What is the name of this geometric structure, where we identify each sphere of vision with the sphere at infinity?

Question

If you consider hyperbolic $n$-space $H^n$, modeled by the open unit ball $B^n \subset \mathbb{R}^n$, then given any two distinct points $x_1$, $x_2$ in $H^n$, there is a natural way of identifying the unit tangent spheres $S_{x_1}$ and $S_{x_2}$ at $x_1$ and $x_2$ respectively. Start at $x_1$. Given a unit tangent vector $v$ at $x_1$, draw the geodesic ray starting at $x_1$ with initial velocity $v$, and define $f_1(v)$ to be the ideal point which is the limiting point of that geodesic ray. Then $f_1: S_{x_1} \to S_\infty$ is a diffeomorphism from $S_{x_1}$ onto the sphere at infinity.

Similarly, one may define the diffeomorphism $f_2: S_{x_2} \to S_\infty$. Then the composition $f_2^{-1} \circ f_1$ is a naturally defined diffeomorphism from $S_{x_1}$ onto $S_{x_2}$.

This is an example where we identify each "sphere of vision" (such as $S_{x_1}$, $S_{x_2}$) with the sphere at infinity.

Another example is Euclidean space $\mathbb{E}^n$. Assume that we compactify $\mathbb{E}^n$ by adding a point at infinity to each oriented direction (thus we add an $n-1$ dimensional sphere at infinity).

In this case, one may trivially identify each sphere of vision $S_x$, for $x \in \mathbb{E}^n$ with the sphere at infinity.

So here are two examples where one may identify each sphere of vision with the sphere at infinity.

Have similar geometric structures been studied before? Does this notion have a name please?

Edit (in reply to @RyanBudney): what I have in mind is something like this. Let $M$ be an $n$-dimensional manifold. Given $p \in M$, define

$$S_p = (T_p(M) \setminus \{0\})/\mathbb{R}_+$$

where $\mathbb{R}_+$ acts on $T_p(M)$ by scaling. Assume that you have a trivialization of the sphere bundle consisting of the unions of $S_p$, for $p \in M$. Then given any $v \in S^{n-1}$, one can associate to it a non-vanishing vector field on $M$, which is defined up to multiplication by a smooth positive function $f: M \to \mathbb{R}_+$. Assuming the flow of this vector field is complete, this defines a foliation of $M$ by (unparametrized) curves.

So far, I have only used the trivialization of the bundle of $S_p$'s, and a completeness assumption. But then I would like to make an additional assumption on $M$, namely that it has a compactification obtained by only adding an $S^{n-1}$ at infinity, such that given $v \in S^{n-1}$, the corresponding curves obtained by integrating the corresponding vector field, all go towards the same point on the sphere at infinity, say $f(v)$, and that the resulting map $f: S^{n-1} \to S^{n-1}_\infty$ is itself a diffeomorphism.

I am guessing one may define a point at infinity as an equivalence class of curves. I have seen this done before, but I don't remember in which article.

Answer 1

This notion of 'the sphere at infinity' is commonly encountered in hyperbolic geometries. Gromov, in particular, has used it in studying the behavior of discrete transformation groups on hyperbolic manifolds and you might also look at the works of Biquard on prescribing the geometry of the boundary at infinity of an Einstein manifold with negative Ricci curvature and the work of Fefferman and Graham and others on 'filling in' the geometry of conformal or CR manifolds so that they become (locally) the boundary of a higher dimensional object.

Meanwhile, there is another way to describe these geometries in terms of what is called 'oriented path geometry'. Essentially, what you are starting with is a map $\pi:S(M^n)\to S^{n-1}$ (where $\xi:S(M)\to M$ is the 'tangent sphere bundle' of $M$ , as the OP described it above) that is smooth and has the property that the restriction $\pi_x:S_x(M)\to S^{n-1}$ is a diffeomorphism for all $x\in M$; in particular, $\pi:S(M)\to S^{n-1}$ is a smooth submersion. Thus, for $r\in S^{n-1}$, the preimage $\pi^{-1}(r)\subset S(M)$ is a smooth section of $S(M)\to M$ that can be represented by a nonvanishing vector field on $M$, unique up to multiplying by a positive function. The integral curves of such a vector field have an intrinsic orientation, so, in this way, we get a $(2n{-}2)$-parameter family of oriented curves, exactly one through each point of $M$ in each (oriented) direction.

In fact, this defines a smooth foliation of $S(M)$ by oriented curves, with the property that, for any leaf $L\subset S(M)$ and any point $\rho\in L$, the oriented tangent to $\xi:L\to M$ at $\xi(\rho)\in M$ is $\rho\in S_{\xi(\rho)}M$ itself. This latter is the very definition of an oriented path geometry. (An 'nonoriented path geometry', also known as a 'path geometry' in the literature, is essentially an oriented path geometry with the property that reversing the orientation of a given oriented path of the oriented path geometry yields another oriented path of the oriented path geometry.)

Path geometries (and, using the same tools, oriented path geometries) have been studied for a long time, with works going back to Lie, Cartan, Tresse, Chern, Hatchroudi, and many others. The basic result is that one can define a canonical Cartan connection for such a geometry and all of the invariants can be read off from the curvature of this connection and its covariant derivatives.

Given an oriented path geometry $\Pi$ on $M$, when a domain $D\subset M$ is suitably '$\Pi$-convex' with a smooth strictly '$\Pi$-convex' boundary $\partial D$, one will have the property that, for every $x\in D$, each oriented path of $\Pi$ leaving $x$ will meet $\partial D$ transversely in a unique point, and this will, in the obvious way, define a map $\pi:S(D)\to\partial D$ define a submersion that gives a diffeomorphism $\pi:S_x(D)\to\partial D$ for all $x\in D$.

Thus, given an oriented path geometry $\Pi$, there will be many structures of the kind you envision, even many satisfying the second property you want, which is that the $S^{n-1}$ can be regarded as the boundary at infinity of the manifold $M$.

However, this also shows how one can define such structures $\pi:S(M)\to S^{n-1}$ so that the $S^{n-1}$ cannot be naturally regarded as a boundary of $M$. For example, take the oriented path geometry of oriented straight lines in the plane, let $M$ be the interior of the unit disk centered at the origin, but let $\Sigma$ be the circle $x^2+y^2 = 2$, and, for nonzero $v$ in $\mathbb{R}^2$, let $\pi(u,v)$ be the point of the form $u+tv$ with $t>0$ that satisfies $u+tv\in\Sigma$. Then $\pi:S(M)\to \Sigma\simeq S^1$ is a smooth submersion that is a diffeomorphism restricted to each fiber $S_u(M)\simeq S^1$, but $\Sigma$ cannot reasonably be regarded as the 'boundary' of $M$.