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.