelms the events on the screen, but all in all, I did a good time with this movie.
It was a surprise, one of the bigger surprises of the year on Netflix and I think one of the surprises. One big reason is that you have a star for the first time because of the talent you have for the first director of feature films, Chris Robinson and then an actor like Paul Walter Hauser who comes in and plays movies like I Tanya Black Clansmen in Cobra too, but you have actors like this, he and Anthony Anderson and this new kid, and they all work very well together and it's just a beautiful story that I'm not sitting here saying that it's the best movie I've seen all year.
There were definitely shortcomings and I do not think everyone will love this movie, but I really enjoyed it and it was surprising to me that neither 66 nor 4 beats were said, I enjoyed it more than I thought I would have thought it would be a commercial ats and then I thought it would be a clichéd and predictable story and it was a bit predictable, but I still had a good time with it.
beats movie
beats studio wireless
beats trailer
beats updater
Previous articleYesterday 2019
Next articleShelley Horror Movie Reviews "Spoiler Free"
Neal L. Thomas June 21, 2019 at 2:23 pm
There were some parts that took me out of the movie and it wasn't perfect but I really liked it for the most part
Earnest August 22, 2019 at 11:29 pm
Crazy Eyes done got outta jail & became a Momma.
Fantastic Beasts: The Crimes of Grindelwald Movie Review
The Grinch Movie Review
Once Upon A Time In Hollywood Movie Review
Ford v Ferrari Movie Review
Top 10 Favorite Movies of 2019
Guava Island Movie Review

redpajama_set_name: RedPajamaCommonCrawl


CHARITY group Ocho Rios Orchid Society is set to host its annual show on February 10, at Grizzlys Plantation Cove in Priory, St Ann.
President Pam Chong told the Jamaica Observer North & East last Wednesday that the aim of the society is to promote the love and growth of orchids while giving back to the community through proceeds from the event.
"We will have vendors selling plants, produce, have a food court and lots of classic cars on display. The show will begin at 10:00 am and should end at 5:00 pm. We are a non-profit organisation and we donate to charity. We donate to various places including Mustard Seed, the churches and St Ann's Bay infirmary among a host of others," Chong said.
Chong is expecting more than 500 lovers of the exotic plant at this year's event, which she said has constantly attracted repeat patrons each year because of its uniqueness and entertainment package.
"Come out and see the beautiful orchids, beautiful people, beautiful cars," Chong said.
The Ocho Rios Orchid Society was formed 27 years ago with the objective of fostering and promoting a love for orchids and to help its members become knowledgeable and proficient in growing the plants.
The society has also offered wide travel experiences to members. Many have travelled to Panama, Thailand, Singapore, Malaysia, Hawaii, Bali and the United States of America visiting nurseries and importing plants.
Members are encouraged to expand their collection and provide at least three plants for display at the annual show.


We could define form-field value trimming same way as the JS-method is defined.
Next message: bugzilla@jessica.w3.org: "[Bug 14338] New: <track> What does line position auto actually do?"
Previous message: bugzilla@jessica.w3.org: "[Bug 12544] <video> MEDIA CONTROLLER requires track kind for in-band tracks"


The sun is shining, the weather is warm, and the summer of Florida Man continues unabated. Deputies with the Marion County Sheriff's Office arrested the driver of a little red Corvette in Ocala, Florida last weekend after he was reportedly seen driving at 100 mph on surface streets, but at least it was easy this time: The driver promptly told them he didn't have a driver's license, he had just taken Percocet and used cocaine, marijuana, and heroin, and he had fake tags on the car to boot.
According to the Villages News, a concerned motorist called police on Sunday evening after seeing 54-year-old David Whitlow speeding and weaving in and out of traffic. Deputies soon spotted the red Corvette and initiated a traffic stop, at which point Whitlow was reportedly slow to react to the emergency lights and struck the curb when he tried to pull over.
Then came the truth vomit. His first words to the officers reportedly were "I don't have one," referring to a driver's license, and he told them he had placed fake license plates on the Corvette because the police had confiscated his real tags after a previous conviction for driving on a suspended license. He then reportedly asked about calling a bail bond service and failed a sobriety test.
Just for good measure, the deputies watched a baggie of marijuana and cocaine fall out of his shorts while they were putting him in the back of the patrol car, and a crack pipe was later found in his pocket at the jail, according to Ocala.com. But just look at that mug shot: Florida Man ain't care.
A unique setup highlighted in a beautiful short film.
He was reportedly upset the utility workers wouldn't move their trucks while working on a line.


The purpose of the PEAR project is to investigate the possibilities of using Augmented Reality technologies to increase public engagement. In contrast to the nowadays "normal" mode of online engagement with public matters that takes place at a distance, we explore more situated contexts where information is provided and interaction takes place at the relevant physical location.
The PEAR framework, short for Augmented Reality for Public Engagement, aims to allow the public to participate in discussions, or be informed about issues. People can physically go to a particular location, and use their mobile devices to access on-site an Augmented Reality (AR) visualization. This visualization can represent results of an online voting process or data collected by various sensors, and is updated live as more people participate or additional data is collected.
Architecture illustration of the developed PEAR system.
PEAR is motivated by the objective to provide people with information and/or debate updates at a specific site to which a public issue is related. It is true that people can also obtain such information online, but this is at a greater, both geographical and psychological, distance. We believe that providing people with access to "live" information in-situ encourages participation and engagement, and also allows them to reflect from a closer, more involved perspective.
A pilot study ran in May-September 2016 as PEAR 4 VXO in collaboration with the municipality of Växjö, related to the ongoing public debate regarding future development of the Ringsberg/Kristineberg area in Växjö. The public was invited to vote about this issue by using specific hashtags on Twitter (e.g. include #parkRK in a tweet to @vaxjokommun).
People could also visit the physical location, use their mobile devices to scan an information poster that was placed there, download the AR app from Google Play, and view (in AR) a visualization that represented how many votes were cast for each option. This visualization was live and updated as long as people kept voting. See the news item New smartphone app aims to increase public engagement in local issues.
A second study was conducted during the 1st International Digital Humanities Symposium, organized at Linnaeus University in November 2016, as PEAR 4 DigiHum.
The AR application is developed using the Unity3D game engine and utilizes the Vuforia SDK for the application's AR features. The server and database, based on Node.js and MongoDB, enable the aggregation of data from social media sources such as Twitter, as well as process and provide the data to the AR application.
The project is carried out within the VRxAR Labs research group.


