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Einstein-type metrics and generalized Ricci solitons on weak $f$-K-contact manifolds

Vladimir Rovenski

Abstract

A weak metric $f$-structure $(f,Q,ξ_i,η^i,g)\ (i=1,\ldots,s)$, generalizes the metric $f$-structure on a smooth manifold, i.e., the complex structure on the contact distribution is replaced with a nonsingular skew-symmetric tensor. We study geometry of a weak $f$-K-contact structure, which is a weak $f$-contact structure, whose characteristic vector fields are Killing. We show that $\ker f$ of a weak $f$-contact manifold defines a $\mathfrak{g}$-foliation with an abelian Lie algebra. Then we characterize weak $f$-K-contact manifolds among all weak metric $f$-manifolds by the property known for $f$-K-contact manifolds, and find when a Riemannian manifold endowed with a set of orthonormal Killing vector fields is a weak $f$-K-contact manifold. We show that for $s>1$, an Einstein weak $f$-K-contact manifold is Ricci flat, then find sufficient conditions for a weak $f$-K-contact manifold with parallel Ricci tensor or with a generalized gradient Ricci soliton structure to be Ricci flat or a quasi Einstein manifold. We prove positive definiteness of the Jacobi operators in the characteristic directions and use this to deform a weak $f$-K-contact structure to an $f$-K-contact structure. We define an $η$-Ricci soliton and $η$-Einstein structures on a weak metric $f$-manifold (which for $s=1$, give the well-known structures on contact metric manifolds) and find sufficient conditions for a compact weak $f$-K-contact manifold with an $η$-Ricci soliton structure of constant scalar curvature to be $η$-Einstein.

Einstein-type metrics and generalized Ricci solitons on weak $f$-K-contact manifolds

Abstract

A weak metric -structure , generalizes the metric -structure on a smooth manifold, i.e., the complex structure on the contact distribution is replaced with a nonsingular skew-symmetric tensor. We study geometry of a weak -K-contact structure, which is a weak -contact structure, whose characteristic vector fields are Killing. We show that of a weak -contact manifold defines a -foliation with an abelian Lie algebra. Then we characterize weak -K-contact manifolds among all weak metric -manifolds by the property known for -K-contact manifolds, and find when a Riemannian manifold endowed with a set of orthonormal Killing vector fields is a weak -K-contact manifold. We show that for , an Einstein weak -K-contact manifold is Ricci flat, then find sufficient conditions for a weak -K-contact manifold with parallel Ricci tensor or with a generalized gradient Ricci soliton structure to be Ricci flat or a quasi Einstein manifold. We prove positive definiteness of the Jacobi operators in the characteristic directions and use this to deform a weak -K-contact structure to an -K-contact structure. We define an -Ricci soliton and -Einstein structures on a weak metric -manifold (which for , give the well-known structures on contact metric manifolds) and find sufficient conditions for a compact weak -K-contact manifold with an -Ricci soliton structure of constant scalar curvature to be -Einstein.
Paper Structure (8 sections, 28 theorems, 96 equations)

This paper contains 8 sections, 28 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weak $f$-contact and weak $f$-K-contact structures
  4. Killing vector fields of unit length
  5. The curvature in the characteristic direction
  6. Weak $f$-K-contact structure as a generalized Ricci soliton
  7. $\eta$-Ricci solitons on compact weak $f$-K-contact manifolds
  8. Conclusion

Key Result

Proposition 3.1

For a weak $f$-contact structure, the tensors $N^{\,(2)}_i$ and $N^{\,(4)}_{ij}$ vanish; moreover, $N^{\,(3)}_i$ vanishes if and only if $\,\xi_i$ is a Killing vector field. In particular, for a weak $f$-K-contact structure, the tensors $N^{\,(2)}_i$, $N^{\,(3)}_i$ and $N^{\,(4)}_{ij}$ vanish.

Theorems & Definitions (54)

  • Remark 2.1: see r-23
  • Remark 2.2
  • Definition 3.1
  • Remark 3.1
  • Proposition 3.1: see Theorem 2.2 in rst-43
  • Proposition 3.2: see Corollary 2.1 in rst-43
  • Corollary 3.1
  • proof
  • Remark 3.2
  • Definition 3.2: see rst-43
  • ...and 44 more