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A new approach to the classification of almost contact metric manifolds via intrinsic endomorphisms

Ilka Agricola, Dario Di Pinto, Giulia Dileo, Marius Kuhrt

TL;DR

The paper introduces an intrinsic, endomorphism-based framework for almost contact metric manifolds, defining $S=\varphi\circ\nabla^g\xi$ and $h=\tfrac12\mathcal L_\xi\varphi$ (with $P=-2h+\varphi S\varphi+S$) to replace the opaque $2^{12}$ Chinea--González classification. It shows that $S$ and $h$ determine the $\mathcal H$-parallel component of $\nabla^g\Phi$ and provide a natural flowchart that decomposes $\mathcal D_2\oplus\mathcal D_3$ into irreducible classes, refining the CG classes and enabling a direct characterization of geometric properties such as $CR$-integrability, normality, anti-normality, and the existence of characteristic connections. The paper also analyzes the intrinsic torsion via the minimal adapted connection, identifies a distinguished subspace $\mathcal C_{\min}$ governing skew torsion, and gives necessary and sufficient conditions for the existence and parallelism of the characteristic connection, including curvature criteria for torsion parallelism. It applies the framework to a range of examples (e.g., $3$-$\alpha$-$Sasaki$, nearly Sasakian/cosymplectic, and 5D quasi-Sasakian) and clarifies how the $S,h$ data control torsion type and the appearance of $\mathcal C_{10},\mathcal C_{11}$ representations. Overall, the approach yields a more natural and comprehensive understanding of almost contact metric structures and their adapted connections.

Abstract

In 1990, D. Chinea and C. Gonzalez gave a classification of almost contact metric manifolds into $2^{12}$ classes, based on the behaviour of the covariant derivative $\nabla^gΦ$ of the fundamental $2$-form $Φ$. This large number makes it difficult to deal with this class of manifolds. We propose a new approach to almost contact metric manifolds by introducing two intrinsic endomorphisms $S$ and $h$, which bear their name from the fact that they are, basically, the entities appearing in the intrinsic torsion. We present a new classification scheme for them by providing a simple flowchart based on algebraic conditions involving $S$ and $h$, which then naturally leads to a regrouping of the Chinea-Gonzalez classes, and, in each step, to a further refinement, eventually ending in the single classes. This method allows a more natural exposition and derivation of both known and new results, like a new characterization of almost contact metric manifolds admitting a characteristic connection in terms of intrinsic endomorphisms. We also describe in detail the remarkable (and still very large) subclass of $\mathcal{H}$-parallel almost contact manifolds, defined by the condition $(\nabla^g_XΦ)(Y,Z)=0$ for all horizontal vector fields, $X,Y,Z\in\mathcal{H}$.

A new approach to the classification of almost contact metric manifolds via intrinsic endomorphisms

TL;DR

The paper introduces an intrinsic, endomorphism-based framework for almost contact metric manifolds, defining and (with ) to replace the opaque Chinea--González classification. It shows that and determine the -parallel component of and provide a natural flowchart that decomposes into irreducible classes, refining the CG classes and enabling a direct characterization of geometric properties such as -integrability, normality, anti-normality, and the existence of characteristic connections. The paper also analyzes the intrinsic torsion via the minimal adapted connection, identifies a distinguished subspace governing skew torsion, and gives necessary and sufficient conditions for the existence and parallelism of the characteristic connection, including curvature criteria for torsion parallelism. It applies the framework to a range of examples (e.g., --, nearly Sasakian/cosymplectic, and 5D quasi-Sasakian) and clarifies how the data control torsion type and the appearance of representations. Overall, the approach yields a more natural and comprehensive understanding of almost contact metric structures and their adapted connections.

Abstract

In 1990, D. Chinea and C. Gonzalez gave a classification of almost contact metric manifolds into classes, based on the behaviour of the covariant derivative of the fundamental -form . This large number makes it difficult to deal with this class of manifolds. We propose a new approach to almost contact metric manifolds by introducing two intrinsic endomorphisms and , which bear their name from the fact that they are, basically, the entities appearing in the intrinsic torsion. We present a new classification scheme for them by providing a simple flowchart based on algebraic conditions involving and , which then naturally leads to a regrouping of the Chinea-Gonzalez classes, and, in each step, to a further refinement, eventually ending in the single classes. This method allows a more natural exposition and derivation of both known and new results, like a new characterization of almost contact metric manifolds admitting a characteristic connection in terms of intrinsic endomorphisms. We also describe in detail the remarkable (and still very large) subclass of -parallel almost contact manifolds, defined by the condition for all horizontal vector fields, .

Paper Structure

This paper contains 18 sections, 34 theorems, 204 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. New tools for the classification of almost contact metric structures
  3. Review of the Chinea-Gonzalez classification
  4. $\mathcal{H}$-parallel almost contact metric manifolds
  5. The intrinsic endomorphisms $(S,h)$
  6. Classification of $\mathcal{H}$-parallel almost contact metric manifolds
  7. The flowchart of $\mathcal{H}$-parallel structures via intrinsic endomorphisms
  8. Special geometric conditions for $\mathcal{H}$-parallel manifolds
  9. $CR$-integrability
  10. Normal structures
  11. Anti-normal structures
  12. Structures with Killing vector field $\xi$
  13. Infinitely many equivalent representations in $\mathcal{C}_{10}\oplus\mathcal{C}_{11}$
  14. Structures with integrable horizontal distribution
  15. The intrinsic torsion
  16. ...and 3 more sections

Key Result

Proposition 2.2

An almost contact metric manifold $(M,\varphi,\xi,\eta,g)$ is $\mathcal{H}$-parallel if and only if where $N_\varphi(X,Y,Z):=g(N_\varphi(X,Y),Z)$.

Figures (2)

  • Figure 1: Flowchart for classifying $\mathcal{H}$-parallel almost contact metric manifolds based on the intrinsic endomorphisms $S$ and $h$ as well as the auxiliary endomorphism $P$.
  • Figure 2: Remarkable classes of type $\mathcal{C}_{\lambda,\mu}$

Theorems & Definitions (83)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Proposition 2.8
  • ...and 73 more