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Variations of curves, Jacobi fields, and the differential of the exponential map

Posted by Akhil Mathew under

differential geometry,

MaBloWriMo | Tags:

calculus of variations,

exponential map,

Jacobi fields |

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It is of interest to consider functions on the space of curves , where is an interval and is a smooth manifold. To study maxima and minima, it is of interest to consider variations of curves, holding the endpoints fixed. Let be smooth. A **variation of ** is a smooth map

with , and for all . For a variation of , define the **variation vector field** (which is an analog of a “tangent vector”)

this is a vector field along . Similarly we can define the “velocity vector field” along . If is provided with a connection, we can define the “acceleration vector field” , where denotes covariant differentiation.

Given a vector field along , we can construct a variation of with as the variation vector field: take .

**Variations of geodesics and Jacobi fields **Let now be a manifold with a symmetric connection .Let be a variation of a geodesic such that for any , is a geodesic as well. Then the variation vector field satisfies a certain differential equation. Now

We have used the symmetry of . Now we can write this as

By geodesy, the first part vanishes, and the second is

We have shown that satisfies the **Jacobi equation**

Any vector field along satisfying this is called a **Jacobi field**.

**The differential of the exponential map **Let , and consider the exponential map where is a neighborhood of the origin in . Let . Now the map takes horizontal lines to geodesics in when are small enough. This can be viewed as a vector field along the geodesic . The variation vector field is thus a Jacobi field, and also at is Note that satisfies and

**Proposition 1** *Suppose are sufficiently small. Let be the Jacobi field along the geodesic with (i.e. using the ODE theorems). Then *

*
*

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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