This is a simple result on the homotopy of two curves with the same endpoints in a Riemannian manifold , which is raised in today’s Riemannian geometry lesson. Professor sketched the proof of the result but there’s one point which seems unclear (at least to me), and I also have some doubt about the statement of the result. I am posting it here in the hope that some of you can actually help me improve the statement of the following result.
Theorem 1 Let be a continuous curve, then there exists (depending on ) such that if is another continuous curve with the same endpoints, satisfying
Then and are homotopic relative to their endpoints. i.e. there exists continuous such that , , for all and for all .
Remark 1 In the lecture, it was said that can be chosen to be half of the injectivity radius of (if it is positive). I am not able to show this, and my cannot be explicitly written down (due to inverse function theorem argument, see below).
Proof: Let us define the map by , where is an open set (which can be chosen to be maximal) containing the zero section of such that the exponential map is well-defined. Then by inverse function theorem, is a diffeomorphism in an open set of , containing . This is because is and the differential of at a “point” is which is invertible. (I’ve omitted some details. )
Let which is an open set in , containing the diagonal . Let be so small that if satisfies (1) then and by shrinking further, that lies in the convex normal neighborhood of for all (i.e. every two points in this neighborhood can be joined by a unique minimal geodesic which is contained in this neighborhood). Then for all fixed , there is a unique minimal geodesic such that . Now we define . Note that we then have , , for all and for all .
It remains to show that is continuous (on the square ). By our construction, the inverse of (which for convenience still denote by ) exists on . So let , where , will be continuous. It is then easy to see that
which is continuous.