text:
← A Look Back at 20 Years of Mankoff's New Yorker
Market Advantage →
THIS WEEK IN COMICS! (3/29/17 – Goodbye, Sweetheart)
BY Joe McCulloch Mar 28, 2017
Above we see my favorite bit of Dominique Goblet's Pretending Is Lying - published in French in 2007 and entirely re-lettered by the artist herself for the 2017 English edition. Goblet also collaborated on the translation itself, with Sophie Yanow; it's a work that benefits from as much of the artist's presence as possible. Immediately, we think of the unique state of the work's visual presentation: how its 'color b&w' pages incorporate the wear of time into the drawing, because some of the boards are a decade older than the eventually published book. And yet, to usurp time is not the only defiance of the work; even Goblet's resistance to narrative chronology, to the spectacle of ages passing, even modest ones, is not the paramount affront.
The tradition of American autobiographical comics is that of centering the author. Harvey Pekar, addressing the reader; the present-tense exclamations of Robert Crumb's myriad neuroses. Goblet's autobiography, to this American, is marked by departure. Primarily, she reacts. To my mind, the narrative genders her associations: with men, she interacts with strong, charismatic, not always sympathetic personalities. There is her father, a drinker and a fast driver with whom she spends roughly half the book trying to visit. There is also her lover, with whom roughly the other half of the book is concerned; in fact, he co-wrote those portions of the text concerning him, and by god is it not disquieting to witness a male leaving some imprint on a formidable female artist's work, his lover's work? God, it unnerves me, though I don't know the details that aren't in the book, and art anyway isn't really really-for-real in touching the banality of creative exchange.
But anyway, up top: they have temporarily broken it off, Goblet and her lover, and her father will not accept her tears. This is the primary crossover between the two streams of male interaction in Pretending Is Lying, and here -- perhaps with Goblet herself -- we see a new facet of her father's blowsiness. "Believe me my dear, nobody's important enough for your tears..." It is not that he wants to overpower her with his personality; this is perhaps what happens regardless, but he does not want it. What he wants, is for her to be as fierce as him, though she will not do it in the same way.
Women, in Goblet's book, are less knowable. An ingenious prelude introduces Goblet's mother as a magician and a charlatan, believed to possess nigh-wizardly powers by the young "Nikske", which means "Little Nothing". Later, the mother is presented as a fearsome God, binding the child Goblet with rope as she screams and cries; it is horrifying, and followed immediately by the mother's wary comfort of the child, who accepts everything as natural. Goblet's own daughter is elsewhere characterized as faintly unknowable in the way small children inevitably are, while her father's lover is drawn as a Munchian grotesque, with whom Goblet comes to quarrel over her daughter, a double-reflection of her mother. The women in the book, thus, are powerful and contentious figures, against whom the artist collides.
To her father, Goblet depicts herself absorbing. Or perhaps she is absorbed. As it happens, when her father drives off with her, drunkenly, dangerously, in chapter 3 of 4, Goblet physically disappears from the book. The final chapter depicts only her lover on-panel, as he seethes over boots, listens to music, tangles with the cat, and from the author's disappearance we understand the absence of her as a presence in his life. Previously, he was characterized as haunted by a ghost, by the indecisive conclusion of his last relationship. Goblet draws the other woman as a ghost, another elusive feminine figure, but she simply declines to draw herself at the end of her book, and her absence is felt, by us and him. Hers is a space in his obnoxious, talkative mind. His cat brings a dead bird and he invents from that a reason to call her. Goblet pulls back from depicting characters to depicting city architecture; François Schuiten appears in the Acknowledgements. She pulls back from architecture to birds, and then to just the sky, a smear of oil against paper, with text booming in space. God, she is there again. She didn't leave.
"I really want to see you!" he says.
"When?"
"now"
PLEASE NOTE: What follows is not a series of capsule reviews but an annotated selection of items listed by Diamond Comic Distributors for release to comic book retailers in North America on the particular Wednesday identified in the column title above. Be aware that some of these comics may be published by Fantagraphics Books, the entity which also administers the posting of this column, and that I also run a podcast with an employee of Nobrow Press. Not every listed item will necessarily arrive at every comic book retailer, in that some items may be delayed and ordered quantities will vary. I have in all likelihood not read any of the comics listed below, in that they are not yet released as of the writing of this column, nor will I necessarily read or purchase every item identified; THIS WEEK IN COMICS! reflects only what I find to be potentially interesting. You could always just buy nothing.
SPOTLIGHT PICKS!
What Parsifal Saw: A new Fantagraphics collection of recent work by Ron Regé, Jr., without a doubt one of the most distinctive cartoonists of his generation. The slim, 80-page color softcover deals in "magical, alchemical, ancient, and mysterious ideas," per the publisher, which as it happens is also issuing a new softcover edition of Regé's 2012 esoteric manifesto The Cartoon Utopia this week, in case you missed it the first time. This is total invocation here - a whirling tour of an author's worldview with no conceivable compromise; $14.99 ($24.99 for Utopia).
Otomo: A Global Tribute To the Mind Behind Akira: TALES OF CAPITALISM - I saw the Japanese edition of this at the same bookstore where I picked up Golgo 13 magazine last week; the ¥ 5,400 cover price translated to north of $70.00 USD off the rack, which was definitely too rich for my blood. Now, like magic, Kodansha has a much less expensive English edition ready at basically the same dimensions: 168 pages at 10.7" x 12.4" in hardcover. You may recall Ōtomo winning the Grand Prix at Angoulême in 2015; this book is essentially a mass-market catalog for an exhibition held the following year in his honor, amounting to a fancy collection of tribute art by international notables. Contributors include Manuele Fior, Masamune Shirow(!), Tanino Liberatore, Taiyō Matsumoto, Tomer & Asaf Hanuka, Naoki Urasawa, Juan Giménez, Stan Sakai, Tsutomu Nihei, Jordi Bernet and the late Jirō Taniguchi, among many others. Will anyone dare throw in a Domu piece? Note also that Kodansha is planning a deluxe all-hardcover box set of the entire Akira series (and the Akira Club art book) in the unaltered right-to-left format for this October; $29.99.
Audubon: On the Wings of the World: If the Otomo book above doesn't fit your criteria for a comic -- which would make since, because it isn't -- then your *formal* Eurocomic pick of the week is this 2006 album from artist Jérémie Royer and writer Fabien Grolleau, profiling the 19th century naturalist of the title, author of the famous The Birds of America. Lots of opportunities for nature drawing, released in English via Nobrow as a 184-page, 8.07" x 10.83" color hardcover; $22.95.
Lobster Johnson: The Pirate's Ghost #1 (of 3): Your no-doubt extremely smooth and pretty mainstream costumed action comic of the week comes from one of the most reliable teams around - writers Mike Mignola & John Arcudi, and artist Tonči Zonjić. With the main Hellboy series wrapped up, this is the only Mignola-branded project where I never miss an issue, as the focus is very tight (early 20th century period antics starring a determinedly un-nuanced pulp avenger), the pace is very fast, and nothing ever really gets in the way of its formidable surface appeal; $3.99.
Space Riders: Galaxy of Brutality #1: In contrast, I don't really know anything about the Black Mask-published Space Riders title ("Capitan Peligro and his fearless crew deal harsh justice to the scum of the galaxy while searching for the hidden truths of the universe!"), under which this appears to be the second miniseries, but I do like what I've seen of the artist, Alexis Ziritt, who works in a kind of tattoo art style buzzing between Mike Allred and Gary Panter. Try and flip through if you see it. Preview; $3.99.
Valerian and Laureline Vol. 14: The Living Weapons (&) Blake & Mortimer Vol. 24: The Testament of William S.: Cinebook has a whole stack of Franco-Belgian translations out via Diamond this week, so I'm gonna highlight a duo of classy youth adventure comics with long histories behind them. The Living Weapons, granted, isn't from *too* many years ago, being a 1990 release from creators Pierre Christin & Jean-Claude Mézières. If you have the old iBooks omnibus Valerian: The New Future Trilogy (2004), this 56-pager is one of the albums in there. The Testament of William S. is far more removed from the source; it's actually the newest Blake and Mortimer investigatory adventure, released in French only last year. The writer is Yves Sente and the artist is André Juillard, working in the tradition of creator Edgar P. Jacobs for 72 pages set in the late 1950s, at which time the actual series was already ten years old; $13.95 (Weapons), $15.95 (Testament).
Scene But Not Heard (&) Beyond Palomar: Here's a pair of books from longtime alt-comics guys that have been around before, and now will be available again. Scene But Not Heard is a 128-page collection of wordless, rather Kurtzmanesque color comics Sam Henderson created for Nickelodeon Magazine, perhaps rousing memories of an era when that very mainstream forum gave a lot of work to small-press cartoonists. Noah Van Sciver contributes a comics-format introduction to the Alternative Comics release, which was initially co-published with Top Shelf, although I don't know what the status is now. Beyond Palomar, meanwhile, is a collection of Gilbert Hernandez comics from Love and Rockets, specifically the one that contains Poison River, one of the holy fucking shit comics of the late '80s/early '90s. It is accompanied by its lifelong companion companion serial, Love and Rockets X, for 256 big pages of challenge and reward; $14.95 (Scene), $16.95 (Beyond).
The Legion of Regrettable Supervillains: Also humor but not exactly comics is this print iteration of an enduring type of online entertainment - the funny commentary on wacky old funnybooks. Actually, I can't remember the last time something like this appeared in print. The author is Jon Morris, a webcomics artist going back quite a ways (he was nominated for an Ignatz in 2001, the year the awards were cancelled), and a comics blogger dating to at least the late 1990s. The present volume is a 256-page Quirk Books survey of oddball rogues, following up on a 2015 look at similarly goofy superheroes; $24.95.
Asian Comics (&) On the Graphic Novel: Finally, here are two University Press of Mississippi books-on-comics now available in softcover editions. Both are of some unusual interest. Asian Comics is a massive (352-page, 8.5" x 11") overview of the comics of 16 nations - not Japan, but China, Hong Kong, Korea, India and others. The author is John A. Lent, founder of the International Journal of Comic Art, and I man whom I suspect has forgotten more comics than any of us have read. On the Graphic Novel is a 375-page Bruce Campbell translation of a text by Santiago García, a Spanish-born writer and translator recently seen with the artist David Rubín on a version of Beowulf released stateside by Image. García "follows the history of the graphic novel from early nineteenth- century European sequential art, through the development of newspaper strips in the United States, to the development of the twentieth-century comic book and its subsequent crisis," as the publisher puts it; $30.00 (each).
FILED UNDER: Dominique Goblet
7 Responses to THIS WEEK IN COMICS! (3/29/17 – Goodbye, Sweetheart)
Knave says:
Wasn't the final issue of Alan Moore's Providence scheduled for release this week? Hmm, could be next week, if it hasn't hit the regularly scheduled delay.
Joe McCulloch says:
I've heard in a few places that it might arrive, though Diamond does not have it listed for release. I'd keep my eyes peeled, though…
David Bitterbaum says:
Black Mask is a really hit-or-miss publisher for me. When they, "Miss," it is bad, but when they hit, they hit hard. Plus, props to a publisher for not hesitating to make a solid political statement that might ruffle feathers–something which big publishers like Marvel and DC like to pretend they are doing but aren't really.
Billy Foss says:
That's an important distinction with Lobster Johnson lacking nuance. While the supporting cast and the villain provide the emotional weight of the story, the Lobster remains stoic, and ever vigilant. It's a very different experience from the typical superhero fare of today with endless navel-gazing, garish colors and/or techniques, and decompressed storylines. Personally, this characterization or lack thereof is a key factor in what makes the series a favorite monthly book. Not to mention Tonci Zonjic's clean lines and pitch blacks, which are the perfect complimentary style, of course.
Dave Knott says:
I took a chance with the previous Space Riders comic and can only give it a mild recommendation.
The art is definitely the selling point for this one, a very nice fusion of mainstream SF and punk aesthetics.
However, the writing is not so great. B-movie trash and not necessarily in a good way.
Archibald says:
Knave & Joe: The final issue of Alan Moore's PROVIDENCE (#12) is scheduled drop (finally!) next Wednesday, April 5th according to the Diamond shipping list for next week:
https://www.previewsworld.com/NewReleases?pub=AVATAR%20PRESS%20INC&releaseDate=4/5/2017
Awesome. Thanks for the heads up, Archbald.


