Show that the convex neighborhood in a Riemannian Manifold are subset contractibles (to any of their points).
A subset $A$ of the differential manifold $M$ is contractible to the point $a\in A$ when, the aplications $id_{A}$ (identity in $A$) and $\gamma_{a}:x\in A\to a\in A$ are homotopic (with a point basis $a$).
I know that "geodesic convexity" is a natural generalization of convexity for sets and functions to Riemannian manifold, so in a set the convexity implies the homotopy. My idea was, take a riemannian metric and take exponential map in convex neighborhood, but I don't see these set are contractible to a point. Any hint!
Let $U$ be a geodesically convex subset of $(M,g)$. Choose a point $p\in U$. For each $q\in U$ there exists a unique minimizing geodesic starting at $p$ and ending at $q$, i.e. there is a unique tangent vector $\hat{q}\in T_pM$ such that $$q = \exp_p\hat{q}.$$ It follows that there is a diffeomorphism $$\hat{U}:=\{\hat{q}\in T_pM\mid q\in U\}\cong U$$ via the exponential map (exercise: check it is indeed smooth with smooth inverse). The contraction is then induced by the map $I\times\hat{U}\to\hat{U}$ given by $$(t,\hat{q})\longmapsto t\hat{q}.$$ Exercise: Show that the set $\hat{U}$ is star-shaped with $0$ as a center. (Hint: a shorter piece of a minimizing geodesic is again a minimizing geodesic.)