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Einstein connection of nonsymmetric pseudo-Riemannian manifold

Vladimir Rovenski, Milan Zlatanović

Abstract

A.~Einstein considered a nonsymmetric (0,2)-tensor $G=g+F$, where $g$ is a pseudo-Riemannian metric and $F\ne0$ is skew-symmetric, and a linear connection $\nabla$ with torsion $T$ such that $(\nabla_X\,G)(Y,Z)=-G(T(X,Y),Z)$. M. Prvanović (1995) obtained the explicit form of the Einstein connection of an almost Hermitian manifold. In this paper, first, we present the result above in coordinate-free form, and then extend it to almost contact metric mani\-folds satisfying the so-called $f^2$-torsion condition. We then derive the Einstein connection of nonsymmetric pseudo-Riemannian, in particular, weak almost Hermitian manifolds $(M,f,g)$, satisfying the $f^2$-torsion condition, where $F(X,Y)=g(X,fY)$, give explicit formulas for the torsion in terms of $\nabla^g F$, $dF$ and a new (1,1)-tensor $\widetilde Q:=-f^2-{\rm Id}$, and show that in the almost Hermitian case, our main result reduces to the coordinate-free form of Prvanović's solution. Finally, we describe special Einstein connections, i.e., the difference tensor has the property~$K_XY=-K_YX$, and indicate the Gray-Hervella classes. Illustrative examples are given, including the construction of a weighted product.

Einstein connection of nonsymmetric pseudo-Riemannian manifold

Abstract

A.~Einstein considered a nonsymmetric (0,2)-tensor , where is a pseudo-Riemannian metric and is skew-symmetric, and a linear connection with torsion such that . M. Prvanović (1995) obtained the explicit form of the Einstein connection of an almost Hermitian manifold. In this paper, first, we present the result above in coordinate-free form, and then extend it to almost contact metric mani\-folds satisfying the so-called -torsion condition. We then derive the Einstein connection of nonsymmetric pseudo-Riemannian, in particular, weak almost Hermitian manifolds , satisfying the -torsion condition, where , give explicit formulas for the torsion in terms of , and a new (1,1)-tensor , and show that in the almost Hermitian case, our main result reduces to the coordinate-free form of Prvanović's solution. Finally, we describe special Einstein connections, i.e., the difference tensor has the property~, and indicate the Gray-Hervella classes. Illustrative examples are given, including the construction of a weighted product.
Paper Structure (6 sections, 16 theorems, 124 equations)

This paper contains 6 sections, 16 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Einstein's nonsymmetric geometry
  3. Explicit form of the Einstein connection of almost Hermitian manifold
  4. Explicit form of the Einstein connection of almost contact metric manifold
  5. Explicit form of the Einstein connection of nonsymmetric pseudo-Riemannian space
  6. The terms $\delta_i(X,Y,Z)$

Key Result

Lemma 2.1

For an Einstein connection $\nabla$, the following conditions are equivalent: (i) $K(X,Y,Z)=-K(X,Z,Y)$, (ii) $\nabla g=0$, (iii) $(\nabla_X F)(Y,Z)=-(\nabla_Y F)(X,Z)$, or $g((\nabla_X f)Z, Y) = -g((\nabla_Y f)Z, X)$.

Theorems & Definitions (39)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Example 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Proposition 2.7: see Proposition 2 of Prvanovic-95
  • ...and 29 more