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Geodesics and the exponential map
*November 4, 2009*

*Posted by Akhil Mathew in differential geometry, MaBloWriMo.*

Tags: connections, exponential map, geodesics, ordinary differential equations

5 comments

Tags: connections, exponential map, geodesics, ordinary differential equations

5 comments

Ok, we know what connections and covariant derivatives are. Now we can use them to get a map from the tangent space at one point to the manifold which is a local isomorphism. This is interesting because it gives a way of saying, “start at point and go five units in the direction of the tangent vector ,” 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 such that the vector field along created by the derivative is parallel. In local coordinates , here’s what this means. Let the Christoffel symbols be . Then using the local formula for covariant differentiation along a curve, we get

so being a geodesic is equivalent to the system of differential equations

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Covariant derivatives and parallelism
*November 1, 2009*

*Posted by Akhil Mathew in differential geometry, MaBloWriMo.*

Tags: connections, covariant derivatives, ordinary differential equations, parallelism

7 comments

Tags: connections, covariant derivatives, ordinary differential equations, parallelism

7 comments

[**Nobody should read this post without reading the excellent comments below. It turns out that thinking more generally (via connections on the pullback bundle) clarifies things. Many thanks to the (anonymous) reader who posted them. – AM, 5/16]**

A couple of days back I covered the definition of a (Koszul) connection. Now I will describe how this gives a way to differentiate vector fields along a curve.

**Covariant Derivatives **

First of all, here is a minor remark I should have made before. Given a connection and a vector field , the operation is linear in over smooth functions—thus it is a tensor (of type (1,1)), and the value at a point can be defined if is replaced by a **tangent vector** at . In other words, we get a map , where denotes the space of vector fields. We’re going to need this below. (more…)