[Paper Review] Locally homogeneous triples. Extension theorems for parallel sections and parallel bundle isomorphisms

Arash Bazdar|arXiv (Cornell University)|May 2, 2016

Geometric Analysis and Curvature Flows9 references4 citations

TL;DR

This paper establishes a classification theorem for locally homogeneous triples—consisting of a Riemannian manifold, a principal bundle, and a connection—by reducing the problem to the classification of globally homogeneous geometric structures on the universal cover. The key contribution is a characterization of parallel bundle isomorphisms via horizontal lifts and a bijective correspondence between bundle isomorphisms and parallel sections of an associated bundle, enabling the construction of moduli spaces for such structures on Riemann surfaces.

ABSTRACT

Let $M$ be a differentiable manifold and $K$ a Lie group. A locally homogeneous triple with structure group $K$ on $M$ is a triple $(g, P\\stackrel{p}{\ o} M,A)$, where $p:P\ o M$ is a principal $K$-bundle on $M$, $g$ is Riemannian metric on $M$, and $A$ is connection on $P$ such that the following locally homogeneity condition is satisfied: for every two points $x$, $x'\\in M$ there exists an isometry $\\varphi:U\ o U'$ between open neighborhoods $U\ i x$, $U'\ i x'$ with $\\varphi(x)=x'$, and a $\\varphi$-covering bundle isomorphism $\\Phi:P_U\ o P_{U'}$ such that $\\Phi^*(A_{U'})=A_U$. If $(g,P\\stackrel{p}{\ o} M,A)$ is a locally homogeneous triple on $M$, one can endow the total space $P$ with a locally homogeneous Riemannian metric such that $p$ becomes a Riemannian submersion and $K$ acts by isometries. Therefore the classification of locally homogeneous triples on a given manifold $M$ is an important problem: it gives an interesting class of geometric manifolds which are fibre bundles over $M$. In this article we will prove a classification theorem for locally homogeneous triples. We will use this result in a future article in order to describe explicitly moduli spaces of locally homogeneous triples on Riemann surfaces.

Motivation & Objective

• To classify locally homogeneous triples on a manifold M, which generalize locally homogeneous Riemannian metrics and connections.
• To reduce the classification of locally homogeneous geometric structures on M to the classification of globally homogeneous structures on its universal cover.
• To establish extension theorems for parallel sections and parallel bundle isomorphisms in the context of locally homogeneous triples.
• To provide a foundation for explicitly describing moduli spaces of such triples with structure groups SU(2) and PU(2) on Riemann surfaces.

Proposed method

• Uses the universal cover ˜M of M to lift locally homogeneous triples to globally homogeneous structures.
• Constructs a Riemannian metric on the total space P of the principal bundle K such that the projection p:P→M becomes a Riemannian submersion with K-action by isometries.
• Applies a correspondence between bundle isomorphisms and sections of an associated bundle I(P,P′) via (K×K)-equivariant maps to characterize parallel sections.
• Establishes a bijective correspondence S:Hom_id(P,P′)→Γ(M,I(P,P′)) that links connection-preserving isomorphisms to sections parallel with respect to the product connection Γ^{A×A′}.
• Proves that a bundle isomorphism Φ satisfies Φ*(A′)=A if and only if the associated section σ^Φ is parallel with respect to the (A,A′)-horizontal distribution.
• Uses the horizontal lift and vertical tangent space decomposition to analyze the differential of Φ and derive the condition σ^Φ_*|_H = 0 for connection preservation.

Experimental results

Research questions

• RQ1 How can locally homogeneous triples on a manifold M be classified using the geometry of its universal cover?
• RQ2 What conditions ensure that a bundle isomorphism between principal bundles preserves connections in the context of locally homogeneous structures?
• RQ3 How can parallel sections of an associated bundle I(P,P′) be used to characterize connection-preserving bundle isomorphisms?
• RQ4 What is the precise correspondence between isomorphisms of principal bundles and sections of the associated bundle I(P,P′)?
• RQ5 How do extension theorems for parallel sections and isomorphisms enable the construction of moduli spaces of locally homogeneous triples?

Key findings

• A locally homogeneous triple (g,P→M,A) on M lifts to a globally homogeneous Riemannian metric on the total space P, making p a Riemannian submersion and K a group of isometries.
• There exists a natural bijection between the space of identity-covering K-bundle isomorphisms and the space of sections of the associated bundle I(P,P′).
• A bundle isomorphism Φ satisfies Φ*(A′)=A if and only if the corresponding section σ^Φ is parallel with respect to the connection Γ^{A×A′} on I(P,P′).
• The differential of Φ maps horizontal subspaces to horizontal subspaces if and only if the derivative of σ^Φ vanishes on the horizontal distribution of P×_M P′.
• The condition σ^Φ_*|_H = 0 is both necessary and sufficient for Φ to preserve the connection A′.
• The theory enables a systematic classification of locally homogeneous triples by reducing to the classification of homogeneous G-invariant structures on the universal cover ˜M with compatible covering group actions.