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Super twisted products

Tong Wu, Yong Wang, Xue Wang

Abstract

In this paper, we define the $W_2$-curvature tensor on super Riemannian manifolds. And we compute the curvature tensor, the Ricci tensor and the $W_2$-curvature tensor on super twisted product spaces. Furthermore, we investigate the $W_2$-curvature flat super twisted product manifolds. And, we get a result that a mixed Ricci-flat super twisted product semi-Riemannian manifold can be expressed as a super warped product semi-Riemannian manifold.

Super twisted products

Abstract

In this paper, we define the -curvature tensor on super Riemannian manifolds. And we compute the curvature tensor, the Ricci tensor and the -curvature tensor on super twisted product spaces. Furthermore, we investigate the -curvature flat super twisted product manifolds. And, we get a result that a mixed Ricci-flat super twisted product semi-Riemannian manifold can be expressed as a super warped product semi-Riemannian manifold.
Paper Structure (4 sections, 8 theorems, 72 equations)

This paper contains 4 sections, 8 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $W_2$-curvature tensor on super twisted products
  4. Mixed Ricci flat super twisted products

Key Result

Theorem 2.7

(Theorem 1 in BG) There is a unique symmetric (torsionless) and metric compatible affine connection $\nabla^L$ on a Riemannian $\mathbb{Z}_2$-manifold $(M, g)$ which satisfies the Koszul formula for all homogeneous $X,Y,Z\in {\rm Vect}(M)$.

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 18 more