Geodesics and the exponential map November 4, 2009Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: connections, exponential map, geodesics, ordinary differential equations
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
The fundamental theorem of ODEs does not, unfortunately, apply to this though. However, there is an elementary trick we can use. Let be functions of and consider the equations
Now this is a system of ODEs, and given a solution to this, we can take as a geodesic. In particular, given initial conditions and (respectively the and ) we can construct a geodesic locally. More precisely, using the local smoothness theorem, we find:
Proposition 1 Fix . Then there is a neighborhood of and a smooth map such that if , then for is a geodesic starting at with initial tangent vector .
There’s a slight problem here I haven’t yet addressed: strictly speaking, the local smoothness theorem only gives the proposition with replaced by for some small. But geodesics can be rescaled—if is a geodesic for the tangent vector , then , which is defined on an interval times as large, is one for the tangent vector . So just shrink by a small constant factor to allow the geodesics to be defined on a larger interval.Another reason that geodesics are important is that, as I will eventually prove, with respect to a Riemannian metric and the associated Levi-Civita connection, geodesics have the chracteristic property of being locally distance-minimizing.
Now fix the map as before and let . Then is a smooth map where is as above, called the exponential map. I claim that the differential of at is the identity, if we identify in the usual way. In particular,
Proposition 2 is a local isomorphism.
To prove the claim, take a tangent vector and consider the corresponding path , in . Then sends this path to by the above rescaling remarks, and this traces out a geodesic in whose tangent vector at is . Thus the assertion on the differential is proved.
Where next? I want to spend a post on the tubular neighborhood theorem just because it’s fun, and provides an illustration of these concepts. Next I think it might be interesting to discuss what the differential of the exponential map is other than the origin. There is a formula of Helgason that describes that for real-analytic manifolds. Then I should get to some Riemannian geometry.