# On weakly reflective PF submanifolds in Hilbert spaces

## TL;DR

This paper introduces the concept of weakly reflective submanifolds into infinite-dimensional Hilbert spaces, revealing numerous examples of homogeneous minimal submanifolds with symmetry properties that differ from finite-dimensional cases.

## Contribution

It extends the notion of weakly reflective submanifolds to proper Fredholm submanifolds in Hilbert spaces and demonstrates their abundance, including fibers of the parallel transport map.

## Key findings

- Existence of many infinite-dimensional weakly reflective PF submanifolds in Hilbert spaces.
- Each fiber of the parallel transport map is weakly reflective.
- Presence of numerous homogeneous minimal submanifolds that are not totally geodesic.

## Abstract

A weakly reflective submanifold is a minimal submanifold of a Riemannian manifold which has a certain symmetry at each point. In this paper we introduce this notion into a class of proper Fredholm (PF) submanifolds in Hilbert spaces and show that there exist so many infinite dimensional weakly reflective PF submanifolds in Hilbert spaces. In particular each fiber of the parallel transport map is shown to be weakly reflective. These imply that in infinite dimensional Hilbert spaces there exist so many homogeneous minimal submanifolds which are not totally geodesic, unlike in the finite dimensional Euclidean case.