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An Abstract Morimoto Theorem for Generalized $F$-structures

Marco Aldi, Daniele Grandini

Abstract

We abstract Morimoto's construction of complex structures on product manifolds to pairs of certain generalized $F$-structures on manifolds that are not necessarily global products. As applications we characterize invariant generalized complex structures on product manifolds in which one factor is a Lie group and we generalize a theorem of Blair, Ludden and Yano on Hermitian bicontact manifolds.

An Abstract Morimoto Theorem for Generalized $F$-structures

Abstract

We abstract Morimoto's construction of complex structures on product manifolds to pairs of certain generalized -structures on manifolds that are not necessarily global products. As applications we characterize invariant generalized complex structures on product manifolds in which one factor is a Lie group and we generalize a theorem of Blair, Ludden and Yano on Hermitian bicontact manifolds.

Paper Structure

This paper contains 10 sections, 23 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Generalized Geometry
  3. Split structures
  4. Normalizers and Normal Pairs
  5. The Abstract Morimoto Theorem
  6. Flat Ehresmann connections
  7. Morimoto products
  8. Products with Lie Groups
  9. Flat principal bundles
  10. Abstract Blair-Ludden-Yano Theorem

Key Result

Lemma 24

Let $J$ be a split generalized CRF-structure on $E\in \mathbb{E}(M)$ and let $\mathbf{u}\in \mathbb{I}(J)$. Then $J(\mathbf{u})\in \mathbb{I}(J)$.

Theorems & Definitions (90)

  • Definition 1
  • Definition 2
  • Remark 3
  • Remark 4
  • Definition 6
  • Definition 7
  • Remark 8
  • Remark 9
  • Definition 10
  • Remark 11
  • ...and 80 more