[Paper Review] Reduction and submanifolds of generalized complex manifolds

Izu Vaisman|ArXiv.org|Nov 1, 2005

Homotopy and Cohomology in Algebraic Topology20 references4 citations

TL;DR

This paper establishes geometric reduction theorems of Marsden-Ratiu and Marsden-Weinstein type for generalized complex, paracomplex, and subtangent structures on differentiable manifolds by leveraging Poisson geometry and Dirac structures. It characterizes submanifolds that inherit induced generalized structures via classical tensor fields, showing that Poisson-Dirac submanifolds with compatible invariant normal bundles yield integrable induced structures.

ABSTRACT

We recall the presentation of the generalized, complex structures by classical tensor fields, while noticing that one has a similar presentation and the same integrability conditions for generalized, paracomplex and subtangent structures. This presentation shows that the generalized, complex, paracomplex and subtangent structures belong to the realm of Poisson geometry. Then, we prove geometric reduction theorems of Marsden-Ratiu and Marsden-Weinstein type for the mentioned generalized structures and give the characterization of the submanifolds that inherit an induced structure via the corresponding classical tensor fields.

Motivation & Objective

• To extend the Marsden-Ratiu and Marsden-Weinstein reduction theorems to generalized complex, paracomplex, and subtangent structures using Poisson geometry.
• To characterize submanifolds that inherit an induced generalized structure from the ambient manifold.
• To clarify the role of Poisson-Dirac submanifolds and invariant normal bundles in the integrability of induced structures.
• To unify the treatment of generalized complex, paracomplex, and subtangent structures through classical tensor fields and Courant brackets.
• To provide a framework for understanding induced generalized structures in terms of Hitchin pairs and Nijenhuis tensors.

Proposed method

• Represents generalized complex, paracomplex, and subtangent structures via classical tensor fields, including Poisson bivectors and Nijenhuis tensors.
• Uses the Courant bracket on the generalized tangent bundle $ T^{\text{big}}M = TM \oplus T^*M $ to define integrability conditions.
• Applies the Marsden-Ratiu reduction framework via a submanifold and control vector bundle to construct reduced generalized structures.
• Applies the Marsden-Weinstein reduction as a special case when the control bundle arises from a Hamiltonian action.
• Characterizes induced structures on submanifolds using pullbacks of Dirac eigenbundles and compatibility with the ambient structure's Courant-Nijenhuis torsion.
• Establishes that integrability of the induced structure follows from the integrability of the ambient structure and the submanifold being Poisson-Dirac and invariant.

Experimental results

Research questions

• RQ1 How can the Marsden-Ratiu and Marsden-Weinstein reduction theorems be generalized to include generalized complex, paracomplex, and subtangent structures?
• RQ2 Which submanifolds of a generalized complex manifold inherit a well-defined induced generalized structure?
• RQ3 What conditions ensure that the induced structure on a submanifold is integrable?
• RQ4 How do Poisson-Dirac submanifolds and invariant normal bundles relate to the existence and integrability of induced generalized structures?
• RQ5 Can the notion of Poisson-Nijenhuis submanifolds be naturally extended to generalized structures using the induced tensor fields?

Key findings

• The generalized, complex, paracomplex, and subtangent structures are unified under a common framework based on classical tensor fields and Poisson geometry.
• The Marsden-Ratiu reduction theorem is generalized to these structures, with the reduced structure inheriting integrability if the original structure and submanifold satisfy appropriate conditions.
• The Marsden-Weinstein reduction is recovered as a special case when the control bundle arises from a Hamiltonian action.
• Submanifolds that inherit an induced generalized structure must be Poisson-Dirac submanifolds, and the induced structure is integrable if the ambient structure is integrable and the submanifold is invariant.
• For non-degenerate generalized c.p.s. structures, the induced structure is integrable if the submanifold is a symplectic submanifold of the underlying Poisson structure.
• The induced generalized structure on a Poisson-Dirac, invariant submanifold is integrable, and the induced Poisson-Nijenhuis hierarchy is compatible with the ambient hierarchy.