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.

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