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Characterization of Sasakian manifolds

Vladimir Rovenski

Abstract

Weak contact metric manifolds, i.e., the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, defined by the author and R. Wolak, allowed us to take a new look at the theory of contact manifolds. In this paper, we continue our study, see arXiv:2312.11411, of a structure of this type, called a weak nearly Sasakian structure, and prove two theorems characterizing Sasakian manifolds. Our main result generalizes the theorem by A. Nicola - G. Dileo - I. Yudin (2018) and provides a new criterion for a weak almost contact metric manifold to be Sasakian.

Characterization of Sasakian manifolds

Abstract

Weak contact metric manifolds, i.e., the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, defined by the author and R. Wolak, allowed us to take a new look at the theory of contact manifolds. In this paper, we continue our study, see arXiv:2312.11411, of a structure of this type, called a weak nearly Sasakian structure, and prove two theorems characterizing Sasakian manifolds. Our main result generalizes the theorem by A. Nicola - G. Dileo - I. Yudin (2018) and provides a new criterion for a weak almost contact metric manifold to be Sasakian.
Paper Structure (4 sections, 11 theorems, 86 equations)

This paper contains 4 sections, 11 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Proofs of Lemmas \ref{['L-R01b']} and \ref{['L-R03']}

Key Result

Lemma 1

For a weak nearly Sasakian manifold $M^{\,2n+1}(\varphi,Q,\xi,\eta,g)$ we obtain Moreover, $h^2\varphi=\varphi h^2$, $h\varphi^2=\varphi^2 h$, $h^2\varphi^2=\varphi^2 h^2$, etc.

Theorems & Definitions (16)

  • Definition 1: see rov-2023
  • Lemma 1: see rov-2023c
  • Lemma 2: see rov-2023c
  • Proposition 1: see also rov-2023c
  • Theorem 1: see rov-2023c
  • Theorem 2
  • proof
  • Lemma 3
  • Lemma 4
  • Remark 1
  • ...and 6 more