Isotropic submanifolds of $T\mathbb{S}^n$ and their focal sets

TL;DR

This paper investigates isotropic submanifolds of the tangent bundle of the sphere, characterizes their focal sets, and extends classical results relating curvature properties of hypersurfaces to their focal sets, generalizing Bianchi's theorem.

Contribution

It introduces a new framework for understanding isotropic submanifolds in $T ext{S}^n$, characterizes their focal sets, and generalizes Bianchi's theorem to higher dimensions and codimensions.

Findings

Characterization of isotropic families of lines orthogonal to submanifolds.

Expression of focal set curvatures in terms of focal point distances.

Extension of Bianchi's theorem relating curvature differences to focal set properties.

Abstract

Families of oriented lines in $\mathbb{R}^{n+1}$ are studied via their identification with submanifolds of $T\mathbb{S}^n$. In particular, families of oriented lines which are orthogonal to submanifolds in $\mathbb{R}^{n+1}$ are shown to characterise those which are isotropic with respect to the canonical sympleptic structure on $T\mathbb{S}^n$. Families of lines that are tangent to a $k$-dimensional submanifold of $\mathbb{R}^{n+1}$ are then studied. For such families, isotropy is shown to be equivalent to the generating vector field being geodesic and hypersurface-orthogonal on the submanifold. The focal set in $\mathbb{R}^{n+1}$ of a family of lines is introduced, extending the classical definition for families normal to hypersurfaces, to general families of lines of arbitrary codimension. A formula is derived that expresses certain sectional curvatures of the focal set in terms of the signed distances between corresponding focal points. We then solve an inverse problem for the focal sets of hypersurfaces and show certain sectional and Ricci curvatures of the focal set are determined by the differences between the hypersurface's radii of curvature. This generalises a Theorem of Bianchi from 1874 - namely that surfaces in $\mathbb{R}^3$ of constant astigmatism have pseudo-spherical focal sets.