A metric tensor assigns a bilinear form to each point on the manifold.
We can then take two vectors in the tangent space and return a scalar.
Metric is positive definite.
The metric isn’t necessarily positive definite.
metric tensor field assigns a metric tensor to each point. metric tensor is defined on the tangent bundle. so we have metric on each tangent bundle, but the metric can change thoughout the manifold
We can work out the length of a path through a Riemann manifold.
The geodesic is the shortest such path.
The Riemann metric between two points is the length of the geodesic.
If we have two vectors in the tangent space of a manifold with a metric tensor, we can get a scalar:
If we transport two vectors along a connection, we have the metric at the new point.
If the connection preserves the metric, then the connection is metric compatible.
For any metric tensor there is only one connection which preserves the metric and is torsion free.
How do we have straight line on a curve? eg going round equator, but not going via uk.
Take start direction and find tangent vectors. geodesic is where tangent vectors stay parallel.