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

Nikos Georgiou, Brendan Guilfoyle, Morgan Robson

TL;DR

The paper characterizes isotropic submanifolds of $T\mathbb{S}^n$ in terms of corresponding families of oriented lines in $\mathbb{R}^{n+1}$ and proves that isotropy is equivalent to lines being orthogonal to some submanifold locally. It then analyzes line families that are tangent to submanifolds, showing isotropy occurs exactly when the generating unit vector field is a gradient with an integrable orthogonal distribution and is geodesic, accompanied by a structure theorem for the resulting orthogonal submanifolds. Focal sets for such line families are introduced, with curvature relations derived: focal sheet sectional curvatures are given by expressions involving the derivatives of the signed distances between focal points, and Ricci curvatures are determined by astigmatism differences. In the hypersurface case, the paper specializes to focal sheets arising from radii of curvature, establishing explicit links between focal geometry, Bianchi-type transformations, and generalized astigmatism, thereby extending 1874 results to higher codimensions. The results provide a unified framework for inverse focal-set problems and reveal deep connections between symplectic geometry, focal set curvature, and integrable systems.

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.

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

TL;DR

The paper characterizes isotropic submanifolds of in terms of corresponding families of oriented lines in and proves that isotropy is equivalent to lines being orthogonal to some submanifold locally. It then analyzes line families that are tangent to submanifolds, showing isotropy occurs exactly when the generating unit vector field is a gradient with an integrable orthogonal distribution and is geodesic, accompanied by a structure theorem for the resulting orthogonal submanifolds. Focal sets for such line families are introduced, with curvature relations derived: focal sheet sectional curvatures are given by expressions involving the derivatives of the signed distances between focal points, and Ricci curvatures are determined by astigmatism differences. In the hypersurface case, the paper specializes to focal sheets arising from radii of curvature, establishing explicit links between focal geometry, Bianchi-type transformations, and generalized astigmatism, thereby extending 1874 results to higher codimensions. The results provide a unified framework for inverse focal-set problems and reveal deep connections between symplectic geometry, focal set curvature, and integrable systems.

Abstract

Families of oriented lines in are studied via their identification with submanifolds of . In particular, families of oriented lines which are orthogonal to submanifolds in are shown to characterise those which are isotropic with respect to the canonical sympleptic structure on . Families of lines that are tangent to a -dimensional submanifold of 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 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 of constant astigmatism have pseudo-spherical focal sets.
Paper Structure (10 sections, 20 theorems, 91 equations)

This paper contains 10 sections, 20 theorems, 91 equations.

Table of Contents

  1. Introduction and Statement of Results
  2. Preliminaries
  3. The Symplectic Form and Splitting of TTSn
  4. Oriented Lines and TSn
  5. Notation and Conventions
  6. Focal Points of Submanifolds in TSn
  7. Lines Normal to Submanifolds
  8. Lines Tangent to Submanifolds
  9. The Curvatures of Focal Sheets
  10. The Hypersurface case

Key Result

Lemma 1

For any $Y\in T_{((\xi_0,\eta_0),r_0)}(T\mathbb{S}^n\times \mathbb{R})\simeq T_{(\xi_0,\eta_0)}T\mathbb{S}^n \oplus T_{r_0}\mathbb{R}$ write $Y=X+\rho{\pdv{}{r}}$ with $X\in T_{(\xi_0,\eta_0)}T\mathbb{S}^n$, $\rho\in\mathbb{R}$, and $r$ the canonical coordinate on $\mathbb{R}$. Then

Theorems & Definitions (49)

  • Lemma 1
  • proof
  • Definition 1
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • Proposition 2
  • proof
  • ...and 39 more