It is of interest to consider functions on the space of curves ${I \rightarrow M}$, where ${I}$ is an interval and ${M}$ is a smooth manifold. To study maxima and minima, it is of interest to consider variations of curves, holding the endpoints fixed. Let ${c:[a,b] \rightarrow M}$ be smooth. A variation of ${c}$ is a smooth map$\displaystyle H: [a,b] \times (-\epsilon, \epsilon) \rightarrow M$

with ${H(t,0) = c(t)}$, and ${H(a,u)=c(a), H(b,u)=c(b)}$ for all ${t,u}$. For a variation ${H}$ of ${c}$, define the variation vector field (which is an analog of a “tangent vector”)

$\displaystyle V(t) = \frac{\partial}{\partial u} H ;$ this is a vector field along ${c}$. Similarly we can define the “velocity vector field” ${\dot{c}}$ along ${c}$. If ${M}$ is provided with a connection, we can define the “acceleration vector field” ${A(t) = \frac{D}{dt} \dot{c}}$, where ${\frac{D}{dt}}$ denotes covariant differentiation.

Given a vector field ${V}$ along ${c}$, we can construct a variation of ${c}$ with ${V}$ as the variation vector field: take ${(t,u) \rightarrow \exp_{c(t)}(u V(t))}$.

Variations of geodesics and Jacobi fields Let ${M}$ now be a manifold with a symmetric connection ${\nabla}$.Let ${H}$ be a variation of a geodesic ${c}$ such that for any ${u \in (-\epsilon,\epsilon)}$, ${t \rightarrow H(t,u)}$ is a geodesic as well. Then the variation vector field satisfies a certain differential equation. Now$\displaystyle \frac{D^2}{dt^2 } V(t) = \frac{D}{dt} \frac{D}{dt} \frac{\partial}{\partial u } H |_{u=0} = \frac{D}{dt} \frac{D}{du} \frac{\partial}{\partial t} H |_{u=0}.$

We have used the symmetry of ${\nabla}$. Now we can write this as

$\displaystyle \frac{D}{du} \frac{D}{d t} \frac{\partial}{\partial t} H |_{u=0} + R\left( \frac{\partial H}{\partial t}, \frac{\partial H}{\partial u}\right) \frac{\partial H}{\partial t} |_{u=0}.$ By geodesy, the first part vanishes, and the second is

$\displaystyle R( \dot{c}(t), V(t)) \dot{c}(t)$ We have shown that ${V}$ satisfies the Jacobi equation

$\displaystyle \boxed{ \frac{D^2}{dt^2 } V(t) = R( \dot{c}(t), V(t)) \dot{c}(t).}$ Any vector field along ${c}$ satisfying this is called a Jacobi field.

The differential of the exponential map Let ${p \in M}$, and consider the exponential map ${\exp_p: U \rightarrow M}$ where ${U}$ is a neighborhood of the origin in ${T_p(M)}$. Let ${X,Y \in T_p(M)}$. Now the map$\displaystyle H: (t,u) \rightarrow \exp_p( tX + ut Y)$ takes horizontal lines to geodesics in ${M}$ when ${t,u}$ are small enough. This can be viewed as a vector field along the geodesic ${t \rightarrow \exp(tX)}$. The variation vector field ${J}$ is thus a Jacobi field, and also at ${t=1}$ is ${(\exp_p)_{*X}(Y) \in T_{\exp_p(X)}(M).}$ Note that ${J}$ satisfies ${J(0)=0}$ and

$\displaystyle \frac{D}{dt} J(t)|_{t=0} = \frac{D}{du} \frac{\partial}{\partial t} H(t,u) |_{t,u=0} = \frac{D}{du} (X + uY)|_{u=0} = Y.$

Proposition 1 Suppose ${X,Y}$ are sufficiently small. Let ${J}$ be the Jacobi field along the geodesic ${\gamma(t) := \exp_p(tX)}$ with ${J(0) = 0, \frac{D}{dt} J(t)|_{t=0} = Y}$ (i.e. using the ODE theorems). Then$\displaystyle J(1)= (\exp_p)_{*X}(Y) \in T_{\exp_p(X)}(M).$

I will next explain how to use this fact to prove the Cartan-Hadamard theorem on manifolds of negative curvature.