CFD trading will be available only in Close Order only.
Trading with Forex instruments will be available with higher spreads and Stops&Limits level.
CFD trading will be in Close Orders only.
Weekly trading Contests starting on 27.12.2010 will be transferred to 03.01.2011.
Finish of the Monthly "from 5$ to 1.000$" competition.
All trading conditions will be restored to normal.
We recommend yo to keep your Margin Level above 100% and set up Stop Loss orders on your open orders.


text:
« Further Details on those Medieval "Christ Mythicists"
Gods (An Anthropology of Religion Perspective) »
Once more on The Ascension of Isaiah and the Cathars
Filed under: Early Christianities, Intertestamental History and Thought
Tags: Ascension of Isaiah, Cathars
One more note on the medieval Cathars and their use of the Ascenion of Isaiah. . . . .
Among the texts that they obtained from the Bogomils was the Vision of Isaiah (chapters 6-11 of Asc.Isa.), a Greek Gnostic text of the first century A.D., which presented a cosmic view of the creation that was in conformity with dualist beliefs. The Cathars did not use the partial Latin translation made in late antiquity, but commissioned a new Latin translation from the Old Slavonic text, a version which the Bogomils had amended to conform with their own teachings.38
(Hamilton, 107f)
The author is relying on R. H. Charles — as per the footnote:
38 The medieval Latin version exists only in a text printed at Venice in 1522 by Antonio de Fantis and reprinted by A. Dillmann, Ascensio Isaiae Aethiopice et Latine (Leipzig: F.A. Brockhaus, 1877), 76–83. It was read by the moderate dualists of Lombardy, Moneta di Cremona, II, ix, 4, ed. Ricchini, p. 218. For the full edition of the texts in all versions: R.H. Charles, The Ascension of Isaiah (London: Adam and Charles Black, 1900).
(Hamilton, 108)
But a question arises. If the Cathars held a belief in an appearance of Jesus into another world beyond ours, where he was both born and crucified, what need would there have been to modify the Asc. Isa. by removing that "little gospel"? Surely it could be understood as happening in that other world. If the original Asc. Isa. lacked that passage depicting Jesus' birth in Bethlehem and eventual crucifixion in Jerusalem it presumably was not because the original audience for the text related in any way to the beliefs in the "other world" later reflected among the Cathars.
Hamilton, Bernard. 2006. "Bogomil Infuences on Western Heresy." In Heresy and the Persecuting Society in the Middle Ages: Essays on the Work of R.I. Moore, edited by Michael Frassetto, 93–114. Leiden ; Boston: Brill Academic Publishers.
Author Neil GodfreyPosted on 2020-07-09 23:58:51 GMT+0000 2020-07-10 01:58:22 GMT+0000 Categories Early Christianities, Intertestamental History and ThoughtTags Ascension of Isaiah, Cathars
6 thoughts on "Once more on The Ascension of Isaiah and the Cathars"
The belief in two crucifixions is one of their more secret teachings. They don't preach it openly, for fear that the people is scandalized.
(from Liber supra stella, ed. in I. von Döllinger, Beiträge, op. cit. p.81)
Neil Godfrey says:
That assertion comes from Sacconi and was quoted in the previous post. But it doesn't remove the problem: secret or not, the question remains unanswered.
In Revelation there is a birth in heaven. Hence the Cathars could accept it in Ascension of Isaiah, if they found it there.
That's not the birth in the Asc. Isa. — the differences are obvious and many. But I'm trying to dig into the evidence, not surmise what "could have been". It's slow going because it appears most of the studies that explore the depths of the niche I want are in Italian or French — and the sources themselves are in Latin. Not insuperable barriers, but they do make for slow progress.
The primary question that I am attempting to explore here is what the Ascension of Isaiah looked like for the Cathars. That has some relevance to the problem of the original character of the document. According to R.H. Charles the Asc Isa was doctored by dualistic groups (let's not call them "heretics") in the Bulgarian/Slavonic region and introduced to others who came to be known as Cathars in northern Italy and southern France. The doctoring is said to have been primarily in the excision of 11:2-22 — the "pocket gospel" depicting the birth of Jesus in Bethlehem and then his death "in Jerusalem".
If there appears to have been no doctrinal necessity for Cathars to have removed that section then that situation has consequences for the argument concerning the originality of 11:2-22.
It is far from a decisive argument. But it is one that deserves some consideration, I think. And besides, there is so much more primary source material concerning the beliefs of the Cathars and their views and use of various documents, the Asc. Isa. included (and they used other canonical and non-canonical sources, too), that unless someone more familiar with all the sources comes along to help me out I am going to take a long time sorting things out, to whatever extent possible, myself.
Why rely on Charles? More recent scholarship supports L2 and S as dependent on a lost Greek version, which did not have the pocket gospel. See M. Knibb in his AoI introduction in The Old Testament Pseudepigrapha, vol 2, pages 145-146. Jonathan Knight also agrees with this.
Previous Previous post: Further Details on those Medieval "Christ Mythicists"
Next Next post: Gods (An Anthropology of Religion Perspective)


By incorporating environmentally friendly aspects into your work ethic, you are communicating a responsible message to your clients and target market.
With our ever changing planet conditions there has been a massive movement to decrease our carbon foot print that we leave in our wake of destruction. Eco-friendly gifts relate a conscious message to your giving process, by saying "we care about our planet." Our large range of recycled, reusable, natural, eco-friendly gifts will assist you in making the correct choice with the right message.
Conveying a positive message is a vital part of any marketing plan. By incorporating environmentally friendly aspects into your work ethic, you are communicating a responsible message to your clients and target market.
Whether you are looking for Eco-friendly gifts that can be put together to create a hamper for a client, or you are looking for Eco-friendly gifts that will offer long term usage, Ignition Marketing has the range and the team to help you. You might have a solid marketing vision in place or you may just be shopping around for a corporate gift. Either way, our eco- friendly gifts are ideal for all industries and preferences.


Cathryn Gross was the person we turned to when we were first starting out. How do you register a business? Do you know an accountant we can talk to? Can we actually afford to do this? So many questions… And she had the answers. Not only did she get us started, Cathryn is a financial adviser who's helped plenty of people whose lives are in a state of change. Empowering women is her speciality, and we're all about that. Please welcome Cathryn to the Three Minute Squiz.
Over breakfast before I make the school lunches. It's the only quiet time of the day.
You ditched investment banking to start your own business. What was behind that?
I can really only join the dots looking back on this. When I quit investment banking, the two things I knew was I wanted to help people and I wanted flexibility around how I did that. Before starting Twelve Wealth, I worked as an executive coach and ran women's leadership programs. I loved working with smart and motivated women to help them achieve their goals and work through professional issues… but I missed finance (who would have guessed!) I realised I could use my finance and coaching skills to start a financial advice practice for women. I had to start my own business because I had a very clear idea of what our service offering needed to be, and none of the financial advice businesses in the market at that time were offering that service, so I built a business I knew women would engage with.
What is the most common thing the people who come to you are worried about?
Their future! Most of my clients are women who are going through divorcees or massive change. They've usually had a whole future mapped out for themselves… then the sands shifted and nothing looks the same anymore. They want help to map out what a new future might look like and how the decisions they make now might impact that future.
If there was one thing people could do to be more financially fit, what would it be?
That's easy. Save first not last!! Get payroll to put a portion of your pay into an account you can't see then start investing it once you have three months of living expenses in cash.
It does! But honestly, it's about achieving their life goals – the finances are just the tools to get them there. I find financial goals are not so exciting, but what we can do with the finances is the feel-good bit.
You have three kids. What's your top tip for other parents looking to take the plunge to start their own business?
First cab off the rank is my mum. She died at just 62yo and took a secret to her grave that we've since uncovered. I'd love to be able to hear her story. Lydia Lassila – the Australian freestyle skier who competed in five (yes five!) Olympics. I'd love to unpack how she has developed such incredible mental strength… something I did not appreciate until seeing her in Australian Survivor. She was by FAR the strongest competitor! Tanya Plibersek because I'm desperate to convince her to be head of the Labor Party and our country in a more collaborative and compassionate way. And finally, Warren Buffet to try and get a few more gems on how to consistently generate returns for our clients at Twelve Wealth.
I just loved Jasper Jones by Craig Silvey. And although I read it at least 20 years ago, The Poisonwood Bible by Barbara Kingsolver.
INXS Kick. Still love it.
My mum and dad were both incredibly reliable people. They always did what they said they would do. So it's not the best piece of advice, but it's the best demonstrated behaviour! I think I'm pretty reliable as a result.
Perfectionism. Near enough is most of the time good enough.
My worst is definitely my impatience. My best…hmmm… I'm incredibly organised, but really just try and be organised one day at a time.
The number of women over 60yo who are living in poverty or without housing security. Many have spent their lives providing love and security for others, then face their golden years with no security of their own. It's not right.


text:
\section{Introduction}
\label{sec:introduction}

Generalized geometry was originally introduced in
\cite{hitchin-CY:2003} as, naively, the differential geometry
resulting from replacing the tangent bundle $T$ of a manifold $M$ with
the sum of the tangent and cotangent bundles, $T\oplus T^*$, which is
naturally endowed with an $\mathrm{SO}(n,n)$-structure. Classical concepts have
then generalized analogues, such as generalized metrics and
generalized Calabi-Yau or generalized complex structures. An
interesting feature of this geometry is that the bundle of
differential forms $\bigwedge^\bullet T^* M$ becomes a bundle of
spinors, in which some of these structures are formulated. For
instance, a generalized Calabi-Yau structure is given by a closed
section of $\bigwedge^{ev} T^* M\otimes {\mathbb{C}}$ or $\bigwedge^{od} T^*
M\otimes {\mathbb{C}}$ consisting of pure spinors.

The generalized tangent space $T\oplus T^*$ can be further modified by
adding new pieces. The simplest one is the rank $1$ trivial bundle
over $M$, which we denote by $1$. Since the natural metric of $T\oplus
T^*\oplus 1$ has signature $(n+1,n)$, the group of symmetries becomes
$\mathrm{SO}(n+1,n)$. As this group is of Lie type $B_n$, we call this
geometry generalized geometry of type $B_n$, from now on
$B_n$-geometry. Correspondingly, ordinary generalized geometry is
called $D_n$-geometry. Exceptional geometries based on the split real
forms $E^n_n$ have also been studied as, for example, in
\cite{hull:2007}.

