La Grassmannienne non-lin\'eaire comme vari\'et\'e fr\'ech\'etique homog\`ene

TL;DR

This paper studies the non-linear Grassmannian of submanifolds within a compact Riemannian manifold, establishing its structure as a smooth Fréchet manifold and exploring its geometric properties and symmetries.

Contribution

It constructs a smooth Fréchet manifold structure on the non-linear Grassmannian and analyzes its geometric features and the action of the diffeomorphism group.

Findings

The space of smooth embeddings forms a principal fiber bundle over the Grassmannian.

Connected components of the Grassmannian are homogeneous under diffeomorphisms.

The Grassmannian has a natural smooth Fréchet manifold structure.

Abstract

Let (M,g) be a compact Riemannian manifold of dimension n. For k \in {0,...,n}, we denote Gr_{k}(M) the set of compact, connected and oriented submanifolds of M of dimension k. This set is called the non-linear Grassmannian. In this article, we endow Gr_{k}(M) with a smooth Fr\'echet manifold structure and investigate its basic geometrical properties. In particular, if \Sigma \in Gr_{k}(M), we show that the space of smooth embeddings Emb(\Sigma,M) is the total space of principal fiber bundle with base space a collection of connected components of Gr_{k}(M). We also show that the connected components of Gr_{k}(M) are homogeneous with respect to the natural action of the group of diffeomorphisms of M.