Source: These notes, from an introductory course on differential geometry.

An isometry between two Riemannian manifolds ${(M,g), (N,h)}$ is a diffeomorphism ${\phi: M \rightarrow N}$ such that ${\phi^* h = g}$. In other words, ${\phi}$ respects the Riemannian structure as well as the differentiable structure.

It follows that if ${d,d'}$ are the metrics on ${M,N}$ induced by the Riemannian metrics ${g,h}$, then ${d'(\phi(x),\phi(y)) = d(x,y)}$ for ${x,y \in M}$—that is, ${\phi}$ is distance-preserving. Interestingly, a version of the converse is true:

Theorem 1 (Myers-Steenrod) If ${\phi: M \rightarrow M}$ is distance-preserving and surjective, then it is an isometry (in particular, it is smooth).

I should observe that the proof applies to the case of a map of Riemannian manifolds ${\phi: M \rightarrow N}$! I don’t know why Helgason states it with the extra hypotheses. Initially I wrote this post with this question unresolved under the goal of figuring out what I had missed and why this hypothesis was necessary by blogging, but I was unable to do so. With that complete, however, a search reveals that it’s stated in the additional generality in Petersen’s book (the relevant parts of which can be viewed at Google Books). However, it’s easier to do the notations in the case of one manifold.

${\phi}$ is evidently a homeomorphism. Now pick ${p \in M}$ and a neighborhood ${D_r(p)}$ such that for any ${q \in D_r(p)}$, there is a unique geodesic in that neighborhood connecting ${p,q}$. Call it ${\gamma}$. I claim that ${\phi \circ \gamma}$ is a geodesic in ${N}$. The geodesic ${\gamma}$ can be assumed to be parametrized by unit length. We have for all ${t}$,

$\displaystyle d(p,q) = d(\gamma(t),p) + d(\gamma(t), q).$

Thus

$\displaystyle d(\phi(p),\phi(q)) = d(\phi(\gamma(t)),\phi(p)) + d(\phi(\gamma(t)), \phi(q)) \ \ (*).$

In other words, we have strict equality in the triangle inequality. (more…)

Ok, recall our goal was to prove Helgason’s formula,

$\displaystyle \boxed{ (d \exp)_{tX}(Y) = \left( \frac{ 1 - e^{\theta( - tX^* )}}{\theta(tX^*)} (Y^*) \right)_{\exp(tX)}.}$

and that we have already shown

$\displaystyle {(d \exp)_{tX}(Y) f = \sum_{n=0}^{\infty} \frac{t^n}{(n+1)!} ( X^{*n} Y^* + X^{*(n-1)} Y^* X^* + \dots + Y^* X^{*n})f(p).}$  (more…)

We showed that the differential of the exponential map ${\exp_p: T_p(M) \rightarrow M}$ for ${M}$ a smooth manifold and ${p \in M}$ is the identity at ${0 \in T_p(M)}$. In the case of analytic manifolds, it is possible to say somewhat more. First of all, if we’re working with real-analytic manifolds, we can say that a connection ${\nabla}$ is analytic if ${\nabla_XY}$ is analytic for analytic vector fields ${X,Y}$. Using the real-analytic versions of the ODE theorem, it follows that ${\exp_p}$ is an analytic morphism.

So, make the above assumptions: analyticity of both the manifold and the connection. Now there is a small disk ${V_p \subset T_p(M)}$ such that ${\exp_p}$ maps ${V_p}$ diffeomorphically onto a neighborhood ${U \subset M}$ containing ${p}$. We will compute ${d(\exp_p)_{X}(Y)}$ when ${X \in V_p}$ is sufficiently small and ${Y \in T_p(M)}$ (recall that we identify ${T_p(M)}$ with its tangent spaces at each point). (more…)

If ${M}$ is a manifold and ${N}$ a compact submanifold, then a tubular neighborhood of ${N}$ consists of an open set ${U \supset N}$ diffeomorphic to a neighborhood of the zero section in some vector bundle ${E}$ over ${N}$, by which $N$ corresponds to the zero section.

Theorem 1 Hypotheses as above, ${N}$ has a tubular neighborhood. (more…)

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…)