$B_n$-geometry was originally introduced by Baraglia in
\cite{baraglia:2011} (Section 2.4). It also arises as a particular
case of the more general situation studied in
\cite{chen-stienon-xu:2009}. Section \ref{sec:Bn-gengeo} of the
present work is devoted to stating the basic features of
$B_n$-geometry.

In Section \ref{sec:G22-str}, we introduce $G_2^2$-structures on an
orientable $3$-manifold $M$ as suggested by
Baraglia. $G_2^2$-structures are defined by analogy with generalized
Calabi-Yau structures. They are given by a closed section $\rho$ of
$\bigwedge^\bullet T^*$ consisting of non-pure spinors. We consider the existence and equivalence of $G_2^2$-structures on compact orientable $3$-manifolds. While
$G_2^2$-structures with non-vanishing degree $0$ component,
$\rho_0\neq 0$, exist on any $3$-manifold, those with $\rho_0 = 0$ only
exist on orientable mapping tori. In fact any mapping torus of an
orientable surface can be endowed with such a $G_2^2$-structure
(Theorem \ref{theo:map-tor-twisted-G22-str}). In Section
\ref{subsec:deformation-G22} we show that the Moser argument in
symplectic geometry can be modified to obtain the result that a small
deformation within the cohomology class does not change the structure
up to generalized diffeomorphism (Theorem
\ref{theo:small-perturbation}). We finish by describing the cone of
$G_2^2$-structures inside $H^\bullet(M)$ in
Theorem~\ref{theo:G22-cone}. \bigskip

The author wishes to thank his supervisor Nigel Hitchin for
introducing him to this subject and for his constant support and
generosity. This work was been possible thanks to a Fellowship for
Graduate Courses in Universities and Colleges funded by Fundaci\'on
Caja Madrid.

\section{$B_n$-generalized geometry}
\label{sec:Bn-gengeo}

\subsection{The Courant algebroid $T\oplus T^*\oplus 1$}
\label{subsec:group-gen-diff}

Let $M$ be a differentiable manifold of dimension $n$ with tangent
bundle $T$ and cotangent bundle $T^*$. Let $1$ denote the trivial
bundle of rank $1$ over $M$. Define the $B_n$-generalized tangent
bundle by $T\oplus T^*\oplus 1$. The sections of this bundle are called
generalized vector fields and are naturally endowed with a signature
$(n+1,n)$ inner product given~by
$$(X+\xi+\lambda,Y+\eta+\mu)=\frac{1}{2}(i_X\eta + i_Y\xi) + \lambda\mu,$$
where $X+\xi+\lambda, Y+\eta+\mu \in \mathcal{C}^\infty(T\oplus T^*\oplus 1)$.
Together with the canonical orientation on $T\oplus T^*\oplus 1$, this
endows $T\oplus T^*\oplus 1$ with the structure of an
$\mathrm{SO}(n+1,n)$-bundle. We introduce a Courant bracket on
$\mathcal{C}^\infty(T\oplus T^*\oplus 1)$ via
\begin{equation*
  \begin{split}
    [X+\xi+\lambda,Y+\eta+\mu] = {} & [X,Y]+\mathcal{L}_X\eta-\mathcal{L}_Y\xi
    -\frac{1}{2}d(i_X \eta-i_Y \xi)\\ & + \mu d\lambda-\lambda d\mu +(i_X
    d\mu - i_Y d\lambda),
  \end{split}
\end{equation*}
so that $(T\oplus T^*\oplus 1, (,), [,])$ is a Courant algebroid in the
sense of \cite{liu-weinstein-xu:1997}.

The infinitesimal orthogonal transformations of $T\oplus T^*\oplus 1$ are
given by the elements
$$\left(
\begin{array}{ccc}
  E & \beta & -2\alpha \\
  B & -E^t & -2A \\
  A & \alpha & 0
\end{array}
\right) \in \mathcal{C}^\infty(\mathfrak{so}(T\oplus T^*\oplus 1))$$ such that
$E\in \End(T)$, $\beta\in \bigwedge^2 T$, $B\in \bigwedge^2 T^*$, the
$B$-field already present in $D_n$-geometry, $\alpha \in T$ and $A\in
T^*$, the $A$-field which will be relevant in $B_n$-geometry. The
exponentiation of a $B+A$-field gives the element
$$(B,A):=\exp(B+A)=\left(
\begin{array}{ccc}
  1 &  &  \\
  B-A\otimes A & 1 & -2A \\
  A &  & 1
\end{array}
\right)\in \mathcal{C}^\infty(\mathrm{SO}(T\oplus T^*\oplus 1)),$$ which acts by
$(B,A)(X+\xi+\lambda) = X+\xi+i_X B-2\lambda A-i_X A\, A+\lambda +i_X
A.$

The composition law of these elements in $\mathcal{C}^\infty(\mathrm{SO}(T\oplus T^*\oplus
1))$ is
$$(B,A)(B',A')= (B+B'+ A\wedge A',A+A').$$
Note that $A$-fields do not commute and their product involves a
$2$-form.

Their action on the Courant bracket is given by the following result.
\begin{proposition}\label{prop:e(B,a)-courant}
  Let $(B,A)\in \mathcal{C}^\infty(\mathrm{SO}(T\oplus T^*\oplus 1))$. For generalized
  vector fields $u=X+\xi+\lambda$ and $v=Y+\eta+\mu$, we have
  \begin{equation*
    \begin{split}
      [(B,A) u ,(B,A) v] = {} & (B,A)[u,v]+i_Yi_X(dB+A\wedge dA) -2i_Y
      i_X dA \cdot A \\ & + i_Y i_X dA + 2(\lambda i_Y dA - \mu i_X
      dA).
    \end{split}
  \end{equation*}
  In particular, the action of $(B,A)$ commutes with the Courant
  bracket if and only if $A$ and $B$ are closed.
\end{proposition}

Define the group
$$\Omega^{2+1}_{cl}(M)=\{(B,A)\in \mathcal{C}^\infty(\mathrm{SO}(T\oplus T^*\oplus 1)) \;|\; B\in \Omega^{2}_{cl}(M),  A\in \Omega^1_{cl}(M)\}.$$
The group $\Omega^2_{cl}(M)$ is a central subgroup in
$\Omega^{2+1}_{cl}(M)$, so $\Omega^{2+1}_{cl}(M)$ is the central
extension $1\to \Omega^{2}_{cl}(M)\to \Omega^{2+1}_{cl}(M) \to
\Omega^{1}_{cl}(M)\to 1.$

\begin{proposition}\label{prop:semidirect-product}
  The group of orthogonal transformations of $T\oplus T^*\oplus 1$
  preserving the Courant bracket is $\mathrm{Diff}(M) \ltimes
  \Omega^{2+1}_{cl}(M)=: \mathrm{GDiff}(M)$, called the group of
  generalized diffeomorphisms of $M$. The product is given by
  \begin{align*}
    (f\ltimes (B,A))\circ (g\ltimes (D,C)) & = fg \ltimes
    (g^*B,g^*A)(D,C) \nonumber \\ & = fg \ltimes (g^*B+D+g^*A \wedge
    C,g^*A+C).
  \end{align*}
\end{proposition}

We describe $\mathfrak{gdiff}(M)$, the Lie algebra of $\mathrm{GDiff}(M)$, by
differentiating the action of a smooth one-parameter family of
generalized diffeomorphisms $F_t=f_t\ltimes (B_t,A_t)$ such that
$F_t\circ F_s=F_{t+s}$ and $F_0=\id$. By Proposition
\ref{prop:semidirect-product} and $F_t\circ F_s=F_{t+s}$ we have the
three equations \begin{align*} f_{t+s} &=f_t\circ f_s, & A_{t+s} &=A_s
  + f_s^*A_t, & B_{t+s} &=B_s + f_s^*B_t + f_s^*A_t\wedge A_s.
\end{align*}

The first equation says that $\{f_t\}$ is a one-parameter subgroup of
diffeomorphisms of $M$. Let $X$ be the corresponding vector field. From the second equation, $A_t=\int_0^t f_s^* a\, ds$,
where $a=\frac{dA_t}{dt}_{|t=0}$. And from the third equation,
$$\frac{dB_t}{dt}_{|t=s}=f_s^*\frac{dB_t}{dt}_{|t=0} + f_s^*\frac{dA_t}{dt}_{|t=0}\wedge A_s,$$ so $B_t=\int_0^t (f_s^* b + f_s^* a\wedge A_s) ds$, where $b=\frac{dB_t}{dt}_{|t=0}$ and $A_s$ depends on $a$.

