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Attached Submanifolds Beyond Symmetric Spaces

Megan M. Kerr, Tracy L. Payne

TL;DR

This work extends Tamaru's theory of attached submanifolds from symmetric spaces to solvable pseudo-Riemannian manifolds of strong Iwasawa type, using root-space data to define attached subalgebras. A central result is that the Ricci endomorphism of the ambient algebra, restricted to the attached subalgebra, coincides with the intrinsic Ricci endomorphism if and only if a Jacobi Star Condition holds, linking curvature to an algebraic criterion. The study shows attached submanifolds are always minimal and provides a precise orthogonality condition for when they are totally geodesic. An explicit Kac-Moody-inspired example demonstrates the construction yields Einstein solvmanifolds with attached submanifolds that are not tied to symmetric spaces, highlighting broader applicability and potential for new Einstein homogeneous spaces.

Abstract

We study submanifold geometry in the presence of symmetry, focusing on submanifolds of solvmanifolds with an unusual property relative to Ricci curvature. We generalize work of H. Tamaru \cite{tamaru-11} in which he explores the geometry of submanifolds of symmetric spaces of noncompact type constructed from parabolic subgroups of the isometry group. He calls these attached submanifolds. The Ricci curvatures of attached submanifolds coincide with the restrictions of the Ricci curvatures of ambient symmetric spaces. We broaden Tamaru's construction by weakening the hypotheses on the ambient space, allowing a pseudo-Riemannian scalar product, and defining attached submanifolds in terms of root spaces. We demonstrate that in this setting, the Ricci curvature restriction property for attached submanifolds holds if and only if the submanifold satisfies an algebraic criterion that we call the Jacobi Star Condition. Like attached submanifolds of symmetric spaces, our attached submanifolds are minimal, and are only totally geodesic under hypotheses analogous to hypotheses in the symmetric space case. Finally, we give an example of a solvmanifold that has an attached submanifold and is not a symmetric space, demonstrating that attached submanifolds are not unique to symmetric spaces.

Attached Submanifolds Beyond Symmetric Spaces

TL;DR

This work extends Tamaru's theory of attached submanifolds from symmetric spaces to solvable pseudo-Riemannian manifolds of strong Iwasawa type, using root-space data to define attached subalgebras. A central result is that the Ricci endomorphism of the ambient algebra, restricted to the attached subalgebra, coincides with the intrinsic Ricci endomorphism if and only if a Jacobi Star Condition holds, linking curvature to an algebraic criterion. The study shows attached submanifolds are always minimal and provides a precise orthogonality condition for when they are totally geodesic. An explicit Kac-Moody-inspired example demonstrates the construction yields Einstein solvmanifolds with attached submanifolds that are not tied to symmetric spaces, highlighting broader applicability and potential for new Einstein homogeneous spaces.

Abstract

We study submanifold geometry in the presence of symmetry, focusing on submanifolds of solvmanifolds with an unusual property relative to Ricci curvature. We generalize work of H. Tamaru \cite{tamaru-11} in which he explores the geometry of submanifolds of symmetric spaces of noncompact type constructed from parabolic subgroups of the isometry group. He calls these attached submanifolds. The Ricci curvatures of attached submanifolds coincide with the restrictions of the Ricci curvatures of ambient symmetric spaces. We broaden Tamaru's construction by weakening the hypotheses on the ambient space, allowing a pseudo-Riemannian scalar product, and defining attached submanifolds in terms of root spaces. We demonstrate that in this setting, the Ricci curvature restriction property for attached submanifolds holds if and only if the submanifold satisfies an algebraic criterion that we call the Jacobi Star Condition. Like attached submanifolds of symmetric spaces, our attached submanifolds are minimal, and are only totally geodesic under hypotheses analogous to hypotheses in the symmetric space case. Finally, we give an example of a solvmanifold that has an attached submanifold and is not a symmetric space, demonstrating that attached submanifolds are not unique to symmetric spaces.
Paper Structure (26 sections, 15 theorems, 78 equations)

This paper contains 26 sections, 15 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Context and motivation.
  3. Tamaru's results on attached submanifolds of symmetric spaces of noncompact type
  4. Main theoretical results
  5. Main Theorem
  6. Minimality
  7. Solvable Metric Lie algebras defined by Kac-Moody algebras
  8. Overview
  9. Preliminaries
  10. Solvable metric Lie algebras
  11. Geometry of solvable metric Lie algebras
  12. Attached subalgebras and submanifolds
  13. The definition of attached subalgebra
  14. Properties of attached subalgebras
  15. The Jacobi Star Condition
  16. ...and 11 more sections

Key Result

Theorem 2.3

alekseevski-75conti-rossi-19 Given a nilpotent pseudo-Riemannian metric Lie algebra $({\mathfrak{n}}, {\langle\cdot {,} \cdot\rangle})$ with an orthonormal basis $\{E_i\}$, the Ricci endomorphism $\operatorname{Ric}^{{\mathfrak{n}}}$ is given by

Theorems & Definitions (33)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 23 more