# Climbing Mount Bourbaki

Thoughts on mathematics

November 26, 2009

## Towards Cartan’s fixed point theorem: Cosine inequalities and derivative of the distance

Posted by Akhil Mathew under differential geometry, MaBloWriMo | Tags: Cartan fixed point theorem, cosine inequality, geodesics, negative curvature |
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I am now aiming to prove an important fixed point theorem:

Theorem 1 (Elie Cartan) Let ${K}$ be a compact Lie group acting by isometries on a simply connected, complete Riemannian manifold ${M}$ of negative curvature. Then there is a common fixed point of all ${k \in K}$.

There are several ingredients in the proof of this result. These will provide examples of the techniques that I have discussed in past posts.

Geodesic triangles

Let ${M}$ be a manifold of negative curvature, and let ${V}$ be a normal neighborhood of ${p \in M}$; this means that ${\exp_p}$ is a diffeomorphism of some neighborhood of ${0 \in T_p(M)}$ onto ${V}$, and any two points in ${V}$ are connected by a unique geodesic. (This always exists by the normal neighborhood theorem, which I never proved. However, in the case of Cartan’s fixed point theorem, we can take ${V=M}$ by Cartan-Hadamard.)

So take ${a=p,b,c\in V}$. Draw the geodesics ${\gamma_{ab}, \gamma_{ac}, \gamma_{bc}}$ between the respective pairs of points, and let ${\Gamma_{ab}, \Gamma_{bc}}$ be the inverse images in ${T_p(M) = T_a(M)}$ under ${\exp_p}$. Note that ${\Gamma_{ab}, \Gamma_{ac}}$ are straight lines, but ${\Gamma_{bc}}$ is not in general. Let ${a',b',c'}$ be the points in ${T_p(M)}$ corresponding to ${a,b,c}$ respectively. Let ${A}$ be the angle between ${\gamma_{ab}, \gamma_{ac}}$; it is equivalently the angle at the origin between the lines ${\Gamma_{ab}, \Gamma_{ac}}$, which is measured through the inner product structure.

Now ${d(a,b) = l(\gamma_{ab}) = l(\Gamma_{ab}) = d(a',b')}$ from the figure and since geodesics travel at unit speed, and similarly for ${d(a,c)}$. Moreover, we have ${d(b,c) = l(\gamma_{bc}) \geq l(\Gamma_{bc}) \leq d(b',c')}$, where the first inequality comes from the fact that ${M}$ has negative curvature and ${\exp_p}$ then increases the lengths of curves; this was established in the proof of the Cartan-Hadamard theorem.

We have evidently by the left-hand-side of the figure

$\displaystyle d(b',c')^2 = d(a',c')^2 + d(a',b')^2 - 2d(a',b')d(a',c') \cos A.$

In particular, all this yields

$\displaystyle \boxed{d(b,c)^2 \geq d(a,c)^2 + d(a,b)^2 - 2d(a,b)d(a,c) \cos A.}$

So we have a cosine inequality.

There is in fact an ordinary plane triangle with sides ${d(a,b), d(b,c),d(a,c)}$, since these satisfy the appropriate inequalities (unless ${a,b,c}$ lie on the same geodesic, which case we exclude). The angles ${A',B',C'}$ of this plane triangle satisfy

$\displaystyle A \leq A'$

by the boxed equality. In particular, if we let ${B}$ (resp. ${C}$) be the angles between the geodesics ${\gamma_{ab}, \gamma_{bc}}$ (resp. ${\gamma_{ac}, \gamma_{bc}}$), then by symmetry and ${A'+B'+C=\pi}$

$\displaystyle \boxed{A + B+ C \leq \pi.}$

This is a fact which I vaguely recall from popular-math books many years back.   The rest is below the fold. (more…)

November 24, 2009

## The second variation formula

Posted by Akhil Mathew under differential geometry, MaBloWriMo | Tags: calculus of variations, curvature tensor, energy integral, geodesics, second variation formula |

I’m going to keep the same notation as before.  In particular, we’re studying how the energy integral behaves with respect to variations of curves.  Now I want to prove the second variation formula when ${c}$ is a geodesic.Now to compute ${\frac{d^2}{d^2 u} E(u)|_{u=0}}$, for further usage. We already showed$\displaystyle E'(u) = \int_I g\left( \frac{D}{dt} \frac{\partial}{\partial u} H(t,u), \frac{\partial}{\partial t} H(t,u) \right)$ Differentiating again yields the messy formula for ${E''(u)}$:

$\displaystyle \int_I g\left( \frac{D}{du} \frac{D}{dt} \frac{\partial}{\partial u} H, \frac{\partial}{\partial t} H \right) + \int_I g\left( \frac{D}{dt} \frac{\partial}{\partial u} H, \frac{D}{du}\frac{\partial}{\partial t} H(t,u) \right).$

Call these ${I_1(u), I_2(u) }$.

