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On maximal totally real embeddings

Nefton Pali

Abstract

9We consider complex structures with totally real zero section of the tangent bundle. We assume that the complex structure tensor is real-analytic along the fibers of the tangent bundle. This assumption is quite natural in view of a well known existence result by Bruhat and Whitney. We provide explicit integrability equations for such complex structures in terms of the fiberwise Taylor expansion. In a particular geometric case considered in the literature, we explicit much further the fiberwise Taylor expansion of the complex structure as well as the integrability equations.

On maximal totally real embeddings

Abstract

9We consider complex structures with totally real zero section of the tangent bundle. We assume that the complex structure tensor is real-analytic along the fibers of the tangent bundle. This assumption is quite natural in view of a well known existence result by Bruhat and Whitney. We provide explicit integrability equations for such complex structures in terms of the fiberwise Taylor expansion. In a particular geometric case considered in the literature, we explicit much further the fiberwise Taylor expansion of the complex structure as well as the integrability equations.

Paper Structure

This paper contains 22 sections, 27 theorems, 339 equations.

Table of Contents

  1. Introduction and statement of the main result
  2. $M$-totally real almost complex structures over $T_M$
  3. The integrability equations for $M$-totally real almost complex structures
  4. Application of the main integrability result
  5. General connections over vector bundles
  6. Basic definitions
  7. The geometric meaning of the curvature field
  8. Comparison of the curvature fields of two connections
  9. First reduction of the integrability equations
  10. Second reduction of the integrability equations
  11. Third reduction of the integrability equations and proof of the main theorem
  12. The symplectic approach
  13. General facts
  14. Proof of corollary \ref{['MainCoroll']}
  15. Proof of the proposition \ref{['Vanish']}
  16. ...and 7 more sections

Key Result

Theorem 1

Let $M$ be a smooth manifold and let $J_A$ with $A = \alpha + i T B$ be an $M$-totally real almost complex structure over an open neighborhood $U \subseteq T_M$ of the image of the zero section. Let also $\nabla$ be a covariant derivative operator acting on the smooth sections of $T_M$. Then $J_A$ i for any point $\eta \in U$, where $\nabla^{\operatorname{End} \left( T_M \right), \pi}$ is the cova

Theorems & Definitions (35)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Proposition 1
  • Conjecture 1
  • Corollary 2
  • Definition 3
  • Lemma 1
  • ...and 25 more