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

TL;DR

This work characterizes when a compact, intrinsically symmetric submanifold $X\subset E$ is extrinsically symmetric by proving an equivalence: $X$ is extrinsically symmetric iff every maximal torus $T_X$ is a Clifford torus in $E$. It introduces and exploits the notion of Clifford type submanifolds, showing that maximal tori are Clifford tori and constructing a Lie-theoretic framework with reflective symmetries to transfer intrinsic symmetry to extrinsic Clifford structure. A key consequence is a geometric proof that compact extrinsically symmetric spaces yield Clifford tori as maximal tori, linking generating circles, rectangular unit lattices, and the ambient geometry. These ideas are then used to give a geometric proof of Harish-Chandra's theorem on strongly orthogonal roots in semisimple Lie algebras, via extrinsically symmetric orbits and polysphere constructions, thereby establishing a deep bridge between differential geometry and 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.

Theorems & Definitions (7)

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• proof : Proof of Theorem \ref{['THM: Maximal tori of ess are Clifford']}
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