Using the convention $\mathcal{L}_X Y=-\frac{d}{dt}_{|t=0} f_{t\, *} Y$ for
the Lie derivative of a vector field $Y$, we see that the
infinitesimal action of the one-parameter subgroup $\{F_t\}$ is
\begin{equation*
  -\frac{d}{dt}_{|t=0} F_{t\, *} (Y+\eta+\mu) = \mathcal{L}_X (Y+\eta+\mu) - i_Y b + 2\mu a - i_Y a,
\end{equation*}
which only depends on the action of $(X,b,a)$. We thus make the
identification $$\mathfrak{gdiff}(M)=\mathcal{C}^\infty(T)\oplus \Omega^2_{cl}(M) \oplus
\Omega^1_{cl}(M).$$

Conversely, given an infinitesimal generalized diffeomorphism
$(X,b,a)$, we can integrate it to a one-parameter subgroup of
generalized diffeomorphisms using the equations above.

\begin{remark}\label{remark:integrating-time-dependent}
  It is also possible to integrate a time-dependent infinitesimal
  generalized diffeomorphism. From $(X_t,b_t,a_t)$, we get
  $B_t=\int_0^t (f_s^* b_s + f_s^* a_s\wedge A_s) ds$ and
  $A_t=\int_0^t f_s^* a_s ds$, using a method analogous to that used to show Proposition 2.3
  in \cite{gualtieri-annals:2011}.
\end{remark}

\begin{remark}\label{rem:Dorfmann-product}
  We map $\mathcal{C}^\infty(T\oplus T^*\oplus 1)$ to $\mathfrak{gdiff}(M)$ by
$$ X + \xi + \lambda \; \mapsto \; (X, d\xi, d\lambda),$$
so that we regard $X+\xi+\lambda$ as defining an infinitesimal
generalized diffeomorphism. Its natural action on sections of $T\oplus
T^*\oplus 1$ gives an action of a generalized vector field on generalized
vector fields, called the Dorfman product
$$ (X + \xi +\lambda) (Y + \eta +\mu) = [X,Y] + \mathcal{L}_X \eta  + i_X d\mu - i_Y d\xi + 2\mu d\lambda - i_Y d\lambda. $$
The antisymmetrization of the Dorfman product gives the Courant
bracket defined above.
\end{remark}

\subsection{Differential forms as spinors}

By analogy with $D_n$-generalized geometry, the differential forms
$\bigwedge^\bullet T^*M$ are a Clifford module over the algebra
$\mathcal{C}^\infty(Cl(T\oplus T^*\oplus 1))$ with an action defined~by
$$(X+\xi+\lambda)\cdot \phi = i_X \phi + \xi\wedge \phi + \lambda \tau \phi,$$
where $\tau \phi = \phi^{+} - \phi^{-}$ for the even $\phi^+$ and odd
$\phi^-$ parts of $\phi$. Thus, $\tau$ defines an involution of
$\bigwedge^\bullet T^*M$. The action defined above satisfies the
Clifford condition $(X+\xi+\lambda)^2\cdot \phi =
(X+\xi+\lambda,X+\xi+\lambda) \phi,$ as $\tau$ anticommutes with
interior and exterior products.

The action of $B$ and $A$ fields, $B,A\in \mathcal{C}^\infty(\mathfrak{so}(T\oplus T^*\oplus
1))$, on $\bigwedge^\bullet T^*M$ via the spinorial representation
$\sigma:\mathcal{C}^\infty(\mathrm{Spin}(T\oplus T^*\oplus 1))\to \Aut \textstyle
(\bigwedge^\bullet T^*M)$ is given by the Lie algebra action $
\sigma_*(B)\phi =-B\wedge \phi$, $\sigma_*(A)\phi =-A\wedge \tau
\phi,$ and the Lie group action
\begin{align*}
  \sigma(\exp B)\phi & = \phi - B\wedge \phi + B^2\wedge \phi + \ldots = e^{-B}\wedge \phi,\\
  \sigma(\exp A)\phi & = \phi -A\wedge \tau \phi = e^{-A\tau}\wedge
  \phi.
\end{align*}
Since $B$ and $A$ commute, the action of a $B+A$-field is given by
$$\sigma(\exp(B+A))\phi=e^{-B}e^{-A\tau}\phi=e^{-A\tau}e^{-B}\phi.$$

The Lie derivative of a spinor with respect to a generalized vector
field $X+\xi+\lambda$, as also for generalized vector fields in Remark
\ref{rem:Dorfmann-product}, is defined by mapping the vector field to
the infinitesimal generalized diffeomorphism $(X,d\xi,d\lambda)\in
\mathfrak{gdiff}(M)$ and differentiating the action of the one-parameter
subgroup $\{F_t\}$ to which it integrates:
$$\mathbf{L}_{X+\xi+\lambda} \phi = -\frac{d}{dt}_{|t=0} F_t \phi = \mathcal{L}_X \phi + d\xi\wedge \phi + d\lambda \tau \phi.$$

The Lie derivative of a spinor satisfies a Cartan formula, where the
interior product is replaced by the Clifford action, $d
((X+\xi+\lambda)\cdot \phi) + (X+\xi+\lambda)\cdot
d\phi=\mathbf{L}_{X+\xi+\lambda}\phi$.

The differential forms $\bigwedge^\bullet T^*M$ are endowed with an
$\mathrm{SO}(T\oplus T^*\oplus 1)$-invariant pairing with values in $\bigwedge^n
T^*M$ coming from the Chevalley pairing on spinors
(\cite{chevalley-spinors:1954}). Let $\alpha$ be the anti-involution
defined by $\alpha(\omega)=(-1)^{{\deg \omega \choose 2}}\omega$ on
forms of pure degree $\omega$ and extended linearly. For $\rk T=\dim
M$ odd, the pairing is given by
\begin{equation*
  \langle \phi, \psi \rangle =  [\alpha(\phi^-)\wedge \psi^+ - \alpha(\phi^+)\wedge \psi^- ]_{top},
\end{equation*}
while for $\rk T=\dim M$ even, it is given by
$$\langle \phi,\psi \rangle = [\alpha(\phi^+)\wedge \psi^+ + \alpha(\phi^-)\wedge \phi^-]_{top}.$$

\begin{remark}\label{rem:pairing-3-man}
  In the case of $3$-manifolds, \begin{align*}
    \langle \phi,\psi \rangle & =[\alpha(\phi^+)\wedge \psi^- - \alpha(\phi^-)\wedge \psi^+]_{top} \\ & =[(\phi_0 - \phi_2)\wedge (\psi_1+\psi_3) - (\phi_1-\phi_3)\wedge(\psi_0+\psi_2)]_{top} = \\
    & = \phi_0 \psi_3 + \psi_0 \phi_3 - \phi_1\wedge \psi_2 - \psi_1
    \wedge \phi_2,
  \end{align*}
  and, in particular, $\langle \phi,\phi \rangle = 2(\phi^0 \phi^3 -
  \phi^1\wedge \phi^2 )$, thus defining a quadratic form of signature
  $(4,4)$.
\end{remark}

\section{$G_2^2$-structures on $3$-manifolds}
\label{sec:G22-str}

In \cite{hitchin-CY:2003}, for $n=2m$, generalized Calabi-Yau
structures are defined by a complex closed form $\phi$ that is either even or
odd which is a pure spinor and satisfies $\langle \phi,\bar{\phi}\rangle \neq
0$. This structure defines a reduction to the stabilizer of the
spinor~field, $\mathrm{SU}(m,m)$.

In the case of a $3$-manifold, we pointwise have a seven-dimensional
generalized tangent space with an inner product of signature $(4,3)$. Its
space of spinors is eight-dimensional and equipped with a signature
$(4,4)$ inner product. In this setting, pure spinors correspond to
null spinors with respect to the inner product, while non-pure spinors
correspond to non-null spinors. Moreover, up to scalar multiplication, there are only
two orbits under the action of $\mathrm{Spin}(4,3)$: the null ones and the
non-null ones. Hence, all non-null spinors have isomorphic
stabilizers. While the stabilizer of a non-zero spinor in $\mathrm{Spin}(7)$
is the compact exceptional Lie group $G_2$, for the group
$\mathrm{Spin}(4,3)$, the stabilizer of a non-null spinor is its non-compact
real form $G_2^2$. The study of the structure given on a $3$-manifold
by a section of $\bigwedge^{\bullet} T^*M$ consisting of closed
non-null spinors motivates the following definition.

\begin{definition}
  A $G_2^2$-generalized structure on a $3$-manifold $M$ is an
  everywhere non-null section of the real spinor bundle, $\rho \in
  \Omega^\bullet(M)$, such that $d\rho = 0$.  For the sake of brevity,
  we call them $G_2^2$-structures.
\end{definition}

\begin{remark}
  Given a section $\rho \in \Omega^\bullet(M)$ consisting of closed
  null spinors, its annihilator $\Ann(\rho)\subset T\oplus T^*\oplus 1$
  defines an integrable real Dirac structure, i.e., a maximal
  isotropic subbundle of $T\oplus T^*\oplus 1$ involutive with respect to
  the Courant bracket. The involutivity is a consequence of the
  closedness of $\rho$, as in Proposition 1 of \cite{hitchin-CY:2003}.
\end{remark}

\subsection{Existence of $G_2^2$-structures}

From the non-nullity condition we have that $\langle \rho,\rho \rangle=
2(\rho_0\rho_3 - \rho_1\wedge\rho_2)$ defines a volume form on $M$, so
$G_2^2$-structures only exist over orientable manifolds. In fact,
given any volume form $\omega$, $c+\omega$ defines a $G_2^2$-structure
for any constant $c\neq 0$. Since $\rho$ is closed, $\rho_0$ must be a constant.

