Principal bundles of embeddings and nonlinear Grassmannians

TL;DR

This paper explores principal bundles of embeddings of compact manifolds with bases as nonlinear Grassmannians, analyzing their infinite-dimensional differential structures in the context of geometric mechanics and dual pairs.

Contribution

It introduces principal bundles of embeddings with nonlinear Grassmannian bases and studies their infinite-dimensional Fréchet manifold structures, linking to geometric mechanics.

Findings

Established principal bundle structures of embeddings over nonlinear Grassmannians.

Analyzed the infinite-dimensional differential structure in the Fréchet category.

Connected the mathematical structures to applications in continuum mechanics.

Abstract

We present several principal bundles of embeddings of compact manifolds (with or without boundary) whose base manifolds are nonlinear Grassmannians. We study their infinite dimensional differential manifold structure in the Fr\'echet category. This study is motivated by the occurrence of such objects in the geometric Lagrangian formulation of free boundary continuum mechanics and in the study of the associated infinite dimensional dual pairs structures.