[Paper Review] Geometrical structures on the prolongation of a pre-Lie algebroid on fibered manifolds and application to Partial Finsler geometry on foliated anchored bundle

Fernand Pelletier|arXiv (Cornell University)|Dec 21, 2014

Advanced Differential Geometry Research18 references3 citations

TL;DR

This paper extends geometrical structures—such as nonlinear connections, Finsler connections, and curvature—on the prolongation of a pre-Lie algebroid over a fibered manifold, demonstrating that these structures on a foliated anchored bundle depend only on the foliation, not on the specific pre-Lie algebroid bracket. The key result is that a Finsler connection on the total space induces the classical Finsler and Chern connections on each leaf of the foliation.

ABSTRACT

A pre-Lie algebroid is an anchored bundle provided with an almost Lie bracket such that the anchor is compatible with the Lie bracket of vector fields. We firstly show how most geometrical structures intensively studied in the framework of Lie algebroid can easily be extended in the pre-Lie algebroid context. The principal purpose of this paper is to show that how all these results only depend of the foliated structure and do not depend of the pre-Lie algebroid structure that we can put on a foliated anchored bundle. As application, we obtain a Finsler connection on a foliated anchored bundle which induces the classical Finsler connection on each leaf and a similar result for the Chern connection.

Motivation & Objective

• To generalize geometrical structures from Lie algebroids to pre-Lie algebroids on fibered manifolds.
• To show that Finsler and Chern connections on a foliated anchored bundle are induced by global structures independent of the pre-Lie algebroid bracket.
• To establish a link between curvature in the total space and curvature on individual leaves of the foliation.
• To define and analyze flag curvature in the context of partial Finsler algebroids and their induced structures on leaves.

Proposed method

• Prolonging an anchored bundle over a fibered manifold to construct a new vector bundle with an induced anchor and almost Lie bracket.
• Defining almost tangent and almost cotangent structures on the prolongation to model nonlinear connections.
• Using semisprays to generate canonical nonlinear connections compatible with symplectic and metric structures.
• Constructing a Lagrangian metric connection via the dynamical derivation and canonical semispray, ensuring compatibility with the symplectic and metric forms.
• Defining partial Finsler structures on the total space and inducing Finsler and Chern connections on each leaf of the foliation.
• Deriving the flag curvature of the partial Finsler algebroid and relating it to the curvature on individual leaves via projection.

Experimental results

Research questions

• RQ1 How can geometrical structures like nonlinear connections and Finsler connections be generalized from Lie algebroids to pre-Lie algebroids?
• RQ2 To what extent do the induced Finsler and Chern connections on a foliated anchored bundle depend on the choice of pre-Lie algebroid bracket?
• RQ3 How is the curvature of the partial Finsler algebroid related to the curvature of the induced Finsler structure on each leaf?
• RQ4 What is the relationship between the flag curvature in the total space and the flag curvature on a leaf of the foliation?
• RQ5 Can a global Finsler connection on the total space induce the classical Finsler and Chern connections on each leaf?

Key findings

• The Finsler and Chern connections on the total space of a foliated anchored bundle induce the classical Finsler and Chern connections on each leaf of the foliation.
• The induced connections on each leaf are independent of the choice of pre-Lie algebroid bracket, depending only on the foliated structure.
• The Riemannian curvature $ R $ and Minkowski curvature $ P $ of the partial Finsler algebroid project to the corresponding curvatures $ R_N $ and $ P_N $ on each leaf $ N $.
• The flag curvature $ K_N(ar{u}, ar{ u}) $ on a leaf $ N $ equals the flag curvature $ K(u, u) $ in the total space, where $ ho_N(u) = ar{u} $ and $ ho_N( u) = ar{ u} $.
• The curvature of the linear connection $ abla^U $ on a section $ U $ of $ ilde{ ho} $ satisfies $ ilde{ ho}(R( ilde{X}, ilde{Y})(x, U(x))) = R^U(X,Y)(x) $, linking total space and leaf curvatures.
• The canonical nonlinear connection associated with a regular Lagrangian $ ilde{ ho} $ is compatible with both the symplectic and metric structures, ensuring uniqueness.