From now on, $M$ will denote a compact orientable $3$-manifold. Let $\mathrm{GDiff}^+(M)$ be the group of orientation-preserving generalized diffeomorphisms.

\begin{proposition} Up to $\mathrm{GDiff}^+(M)$-equivalence, a $G_2^2$-structure
  $\rho$ with $\rho_0\neq 0$ on $M$
  \begin{enumerate}
  \item is of the form $c+\omega$ for $c\neq 0$ and $\omega$ a volume
    form, and
  \item is completely determined by the cohomology
    classes $$([\rho_0],[\langle \rho,\rho \rangle])\in (H^0(M,{\mathbb{R}})\setminus
    \{0\}) \oplus (H^3(M,{\mathbb{R}})\setminus \{0\}).$$
  \end{enumerate}
\end{proposition}

\begin{proof}
  Let $\rho = \rho_0 + \rho_1 + \rho_2 + \rho_3$ be a
  $G_2^2$-structure with $\rho_0\neq 0$. It is equivalent, by the
  action of the closed $(B+A)$-field $\left(-\frac{\rho_1}{\rho_0},
    -\frac{\rho_2}{\rho_0}\right)$ to $$\rho_0 +
  \frac{1}{\rho_0}(\rho_0\rho_3 - \rho_1\wedge \rho_2)=\rho_0 +
  \frac{1}{2\rho_0}\langle \rho,\rho \rangle,$$ which is of the form
  $c+\omega$ for $c\neq 0$ and $\omega$ a volume form, as stated in
  the first part.  By Moser's theorem (\cite{moser-1965}), any two volume
  forms in the same cohomology class are diffeomorphic.\end{proof}

We deal now with the existence of $G_2^2$-structures with $\rho_0= 0$.
\begin{proposition}\label{prop:G22-str-rho0=0}
  If a compact $3$-manifold is endowed with a $G_2^2$-structure such that
  $\rho_0=0$, then it is diffeomorphic to the mapping torus of a
  symplectic surface by a symplectomorphism.  Conversely, any such
  mapping torus can be endowed with a $G_2^2$-structure with
  $\rho_0=0$.
\end{proposition}

\begin{proof}
  From $\rho_0 = 0$ and $\langle \rho,\rho \rangle \neq 0$ we get $\rho_1\wedge
  \rho_2\neq 0$, so we have nowhere vanishing closed $1$-forms and $2$-forms
  $\rho_1$ and $\rho_2$. We can perform a small deformation on
  $\rho_1$ to give it rational periods (as shown for instance in
  \cite{tischler:1970}). A suitable multiple has integral periods and
  defines a fibration $\pi:M\to \mathrm{S}^1$. To define $\pi$, take a base
  point $m\in M$ and let $\pi(x)=e^{2\pi i \int_{c(t)} \rho_1 dt}$
  where $c(t)$ is any curve joining $m$ and $x$. Let $X$ be the unique
  vector field satisfying $i_X\rho_2=0$ and $i_X \rho_1=1$ (so it
  is transversal to the fibration, $d\pi(X)\neq 0$).  Integrate the
  vector field $X$ to a one-parameter subgroup of diffeomorphisms
  $\{f_t\}$ such that $f_0=id$. Let $S$ be the fibre over the point
  $m\in M$. By the transversality, we have that $M$ is diffeomorphic
  to the mapping torus of $f_1$, i.e., the manifold $$\frac{S \times
    [0,1]}{\{(x,0) \sim (f_1(x),1)\}_{x\in S}}.$$ The diffeomorphism
  is given by $[(y,t)]\mapsto f_t(y)\in M$.  Furthermore, $\mathcal{L}_X
  \rho_2=d( i_X\rho_2)=0$, so $f_t^* \rho_2 = \rho_2$ and the fibres
  have a symplectic structure given by the restriction of $\rho_2$,
  which is closed and non-degenerate in every fibre $f_t(S)$. Thus,
  $S$ is a symplectic manifold and $f_1$ is a symplectomorphism.

  For the second part, let $M_f$ be the mapping torus of an orientable
  surface $(S,\omega)$ by a symplectomorphism $f$. We define a
  $2$-form $\rho_2$ on $M_f$ as the form which is fibrewise
  $\omega$. The form $\rho_2$ is well defined since
  $f^*\omega=\omega$. Let $\rho_1$ be the pullback of a non-vanishing
  $1$-form over the circle. The form $\rho_1 + \rho_2$ then defines a
  $G_2^2$-structure on $M_f$.
\end{proof}

\begin{lemma}\label{lemma:map-torus-difeo}
  The mapping torus of an orientable surface $S$ by an orientation-preserving diffeomorphism is diffeomorphic to the mapping torus of
  $S$ by a symplectomorphism.
\end{lemma}

\begin{proof}
  Let $f$ be the orientation-preserving diffeomorphism and let
  $\omega$ be a volume form of the surface $S$. The $2$-forms
  $f^*\omega$ and $\omega$ have the same volume and hence define the
  same cohomology class in $H^2(S,{\mathbb{R}})$. We apply Moser's argument
  (\cite{moser-1965}) to the family $\omega_t=t\omega + (1-t)
  f^*\omega$, so we get a family of diffeomorphisms $\{\phi_t\}$, with
  $\phi_0=\id$, such that $\phi_t^* \omega_t=\omega$. Then, we have
  that $(\phi_1\circ f)^*=\phi_1^* f^*\omega=\omega$, i.e.,
  $\phi_1\circ f$ is a symplectomorphism, and $\{\phi_t\circ f\}$
  defines a diffeotopy between $f$ and $\phi_1\circ f$ which makes the
  mapping torus of $f$ diffeomorphic to the mapping torus of
  $\phi_1\circ f$.
\end{proof}

The following theorem is a consequence of the two previous results.

\begin{theorem}\label{theo:map-tor-twisted-G22-str}
  A compact $3$-manifold $M$ admits a $G_2^2$-structure with $\rho_0=0$ if and
  only if $M$ is the mapping torus of an orientable surface by an
  orientation-preserving diffeomorphism.
\end{theorem}

\begin{remark}
  From a $G_2^2$-structure with $\rho_0= 0$ on a $3$-manifold $M$ we
  define a symplectic structure on $M\times \mathrm{S}^1$ by $\rho_2 +
  \rho_1 \wedge d\theta$, where $d\theta$ denotes the usual $1$-form
  on $\mathrm{S}^1$ and we really mean the pullbacks of forms on $M$ and
  $\mathrm{S}^1$ to $M\times \mathrm{S}^1$. More generally, the condition that a
  $3$-manifold $M$ fibres over the circle is equivalent to the
  existence of a symplectic structure on $M\times S^1$, as addressed
  in \cite{friedl-vidussi:2011}.
\end{remark}

\begin{remark}
  After acting by a generalized diffeomorphism, a $G_2^2$-structure
  $\rho$ with $\rho_0=0$ can be written as $\rho_1 + \rho_2$. This is
  a co-symplectic structure on the $3$-manifold in the sense of
  \cite{libermann:1959}. In this context, statements similar to the
  ones in this section have been obtained in \cite{li-hongjun:2008}.
\end{remark}

\subsection{Deformation of $G_2^2$-structures}
\label{subsec:deformation-G22}

Inspired by the Moser argument for symplectic geometry, we study whether a
small perturbation of a $G_2^2$-structure (on a compact $3$-manifold $M$) within its cohomology class
may change the $G_2^2$-structure up to equivalence by
$$\mathrm{GDiff}_0(M) = \{ f\ltimes (B,A) \in \mathrm{GDiff}(M) \;|\; f\in \mathrm{Diff}_0(M), B \textrm{ and } A \textrm{ are exact} \}.$$

Let $\rho^0, \rho^1\in \Omega^\bullet(M)$ be two
$G_2^2$-structures representing the same cohomology class,
$\rho^1-\rho^0=d\phi$, and sufficiently close to have that each form
$\rho^t=\rho^0 + t (\rho^1-\rho^0)$ is a $G_2^2$-structure, i.e., $\langle
\rho^t,\rho^t \rangle \neq 0$, for $0\leq t\leq 1$. We would like to have
a one-parameter family of generalized diffeomorphisms $\{ F_t \}$ such
that $F_t^* \rho^t=\rho^0$, making equivalent all the
$G_2^2$-structures between $\rho^0$ and $\rho^1$. We will be looking
for $\{F_t\}$ coming from a time-dependent generalized vector field
$\{X_t+\xi_t+\lambda_t\}$. By differentiating $F_t^* \rho^t=\rho^0$
and using Cartan's formula, we then have
$$ 0 = \frac{d}{dt} [F_t^* \rho^t] = F_t^* \left[ \frac{d\rho_t}{dt} + \mathbf{L}_{X_t+ \xi_t + \lambda_t} \rho^t \right] = F_t^*[ d\phi + d((X_t+\xi_t+\lambda_t)\cdot \rho^t) ]=0.$$
So, in order to find such generalized vector fields it will suffice to
solve the equation $d ((X_t+\xi_t+\lambda_t)\cdot \rho^t)) =
d(-\phi)$, or equivalently, to solve the equation
$(X_t+\xi_t+\lambda_t)\cdot \rho^t = -\phi$ where we are allowed to
modify $\phi$ by the addition of a closed form depending on $t$. This
latter equation corresponds to $\phi$ being in the image of the
Clifford product of the sections of the rank $7$ vector bundle $T\oplus
T^*\oplus 1$ by $\rho^t$. The spinor $\rho^t$ defines a map $T\oplus T^*\oplus
1 \to \bigwedge^\bullet T^*M$. Since $\rho^t$ is non-null, this map is
injective (the annihilator of a non-null spinor is trivial). From the
antisymmetry of the Clifford product with respect to the pairing, $\langle
v_m\cdot \rho^t_m, \rho^t_m \rangle_m = 0$, where $v_m$ and $\psi_m$ lie
over $m\in M$, and the image is $\{\rho^t\}^\perp=\{ \psi \in
\bigwedge^\bullet T^*M \;|\; \langle \rho^t,\psi \rangle = 0 \}$. Thus, $\rho^t$
defines an isomorphism between the rank $7$ vector bundles $T\oplus
T^*\oplus 1$ and $\{\rho^t\}^\perp$. Consequently, for the equation
$(X_t+\xi_t+\lambda_t)\cdot \rho^t=-\phi$ to have a solution and then
apply the Moser argument, we must have $\phi\in
\mathcal{C}^\infty(\{\rho^t\}^\perp)$.