${I_2}$

Now ${I_2(0)}$ is the easiest, since by symmetry of the Levi-Civita connection we get$\displaystyle I_2(0) = \int g\left( \frac{D}{dt} \frac{\partial}{\partial u} H(t,u), \frac{D}{dt}\frac{\partial}{\partial u} H(t,u) \right) = \int g\left( \frac{D}{dt} V, \frac{D}{dt} V \right).$ For vector fields along ${c}$ ${E,F}$ with ${E(a)=F(a)=E(b)=F(b)=0}$, we have$\displaystyle \int g\left( \frac{D}{dt} E, \frac{D}{dt} F \right) = - \int g\left( \frac{D^2}{dt^2} E , F \right).$ This is essentially a forum of integration by parts. Indeed, the difference between the two terms is

$\displaystyle \frac{d}{dt} g\left( \frac{D}{dt} E, F \right).$

So if we plug this in we get

$\displaystyle \boxed{ I_2(0) = -\int g\left( \frac{D^2}{dt^2} V , V \right).}$

${I_1}$

Next, we can write

$\displaystyle I_1(0) = \int_I g\left( \frac{D}{du} \frac{D}{dt} \frac{\partial}{\partial u} H(t,u) |_{u=0}, \dot{c}(t) \right)$ Now ${R}$ measures the difference from commutation of ${\frac{D}{dt}, \frac{D}{du}}$. In particular this equals

$\displaystyle \int_I g\left( \frac{D}{dt} \frac{D}{du} \frac{\partial}{\partial u} H(t,u) |_{u=0}, \dot{c}(t) \right) + \int_I g\left( R(V(t), \dot{c}(t)) V(t), \dot{c}(t)) \right).$

By antisymmetry of the curvature tensor (twice!) the second term becomes

$\displaystyle \int_I g\left( R( \dot{c}(t), V(t)) \dot{c}(t), V(t), \right).$

Now we look at the first term, which we can write as

$\displaystyle \int_I \frac{d}{dt} g\left( \frac{D}{du} \frac{\partial}{\partial u} H(t,u), \dot{c}(t)\right)$

since ${\ddot{c} \equiv 0}$. But this is clearly zero because ${H}$ is constant on the vertical lines ${t=a,t=b}$. If we put everything together we obtain the following “second variation formula:”

Theorem 1 If ${c}$ is a geodesic, then$\displaystyle \boxed{\frac{d^2}{du^2}|_{u=0} E(u) = \int_I g\left( R( \dot{c}(t), V(t)) \dot{c}(t) - \frac{D^2 V}{Dt^2}, V(t) \right).}$

Evidently that was some tedious work, and the question arises: Why does all this matter? The next goal is to use this to show when a geodesic cannot minimize the energy integral—which means, in particular, that it doesn’t minimize length. Then we will obtain global comparison-theoretic results.

November 15, 2009

## Hopf-Rinow II and an application

Posted by Akhil Mathew under differential geometry, MaBloWriMo | Tags: geodesic completeness, geodesics, homotopy, Hopf-Rinow theorem |
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Now, let’s finish the proof of the Hopf-Rinow theorem (the first one) started yesterday. We need to show that given a Riemannian manifold ${(M,g)}$ which is a metric space ${d}$, the existence of arbitrary geodesics from ${p}$ implies that ${M}$ is complete with respect to ${d}$. Actually, this is slightly stronger than what H-R states: geodesic completeness at one point ${p}$ implies completeness.

The first thing to notice is that ${\exp: T_p(M) \rightarrow M}$ is smooth by the global smoothness theorem and the assumption that arbitrary geodesics from ${p}$ exist. Moreover, it is surjective by the second Hopf-Rinow theorem.

