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: , , , |
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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: , , , , |
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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: , , , |
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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: , , , |
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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: , |
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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: , , , |
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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…)