The map $f:T_1M\to \hat{\mathbb{R}}$ is continuous.

Question

Let $(M,d)$ be a complete Riemannian manifold and let $T_1M=\{v\in TM: \|v\|=1\}$. Define a map, $$s:T_1M\to \hat{\mathbb{R}},~~ s(v)= \sup\{t:d(\pi(v),\operatorname{exp}(tv))=t\}$$ where $\pi$ is the projection map from the the tangent bundle to $M$ $(\pi(p,v)=p,~(p,v)\in TM)$. I need to prove that the map $s$ is continuous and $\hat{\mathbb{R}}$ is the one point compactification of $\mathbb{R}$.

Any reference or help will be appreciated.

Answer 1

This is an extended comment.

I would call your function $s$ the cut distance function on the unit sphere bundle $T_1M$.

The claim is not true without further assumptions. Consider, for example, a banana-shaped open set $M\subset\mathbb R^2$ with the Euclidean metric. When the geodesic starting at $v$ is tangent to a concave part of the boundary, problems arise. If you keep the base point fixed and approach the direction $v$ from different directions, you get different limits. The basic problem is that the exponential map is not defined for all time.

This example and your definition of the function $s$ seems to suggest the assumption that all geodesics are defined for all time. In other words, $M$ is assumed (geodesically or metrically; cf. Hopf–Rinow) complete.

If it is true, perhaps something like this would work: Suppose $s$ was discontinuous at a point $v\in T_1M$ where $s(v)<\infty$. Then there is a sequence of points $v_k\in T_1M$ and $\epsilon>0$ so that $v_k\to v$ and $s(v_k)\geq s(v)+\epsilon$ for all $k$ or $s(v_k)\leq s(v)-\epsilon$ for all $k$. (To see this, recall that continuity means that if $v_k\to v$, then $s(v_k)\to s(v)$. If this fails, there is a sequence so that the images do not converge to the right limit or at all; then take a suitable subsequence.)

Answer 2

Are you so sure this is continuous? The relevant notion here is the cut locus. Your function is asking how long can a geodesic get before it stops being minimizing. The cut locus of a point $p$ in a Riemannian manifold $M$ is the set consisting of all points $\gamma(t_0)$ on geodesics $\gamma$ starting at $p$ such that $\gamma$ is distance-minimizing up to $t_0$, but not further. In other words, $t_0=\sup\{t:d(p,\gamma(t))=t\}$. If $\gamma_v$ denotes the geodesic with initial point $\pi(v)$ and initial velocity $v$, then $s(v)$ is just the length of the portion $\gamma_v$ between its start and the cut locus.

Here's a picture of a cut locus from a vertex of a cube. Of course this isn't smooth, but one should be able to smooth things out a bit and keep the picture more or less the same. But now you see if you start from point 0 and aim at point 4, the distance to the cut locus is drastically different then if you aim a little bit left or right.