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All invariant contact metric structures on tangent sphere bundles of compact rank-one symmetric spaces

J. C. González-Dávila

TL;DR

This work completes a systematic classification of $G$-invariant contact metric structures on the tangent sphere bundles $T_{r}(G/K)$ of all compact rank-one symmetric spaces, clarifying when such structures are $K$-contact, Sasakian, or $3$-Sasakian. It establishes that $T_{r}( ext{CP}^n)$ uniquely supports a $3$-Sasakian metric and a unique orthogonal Sasakian-Einstein structure, while $T_{r} ext{S}^n$ and $T_{r} ext{RP}^n$ admit a unique invariant Einstein (Sasakian-Einstein) metric; for CP$^n$ the moduli of invariant structures split into AI–AIII, BI–BIII, and C types with explicit data. Moreover, every invariant contact metric (and Sasakian/Sasakian-Einstein/3-Sasakian) structure on $T_{1}(G/K)$ extends to a corresponding almost Kahler, Kahler, Kahler Ricci-flat, or hyperKähler structure on the punctured bundle $D(G/K)$, linking contact geometry on the unit sphere bundle to holonomy reductions of the metric cone. The paper also provides detailed Einstein- and curvature-related results for the tangent sphere bundles of spheres and real projective spaces, including precise parameterizations of orthogonal metrics and uniqueness up to scale. These findings illuminate the interaction between homogeneous geometry, contact metric theory, and special holonomy in low-dimensional symmetric spaces with explicit constructions.

Abstract

All invariant contact metric structures on tangent sphere bundles of each compact rank-one symmetric space are obtained explicitly, distinguishing for the orthogonal case those that are K-contact, Sasakian or 3-Sasakian. Only the tangent sphere bundle of complex projective spaces admits 3-Sasakian metrics and there exists a unique orthogonal Sasakian-Einstein metric. Furthermore, there is a unique invariant contact metric that is Einstein, in fact Sasakian-Einstein, on tangent sphere bundles of spheres and real projective spaces. Each invariant contact metric, Sasakian, Sasakian-Einstein or 3-Sasakian structure on the unit tangent sphere of any compact rank-one symmetric space is extended, respectively, to an invariant almost Kahler, Kahler, Kahler Ricci-flat or hyperKahler structure on the punctured tangent bundle.

All invariant contact metric structures on tangent sphere bundles of compact rank-one symmetric spaces

TL;DR

This work completes a systematic classification of -invariant contact metric structures on the tangent sphere bundles of all compact rank-one symmetric spaces, clarifying when such structures are -contact, Sasakian, or -Sasakian. It establishes that uniquely supports a -Sasakian metric and a unique orthogonal Sasakian-Einstein structure, while and admit a unique invariant Einstein (Sasakian-Einstein) metric; for CP the moduli of invariant structures split into AI–AIII, BI–BIII, and C types with explicit data. Moreover, every invariant contact metric (and Sasakian/Sasakian-Einstein/3-Sasakian) structure on extends to a corresponding almost Kahler, Kahler, Kahler Ricci-flat, or hyperKähler structure on the punctured bundle , linking contact geometry on the unit sphere bundle to holonomy reductions of the metric cone. The paper also provides detailed Einstein- and curvature-related results for the tangent sphere bundles of spheres and real projective spaces, including precise parameterizations of orthogonal metrics and uniqueness up to scale. These findings illuminate the interaction between homogeneous geometry, contact metric theory, and special holonomy in low-dimensional symmetric spaces with explicit constructions.

Abstract

All invariant contact metric structures on tangent sphere bundles of each compact rank-one symmetric space are obtained explicitly, distinguishing for the orthogonal case those that are K-contact, Sasakian or 3-Sasakian. Only the tangent sphere bundle of complex projective spaces admits 3-Sasakian metrics and there exists a unique orthogonal Sasakian-Einstein metric. Furthermore, there is a unique invariant contact metric that is Einstein, in fact Sasakian-Einstein, on tangent sphere bundles of spheres and real projective spaces. Each invariant contact metric, Sasakian, Sasakian-Einstein or 3-Sasakian structure on the unit tangent sphere of any compact rank-one symmetric space is extended, respectively, to an invariant almost Kahler, Kahler, Kahler Ricci-flat or hyperKahler structure on the punctured tangent bundle.
Paper Structure (7 sections, 25 theorems, 107 equations)

This paper contains 7 sections, 25 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homogeneous almost contact metric manifolds
  4. The tangent sphere bundle of a compact rank-one symmetric space
  5. Invariant contact metric structures
  6. Invariant Sasakian and 3-Sasakian structures
  7. Invariant Einstein metrics on $T_{r}{\mathbb S}^{n}$ and $T_{r}{\mathbb R}{\mathbf P}^{n}$

Key Result

Proposition 2.1

A Riemannian manifold $(M,g)$ is contact metric if and only if the metric cone $({\mathcal{C}}(M),\overline{g})$ is almost Kähler. Moreover, the symplectic $2$-form $F$ in ${\mathcal{C}}(M)$ is the exact form $F = d(r^{2}\eta),$ where $\eta$ is the contact form of $M.$

Theorems & Definitions (46)

  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Definition 4.3
  • ...and 36 more