# Isometry groups of generalized Stiefel manifolds

## TL;DR

This paper explicitly computes the isometry groups of generalized Stiefel manifolds, which are manifolds of orthonormal frames in spaces with non-degenerate bilinear or hermitian forms, up to connected components.

## Contribution

It provides an explicit description of the isometry groups of generalized Stiefel manifolds using automorphisms of an associated non-associative algebra.

## Key findings

- Explicit form of isometry groups up to connected components
- Automorphism groups of the associated algebra are computed
- Method applies to manifolds with bilinear or hermitian forms

## Abstract

A generalized Stiefel manifold is the manifold of orthonormal frames in a vector space with a non-degenerated bilinear or hermitian form. In this article, the Isometry group of the generalized Stiefel manifolds are computed at least up to connected components in an explicit form. This is done by considering a natural non-associative algebra $\mathfrak{m}$ associated to the affine structure of the Stiefel manifold, computing explicitly the automorphism group $Aut(\mathfrak{m})$ and inducing this computation to the Isometry group.