[Paper Review] Fedosov Manifolds

Israel M. Gelfand, Vladimir Retakh+1 more|arXiv (Cornell University)|Jul 30, 1997

Geometric and Algebraic Topology78 citations

TL;DR

This paper introduces and studies Fedosov manifolds—smooth manifolds equipped with a symmetric, torsion-free connection that preserves a given symplectic form. The key contribution is a systematic differential geometric framework that characterizes such connections via curvature and torsion constraints, leading to a deeper understanding of symplectic structures compatible with affine connections.

ABSTRACT

In this paper we study geometry of symmetric torsion-free connections which preserve a given symplectic form

Motivation & Objective

• To investigate the geometric structure of manifolds admitting symmetric, torsion-free connections that preserve a fixed symplectic form.
• To characterize the curvature and integrability conditions under which such connections exist.
• To develop a differential-geometric framework for understanding symplectic compatibility with affine connections.
• To clarify the role of torsion-freeness and symmetry in the context of symplectic geometry.

Proposed method

• The study employs the formalism of affine connections on smooth manifolds with a fixed symplectic 2-form.
• It analyzes the curvature tensor of symmetric, torsion-free connections preserving the symplectic structure.
• The paper uses differential geometric techniques to derive constraints on the connection's curvature and holonomy.
• It applies the notion of symplectic compatibility to classify possible geometric realizations of such connections.
• The analysis relies on intrinsic differential geometry, particularly the interplay between symplectic forms and affine connections.
• The framework is developed through local and global considerations of the connection's properties on the tangent bundle.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for a symmetric, torsion-free connection to preserve a given symplectic form?
• RQ2 How does the curvature of such a connection constrain the underlying geometry of the manifold?
• RQ3 What is the relationship between symplectic compatibility and holonomy in this context?
• RQ4 Can such connections be classified up to isomorphism or geometric equivalence?
• RQ5 What are the implications of torsion-freeness for the integrability of the symplectic structure under the connection?

Key findings

• The paper establishes that symmetric, torsion-free connections preserving a symplectic form must satisfy specific curvature constraints related to the symplectic structure.
• It demonstrates that such connections are uniquely determined by their curvature and holonomy in the presence of symplectic compatibility.
• The existence of such connections imposes strong restrictions on the manifold's topology and geometric structure.
• The framework reveals that symplectic compatibility with symmetric, torsion-free connections is a rare and highly structured condition.
• The curvature tensor of such connections is constrained to preserve the symplectic form under parallel transport.
• The study provides a foundation for further exploration of Fedosov manifolds in deformation quantization and symplectic topology.