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Geodesics and the exponential map November 4, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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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…)

Covariant derivatives and parallelism November 1, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: , , ,

[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 {\nabla} and a vector field {Y}, the operation {X \rightarrow \nabla_X Y} is linear in {X} over smooth functions—thus it is a tensor (of type (1,1)), and the value at a point {p} can be defined if {X} is replaced by a tangent vector at {p}. In other words, we get a map {T(M)_p \times \Gamma(TM) \rightarrow T(M)_p}, where {\Gamma(TM)} denotes the space of vector fields. We’re going to need this below. (more…)