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Geometry of weak metric $f$-manifolds: a survey

Vladimir Rovenski

TL;DR

The article surveys weak metric $f$-manifolds, a generalization of Yano's $f$-structure achieved by replacing the contact distribution's complex structure with a nonsingular skew-symmetric tensor. It develops a hierarchy of distinguished classes (weak ${K}$, weak ${ m S}$, weak ${ m C}$, weak $f$-K-contact, and weak $eta$-Kenmotsu $f$-manifolds) and analyzes their normality, foliation structure, and curvature, including extensive results on Killing fields, totally geodesic foliations, and soliton-type metrics. Key contributions include new characterizations of weak $f$-K-contact structures, deformation to classical $f$-K-contact geometry, and explicit curvature and Ricci identities for compact and warped-product constructions, often leading to $ exteta$-Einstein or Einstein-type outcomes under soliton hypotheses. The work provides obstructions via Adams numbers, integrates Ricci-type soliton theory into the weak framework, and lays groundwork for further geometric and topological applications and extensions beyond constant $eta$. Overall, the framework offers a cohesive approach to understanding geometric and topological rigidity, foliations, and curvature phenomena in weak metric $f$-manifolds with potential for broad influence in differential geometry.

Abstract

A weak $f$-structure on a smooth manifold, introduced by the author and R. Wolak (2022), generalizes K. Yano's (1961) $f$-structure. This generalization allows us to revisit classical theory and discover new applications related to Killing vector fields, totally geodesic foliations, Ricci-type solitons, and Einstein-type metrics. This article reviews the results on weak metric $f$-manifolds, where the complex structure on the contact distribution of a metric $f$-structure is replaced with a nonsingular skew-symmetric tensor, and explores its distinguished classes.

Geometry of weak metric $f$-manifolds: a survey

TL;DR

The article surveys weak metric -manifolds, a generalization of Yano's -structure achieved by replacing the contact distribution's complex structure with a nonsingular skew-symmetric tensor. It develops a hierarchy of distinguished classes (weak , weak , weak , weak -K-contact, and weak -Kenmotsu -manifolds) and analyzes their normality, foliation structure, and curvature, including extensive results on Killing fields, totally geodesic foliations, and soliton-type metrics. Key contributions include new characterizations of weak -K-contact structures, deformation to classical -K-contact geometry, and explicit curvature and Ricci identities for compact and warped-product constructions, often leading to -Einstein or Einstein-type outcomes under soliton hypotheses. The work provides obstructions via Adams numbers, integrates Ricci-type soliton theory into the weak framework, and lays groundwork for further geometric and topological applications and extensions beyond constant . Overall, the framework offers a cohesive approach to understanding geometric and topological rigidity, foliations, and curvature phenomena in weak metric -manifolds with potential for broad influence in differential geometry.

Abstract

A weak -structure on a smooth manifold, introduced by the author and R. Wolak (2022), generalizes K. Yano's (1961) -structure. This generalization allows us to revisit classical theory and discover new applications related to Killing vector fields, totally geodesic foliations, Ricci-type solitons, and Einstein-type metrics. This article reviews the results on weak metric -manifolds, where the complex structure on the contact distribution of a metric -structure is replaced with a nonsingular skew-symmetric tensor, and explores its distinguished classes.
Paper Structure (10 sections, 46 theorems, 115 equations)

This paper contains 10 sections, 46 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometry of weak almost ${\cal S}$-manifolds
  4. Geometry of weak ${\cal S}$--manifolds and weak ${\cal C}$-manifolds
  5. Geometry of weak $f$-K-contact manifolds
  6. Weak $f$-K-contact structure equipped with a generalized Ricci soliton
  7. Compact weak $f$-K-contact manifold equipped with an $\eta$-Ricci soliton
  8. Geometry of weak $\beta$-Kenmotsu $f$-manifolds
  9. $\eta$-Ricci solitons on weak $\beta$-Kenmotsu $f$-manifolds
  10. Conclusions and future directions

Key Result

Proposition 1

The normality condition for a weak metric $f$-structure implies Moreover, ${\cal D}^\bot$ is a totally geodesic distribution.

Theorems & Definitions (83)

  • Definition 1
  • Remark 1
  • Remark 2
  • Proposition 1: see rst-43
  • Proposition 2: see rst-43
  • Definition 2
  • Remark 3
  • Proposition 3: see Theorem 2.2 in rst-43
  • Corollary 1
  • Proposition 4
  • ...and 73 more