Our main goal today is:
Theorem 1 Given , there is a neighborhood containing such that geodesics from to every point of exist and also such that given a path inside from to , we have
with equality holding if and only if is a reparametrization of .
In other words, geodesics are locally path-minimizing. Not necessarily globally–a great circle is a geodesic on a sphere with the Riemannian metric coming from the embedding in , but it need not be the shortest path between two points.
To prove this will require a bit of work. Here is a warm-up lemma we shall need.
Lemma 2Let be a curve, and vector fields along . Then
To prove this, write where the are parallel and orthonormal at (hence along ). Then
by the rules for connections. Then the statement of the lemma becomes merely the product rule.
A corollary of this lemma is that geodesics have constant speed .
Now we move on to proving the theorem. First of all, let’s choose such that it is the diffeomorphic image of the unit ball under the exponential map ; this is because the exponential map’s differential at zero is the identity. Then every path as above can be written in the form , where and (with the norm on coming from the inner product ). Now this is a geodesic up to reparametrization iff is constant.
Motivated by this, consider the map where is a small interval containing the origin, with
Lemma 3 (Gauss) We have .
Let trace out a path in which is also a 1-dimensional closed submanifold with tangent vector at . Now is a function of , which is
evaluated at . So it will be enough to prove that the vector field along the surface vanishes. Take the partial derivative with respect to , using the first lemma:
The first term vanishes by definition of a geodesic—the image of a line in with a fixed point of gets sent via to a geodesic. As for the second, we can use the Clairaut-like theorem for symmetric connections to get that this equals
Now in here we can pull out the to get
(Recall that geodesics move at constant speed .) Going back a few equations to (2) shows that is constant in . Since for all because of the term (all geodesics start at !), we find that . This implies the lemma.
With notation as in (1), we have by the Gauss lemma
This will be minimized precisely when , which is when is a geodesic.