Extrinsically Symmetric Spaces, Submanifolds of Clifford Type and a Theorem of Harish-Chandra

TL;DR

This paper characterizes extrinsically symmetric submanifolds in Euclidean space via Clifford tori and applies this to provide a geometric proof of Harish-Chandra's theorem on strongly orthogonal roots in Lie algebras.

Contribution

It establishes a new characterization of extrinsically symmetric submanifolds using Clifford tori and connects this to a classical Lie algebra result.

Findings

Compact intrinsically symmetric submanifolds are extrinsically symmetric iff their maximal tori are Clifford tori.

Provides a geometric proof of Harish-Chandra's theorem on strongly orthogonal roots.

Links geometric properties of submanifolds to algebraic structures in Lie theory.

Abstract

We prove that a compact, intrinsically symmetric submanifold of a Euclidean space is extrinsically symmetric if and only if its maximal tori are Clifford tori in the ambient space. Moreover, we show that this result can be used to give a geometric proof of a result of Harish-Chandra on strongly orthogonal roots in semisimple Lie algebras.