\begin{proposition}\label{prop:Moser-G22-neq0}
  Any sufficiently small perturbation $\{\rho^t\}$ within the
  cohomology class of a $G_2^2$-structure $\rho^0$ such that
  $\rho^0_0\neq 0$ is equivalent to $\rho^0$ under the action of the group
  $\mathrm{GDiff}_0(M)$.
\end{proposition}

\begin{proof}
  We have that $\rho^t_0=\rho^0_0\neq 0$. Since we can add any closed
  form to $\phi$, we can arbitrarily modify its degree $3$ part. The
  Moser argument applies by setting $\phi^t_3 = - \frac{1}{\rho^0_0}
  \langle \rho^t, \phi_o + \phi_1 + \phi_2 \rangle$, so that we have $\langle \rho^t,
  \phi^t \rangle = 0$.
\end{proof}

When $\rho_0 =0$, the result remains true but involves some
technicalities.

\begin{lemma}\label{lemma:Moser-G22}
  Let $\rho$ be a $G_2^2$-structure with $\rho_0=0$ and $[\rho_1]\in
  H^1(M,{\mathbb{Q}})$. There exists an operator $R:\Omega^\bullet(M)\to
  \Omega^\bullet_{cl}(M)$ such that $\phi+R\phi\in
  \mathcal{C}^\infty(\{\rho\}^\perp)$.
\end{lemma}

\begin{proof}
  By considering a multiple of $\rho$ we can consider $[\rho_1] \in
  H^1(M,{\mathbb{Z}})$. By Proposition \ref{prop:G22-str-rho0=0}, $M$ fibres
  over the circle with fibre $S$. First, define the constant $c=[\langle
  \rho,\phi\rangle] / [\rho_1\wedge \rho_2]$. Add the closed form
  $c\rho_2$ to $\phi$; then the cohomology class of $\langle \rho,
  \phi+c\rho_2\rangle$ is trivial. Thus, $\langle \rho, \phi+c\rho_2\rangle =
  d\alpha$ for some $2$-form $\alpha$. Choose a metric on $M$. Using
  the Hodge decomposition, the codifferential $d^*$ and the Green
  operator $G$, we may take $\alpha=d^*G\langle \rho,\phi'\rangle$. Integrate
  $\alpha$ over the fibres to get a function $g$ on the circle. Since
  $\rho_1\wedge\rho_2\neq 0$, the fibres are homologous and $\rho_2$
  is closed, then $\int_S \rho_2=c'\neq 0$ for any fibre $S$. Let
  $f=g/c'$. The $2$-form $\alpha_0=\alpha - f \rho_2$ has zero
  integral along the fibres. The metric on $M$ induces a metric on any
  fibre $S$, for which we define the codifferential $d^*_{S}$,
  harmonic operator $H_{S}$ and Green operator $G_{S}$ such
  that $$\alpha_{0|S}=H_{S}\alpha_{0|S}+d_{S}(d_{S}^* G_{S}
  \alpha_{0|S}) + d^*_{S}(d_{S} G_{S} \alpha_{0|S}).$$ For degree
  reasons, $d_{S} G_{S} \alpha_{0|S}=0$, and from $\int_{S}
  \alpha_{0|S}=0$, $H_{S}\alpha_{0|S}=0$. We then have, over each
  fibre $S$, $\alpha_{0|S}=d_{S}\beta$ where $\beta=d_{S}^* G_{S}
  \alpha_{0|S}$. Since the metric on $M$ determines a smoothly varying
  family of metrics over the fibres, we have a globally smooth
  $1$-form $\beta$ such that $\alpha_0-d\beta$ is zero restricted to a
  fibre.

  Let $X$ be the vector field transversal to the fibration such that
  $i_X \rho_1=1$, and let $\gamma= - i_X(\alpha_0 - d\beta)$. We have
  that $\alpha_0 - d\beta = \gamma \wedge \rho_1$. By differentiating
  this expression we get $$d\alpha = d(\alpha_0 + f\rho_2) = df \wedge
  \rho_2 + \rho_1 \wedge d\gamma.$$

  Define ${\rm R}\phi=c\rho_2 + df + d\gamma\in
  \Omega^\bullet_{cl}(M)$. Since $c$, $f$ and $\gamma$ have been
  uniquely defined, ${\rm R}$ defines an operator on differential
  forms. We have by construction that $\langle \rho, \phi + {\rm R}\phi
  \rangle = 0$, i.e., $\phi + {\rm R}\phi\in
  \mathcal{C}^\infty(\{\rho\}^\perp)$.\end{proof}

Let ${\rm Q}\phi \in \mathcal{C}^\infty(T\oplus T^*\oplus 1)$ be the unique
generalized vector field such that ${\rm Q}\phi \cdot \rho = - (\phi +
{\rm R}\phi)$. Thus ${\rm Q}$ defines an operator
$\Omega^\bullet(M)\to\mathcal{C}^\infty(T\oplus T^*\oplus 1)$.

  \begin{proposition}\label{prop:Moser-G22-0}
    Any sufficiently small perturbation $\{\rho^t\}$ within the
    cohomology class of a $G_2^2$-structure $\rho^0$ such that
    $\rho^0_0=0$ is equivalent to $\rho^0$ by $\mathrm{GDiff}_0(M)$.
  \end{proposition}

  \begin{proof}

    When $[\rho^0_1]\in H^1(M,{\mathbb{Q}})$, we use Lemma \ref{lemma:Moser-G22}
    to produce an operator ${\rm R}_t$ for each $\rho^t$ and we define
    $\phi^t=\phi+{\rm R}_t \phi$, so that $\langle \rho^t, \phi^t\rangle=0$
    and the Moser argument applies.

    For the general case, we prove an analogous result in a
    neighbourhood of a $G_2^2$-structure with rational degree $1$ part
    and use a density argument. We drop the superindex $t$ for the
    sake of brevity. Consider $\rho + \lambda \beta$, with $\lambda>0$
    and $\beta\in\Omega^\bullet_{cl}(M)$ such that $\beta_0=0$. We
    want to solve the equation $u\cdot (\rho+\lambda \beta) = -\phi$
    up to addition of closed forms. To do that, consider
    \begin{equation}\label{eq1}
      (u_0+\lambda u_1 + \lambda^2 u_2 + \ldots )\cdot (\rho+\lambda\beta) = -(\phi + {\rm R}\phi + \lambda \gamma_1 + \lambda^2 \gamma_2 + \ldots),\tag{$\star$}
    \end{equation}
    for closed forms $\gamma_i$. We solve it iteratively, starting
    with $u_0\cdot \rho = -\phi + {\rm R}\phi$, which has solution
    $u_0={\rm Q}\phi$. We then have $u_1\cdot \rho = -({\rm
      Q}\phi\cdot \beta + \gamma_1)$. We define the operator ${\rm
      P}:\Omega^\bullet(M)\to \Omega^\bullet(M)$ by ${\rm P}\phi={\rm
      Q}\phi\cdot \beta$ and consider $\gamma_1 = {\rm RP}\phi$. The
    equation becomes $u_1\cdot \rho = -({\rm P}\phi+{\rm RP}\phi)$,
    whose solution is $u_1={\rm QP}\phi$. For $j\geq 2$ we have
    $u_j\cdot \rho = - u_{j-1} \cdot \beta + \gamma_j = - {\rm
      P}^{j}\phi + \gamma_j$. By taking $\gamma_j={\rm RP}^j\phi$, the
    solution is given by $u_j={\rm QP}^j\phi$. We thus obtain a formal
    solution of \eqref{eq1} by
$${\rm Q}(\phi+\lambda {\rm P}\phi+\lambda^2{\rm P}^2\phi+\ldots)\cdot (\rho+\lambda\beta)=-\phi+{\rm R}(\phi+\lambda {\rm P}\phi +\lambda^2 {\rm P}^2\phi +\ldots).$$
To see the convergence of the series $\phi+\sum_{j=1}^\infty \lambda^j
{\rm P}^j\phi$ for $\lambda$ sufficiently small, we consider Sobolev
spaces ${\rm H}_s(T\oplus T^*\oplus 1)$ and ${\rm
  H}_s(\bigwedge^\bullet(M))$ with norms $||\;||_s$. Since the
operator ${\rm Q}$ is defined in terms of the Green operator and
integration over the fibres, it is bounded, and so is the operator
${\rm P}$. For $s$ sufficiently large and any $\beta$ such that
$||v\cdot \beta||_s\leq ||v||_s$, there exists some constant $C_s$
such that $||{\rm P}\phi||_s\leq C_s ||\phi||_s$.

Take $\lambda$ such that $0<\lambda<\frac{1}{2C_s}$. Then,
$\phi+\sum_{j=1}^\infty \lambda^j {\rm P}^j\phi$ is a Cauchy sequence
and converges to a form $\Phi\in {\rm
  H}_s(\bigwedge^\bullet(M))$. Equation \eqref{eq1} becomes $u\cdot
(\rho+\lambda\beta)=-(\phi+{\rm R}\Phi)$ and a solution is given by
${\rm Q}\Phi \in {\rm H}_s(T\oplus T^*\oplus 1)$.