Now fix a ${d}$-Cauchy sequence ${q_n \in M}$. We will show that it converges. Draw minimal geodesics ${\gamma_n}$ travelling at unit speed with

$\displaystyle \gamma_n(0)=p, \quad \gamma_n( d(p,q_n)) = q_n.$  (more…)

November 14, 2009

## The Hopf-Rinow theorems and geodesic completeness

Posted by Akhil Mathew under differential geometry, MaBloWriMo | Tags: completeness, geodesic completeness, geodesics, Hopf-Rinow theorem |
1 Comment

Ok, yesterday I covered the basic fact that given a Riemannian manifold ${(M,g)}$, the geodesics on ${M}$ (with respect to the Levi-Civita connection) locally minimize length. Today I will talk about the phenomenon of “geodesic completeness.”

Henceforth, all manifolds are assumed connected.

The first basic remark to make is the following. If ${c: I \rightarrow M}$ is a piecewise ${C^1}$-path between ${p,q}$ and has the smallest length among piecewise ${C^1}$ paths, then ${c}$ is, up to reparametrization, a geodesic (in particular smooth). The way to see this is to pick ${a,b \in I}$ very close to each other, so that ${c([a,b])}$ is contained in a neighborhood of ${c\left( \frac{a+b}{2}\right)}$ satisfying the conditions of yesterday’s theorem; then ${c|_{[a,b]}}$ must be length-minimizing, so it is a geodesic. We thus see that ${c}$ is locally a geodesic, hence globally.

Say that ${M}$ is geodesically complete if ${\exp}$ can be defined on all of ${TM}$; in other words, a geodesic ${\gamma}$ can be continued to ${(-\infty,\infty)}$. The name is justified by the following theorem:

Theorem 1 (Hopf-Rinow)

The following are equivalent:

• ${M}$ is geodesically complete.
• In the metric ${d}$ on ${M}$ induced by ${g}$ (see here), ${M}$ is a complete metric space (more…)

November 13, 2009

## Geodesics are locally length-minimizing

Posted by Akhil Mathew under differential geometry, MaBloWriMo | Tags: Gauss lemma, geodesics |

Fix a Riemannian manifold with metric ${g}$ and Levi-Civita connection ${\nabla}$. Then we can talk about geodesics on ${M}$ with respect to ${\nabla}$. We can also talk about the length of a piecewise smooth curve ${c: I \rightarrow M}$ as

$\displaystyle l(c) := \int g(c'(t),c'(t))^{1/2} dt .$

Our main goal today is:

Theorem 1 Given ${p \in M}$, there is a neighborhood ${U}$ containing ${p}$ such that geodesics from ${p}$ to every point of ${U}$ exist and also such that given a path ${c}$ inside ${U}$ from ${p}$ to ${q}$, we have

$\displaystyle l(\gamma_{pq}) \leq l(c)$

with equality holding if and only if ${c}$ is a reparametrization of ${\gamma_{pq}}$.

In other words, geodesics are locally path-minimizing.   Not necessarily globally–a great circle is a geodesic on a sphere with the Riemannian metric coming from the embedding in $\mathbb{R}^3$, but it need not be the shortest path between two points. (more…)

November 4, 2009

## Geodesics and the exponential map

Posted by Akhil Mathew under differential geometry, MaBloWriMo | Tags: connections, exponential map, geodesics, ordinary differential equations |
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Ok, we know what connections and covariant derivatives are. Now we can use them to get a map from the tangent space ${T_p(M)}$ at one point to the manifold ${M}$ which is a local isomorphism. This is interesting because it gives a way of saying, “start at point ${p}$ and go five units in the direction of the tangent vector ${v}$,” in a rigorous sense, and will be useful in proofs of things like the tubular neighborhood theorem—which I’ll get to shortly.

Anyway, first I need to talk about geodesics. A geodesic is a curve ${c}$ such that the vector field along ${c=(c_1, \dots, c_n)}$ created by the derivative ${c'}$ is parallel. In local coordinates ${x_1, \dots, x_n}$, here’s what this means. Let the Christoffel symbols be ${\Gamma^k_{ij}}$. Then using the local formula for covariant differentiation along a curve, we get

$\displaystyle D(c')(t) = \sum_j \left( c_j''(t) + \sum_{i,k} c_i'(t) c_k'(t) \Gamma^j_{ij}(c(t)) \right) \partial_j,$

so ${c}$ being a geodesic is equivalent to the system of differential equations

$\displaystyle c_j''(t) + \sum_{i,k} c_i'(t) c_k'(t) \Gamma^j_{ij}(c(t)) = 0, \ 1 \leq j \leq n.$ (more…)
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