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On Products of Generalized Geometries

Ralph R. Gomez, Janet Talvacchia

Abstract

In this paper we address what generalized geometric structures are possible on products of spaces that each admit generalized geometries. In particular we consider, first, the product of two odd dimensional spaces that each admit a generalized almost contact structure, and then subsequently, the product of an odd dimensional space that admits a generalized almost contact structure and an even dimensional space that admits a generalized almost complex structure. We also draw attention to the relationship of the Courant bracket to the classical notion of normality for almost contact structures.

On Products of Generalized Geometries

Abstract

In this paper we address what generalized geometric structures are possible on products of spaces that each admit generalized geometries. In particular we consider, first, the product of two odd dimensional spaces that each admit a generalized almost contact structure, and then subsequently, the product of an odd dimensional space that admits a generalized almost contact structure and an even dimensional space that admits a generalized almost complex structure. We also draw attention to the relationship of the Courant bracket to the classical notion of normality for almost contact structures.

Paper Structure

This paper contains 4 sections, 6 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Background on Generalized Complex Structures and Generalized Contact Structures
  3. Products of Manifolds Admitting Generalized Structures
  4. Normal Almost Contact Structures and a Geometric Interpretation of $[[E_+ ,E_- ]] = 0$.

Key Result

Theorem 1

Let $M_1$ and $M_2$ be odd dimensional smooth manifolds each with a generalized almost contact structures $(\Phi_i, E_{+,i}, E_{-,i}) ,\ \ i=1,2$. Then $M_1\times M_2$ admits a generalized almost complex structure $\mathcal{J}$. Further $\mathcal{J}$ is a generalized complex structure if and only

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Theorem \ref{T1}
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 4 more