We have that for any $\rho$ such that $[\rho_1]\in H^1(M,{\mathbb{Q}})$, there
exists a neighbourhood for which there is a solution in ${\rm H}_s(T\oplus
T^*\oplus 1)$. Since $\phi\in\Omega^\bullet(M)$ belongs to ${\rm
  H}_s(\bigwedge^\bullet(M))$ for any $s$, we have that the solution
belongs to ${\rm H}_s$ for any $s$. Thus, the series defines $\Phi\in
\mathcal{C}^\infty(\bigwedge^\bullet(M))$, we have that ${\rm Q}\Phi\in \mathcal{C}^\infty(T\oplus
T^*\oplus 1)$ is a solution of $u\cdot \rho^t = -\phi$ up to closed
forms, and the Moser argument applies. Since there exists a solution
in an open neighbourhood of any rational form, by density of the
rational forms, there exists a solution for any closed form $\rho$ and
the Moser argument applies.
\end{proof}

We summarize Propositions \ref{prop:Moser-G22-neq0} and
\ref{prop:Moser-G22-0} in the following theorem.

\begin{theorem}\label{theo:small-perturbation}
  Any sufficiently small perturbation $\{\rho^t\}$ within the
  cohomology class of a $G_2^2$-structure $\rho^0$ is equivalent to
  $\rho^0$ by $\mathrm{GDiff}_0(M)$.
\end{theorem}

\newpage

\subsection{The cone of $G_2^2$-structures}
\label{subsec:cone-G22}

Inspired by the cones of K\"ahler and symplectic structures inside the
\mbox{second} cohomology group of a manifold, we raise a similar
question for $G_2^2$-structures on compact $3$-manifolds. What are the cohomology classes
$[\rho]\in H^\bullet(M,{\mathbb{R}})$ which have a representative in
$\Omega^\bullet(M,{\mathbb{R}})$ defining a $G_2^2$-structure compatible with
the orientation of $M$? From the homogeneity of the condition $\langle
\rho, \rho \rangle> 0$, it is clear that these elements form an open cone
in $H^\bullet(M,{\mathbb{R}})$.

Consider a mixed degree cohomology class $[\rho]\in H^\bullet(M,{\mathbb{R}})$
satisfying $[\rho_o][\rho_3]-[\rho_1][\rho_2]> 0 \in
H^\bullet(M,{\mathbb{R}})$. In the case that $[\rho_0]\neq 0$, i.e., $\rho_0\neq
0$, consider a non-vanishing form $\omega$ representing the degree $3$
class $[\rho_o \rho_3 - \rho_1 \wedge \rho_2]$. Define
$\rho'=\rho_0+\rho_1+\rho_2 + \frac{1}{\rho_0}(\omega + \rho_1 \wedge
\rho_2)$, which satisfies $\langle \rho',\rho'\rangle=2\omega$ and is thus a
$G_2^2$-structure representing $[\rho]$.

On the other hand, for a class $[\rho]$ with $[\rho_0]=0$, i.e.,
$\rho_0= 0$, the condition $[\langle \rho,\rho\rangle]=-2[\rho_1][\rho_2]> 0$
must be satisfied. Moreover, $[\rho_1]$ and $[\rho_2]$ must be
represented by non-vanishing forms.  From Theorem 5 in
\cite{thurston-memoirs:1986}, the set of cohomology classes $C_1$ in
$H^1(M,{\mathbb{R}})$ which can be represented by a non-singular closed $1$-form
constitutes an open set described as follows. Define the norm $X$ for $\omega
\in H^2(M,{\mathbb{R}})$ as the infimum of the negative parts of the Euler
characteristics of embedded surfaces defining $\omega$, and extend
this definition to $H^1(M,{\mathbb{R}})$ using Poincar\'e duality. Namely, the
norm of a $1$-form $\phi$ in $M$
is $$||\phi||_X=\textrm{min}\{\chi_-(S)\;|\; S\subset M \textrm{
  properly embedded surface dual to } \phi \},$$ where
$\chi_-(S)=\textrm{max}\{-\chi(S),0\}$. The unit ball for this norm is
a polytope called the Thurston ball $B_X$. The set of $1$-cohomology
classes $C_1$ represented by non-vanishing $1$-forms consists of the
union of the cones on some open faces, so-called fibred faces, of the
Thurston ball, minus the origin.

For each element $\alpha=[a]\in C_1$, given by a non-singular $a$, take
$h\in H^2(M,{\mathbb{R}})$ such that $h\cup \alpha>0$. Lemma 2.2 in
\cite{friedl-vidussi:2012} ensures that we can always find a
representative $\Omega$ of the class $h$, such that $\Omega\wedge
a>0$. Hence, if we define $$C=\{ (\alpha,\beta)\in C_1\oplus H^2(M,R) \;|\;
\alpha\cup \beta< 0 \},$$ we have that the cone of $G_2^2$-structures with
$\rho_0=0$ in $H^\bullet(M,{\mathbb{R}})$ is given by $C \oplus H^3(M,{\mathbb{R}}).$ To sum
up, we have the following theorem.

\begin{theorem}\label{theo:G22-cone}
  The cone of $G_2^2$-structures, or $G_2^2$-cone, is given by
$$ \{ [\rho]\in H^\bullet(M,{\mathbb{R}}) \;|\; [\rho_0]\neq 0 \textrm{ and } [\rho_0][\rho_3] - [\rho_1][\rho_2]>0 \} \bigcup \left(C\oplus H^3(M,{\mathbb{R}}) \right).$$
\end{theorem}

\newpage

\bibliographystyle{alpha}

redpajama_set_name: RedPajamaArXiv


Connie Wilder opens a small sandwich shop in downtown Brenham, Texas featuring quiche, salads, and sweets such as Blue Bell Ice Cream and candy. The "Must Be Heaven" name is derived from a 1970's Blue Bell Ice Cream advertisement which answers the question as to why their ice cream tastes so good! The answer is..."because the cows think Brenham must be heaven."
Shirley Syptak parts from her nursing vocations to take over ownership of Must Be Heaven. Under her lead, the menu is expanded to include homemade pies.
Must Be Heaven moves into its current location and the former location of Western Auto at 107 W Alamo St.
Traci & Charlie Pyle purchase the Brenham location as the first franchise from Shirley Syptak.
Navasota, Texas store is relocated to Bryan, Texas near St. Joseph's Hospital.
The Morrows purchase the Bryan Must Be Heaven.
Must Be Heaven - Brenham is recognized by the Texas Downtown Association as the "Best Downtown Business" in the state at annual conference in Temple, Texas.
The Morrows again expand their franchise presence by opening a third location in a revitalized downtown Bryan, Texas.
Must Be Heaven celebrates its 30th Anniversary.
Traci & Charlie Pyle gain a controlling interest in the Must Be Heaven parent company and begin searching for qualified franchise candidates and select locations within the state of Texas.


CYCO Supa Stiky (0.6-3-2) is an additive used during the flowering cycle of plant growth that aids in fruit size and quality and increases plant resin production. A plentiful source of 99.8 percent pure magnesium, Supa Stiky helps in the construction of components crucial to the formation of essential oil compounds in the plant. Safe for food production; enjoy increased aroma and flavor in plants.
CYCO Grow A (2-0-0) and B (2-2-6) give support to a plant's natural growth by supplying a range of macro and micro nutrients, which are needed to give a plant the best possible start in its early development. Calcium, iron, manganese, copper, sodium and zinc are just a few of the elements in CYCO Grow that combine to support growth, cell development, and fruit quality. A and B formulas are sold separately.
CYCO Ryzofuel (0-0-0.2), Australia's leading root stimulant designed for indoor and outdoor plants and seedlings, will rapidly accelerate new root growth. Add to any fertilizer program to achieve greener vigorous plant growth, set more flower sites, stimulate advanced flower growth, encourage more robust flowers and fruit, and increase new leaves and overall biomass. Use in foliar spraying to revive or help suffering potted plants, transplants, ornamentals, flowers, shrubs, natives, trees, seedlings and cuttings.


text:
The North Beach in Burgas is located along the fresh Seaside Park and is an attractive and cosy place for relaxation, sport and entertainment.
The length of the beach strip is 1700 metres and the average width is up to 38 m. The sand is level along its full length and it is cleaned daily.
The beach has a number of restaurants, bars and pubs which welcome its guests with pleasant music and delicious meals, both during the day and in the evening. They will serve you good food and cold drinks and will make sure you're in a good mood and you don't even have to leave the beach.
The north beach has two football and four volleyball fields, three fan tribunes and one playground. There are 14 lifeguard posts taking care of swimmers' safety and the same number of changing facilities and showers. In short, the beach guarantees a comfortable and cosy relaxation time.
Guest House Zemalna
Guest House "Zemalna" in the village of Brashlian is a restored one-storey house in the beautiful Strandja Mountain. The...
Velevi Guest house
Located 100 metre s from the beach in Kraimorie District near Burgas, Velevi Guest House provides an on-site restaurant,...
East Catholic Church
The church is part of the impressive building of the French Boarding School for Girls on Ivan Vazov Blvd....
Youth Cultural center
Contact us: Website: mladite.eu/ E-mail: mkc_bs@abv.bg Phone: +359 56 840 790; 56 84061
Tsar Boris Park
The Tsar Boris Park is located in Vazrazhdane, only a few minutes from the city centre. It has a...
"Lazur" Stadium
"Lazur" Stadium was built in 1967 and has had two major reconstructions throughout the years; currently it's one of...
Familien Pension Meeresters
Set in Burgas, 150 metres from the Sea Garden and 200 metres from the beach, Familien Pension Meeresstern features...
Athletic Fitness
For health, beautiful silhoette and good mood. Contact us: Website: athletic-bg.com/ Phone: +359 878 443 186
Youth Culture Center
Ever since it was created in 1992, the Youth Culture Centre turned into a place where young talents can...
Zoo "Peacocks"
Contact us: Website: paunite.ovo.bg/ E-mail: zookompleks_paunite@abv.bg Phone: +359 887 334466
View other categories
Archaelogical sites
Iconic city landmarks
Sport places
Museum, galleries, culture
Around Burgas
Parks and lakes
Guest houses and hostels
Hotels **
Hotels ***
Hotels ****
Hotels *****
Boutique Hotel & Villa Promenade. Website by 2GOOD Technologies Ltd.