Isometries of extrinsic symmetric spaces

Abstract

We show that every isometry of an extrinsic symmetric space extends to an isometry of its ambient euclidean space. As a consequence, any isometry of a real form of a hermitian symmetric space extends to a holomorphic isometry of the ambient hermitian symmetric space. Moreover, every fixed point component of an isometry of a symmetric R-space is a symmetric R-space itself.

Equivariant embeddings of symmetric spaces

“…We show first that a compact symmetric space X without polars is a Riemannian product of a simply connected symmetric space and possibly a torus [8, Lemma 8]. Then by [8,Thm. 3] the torus part of X is rectangular (a Riemannian product of circles) once X is extrinsic symmetric.…”

“…But her investigations opened us the door to new research on this interesting class of symmetric submanifolds, minimally embedded in the sphere. (Joined work with E. Heintze, P. Quast [7], M.S. Tanaka [8].…”

“…Loos' proof [13] used an algebraic structure called Jordan triple systems, and extended computations were needed to verify the defining identities. In [8,7] we obtained a new proof which is less computational and includes an anwer to Question (2). We will give a sketch in the following two sections.…”

“…3 Dimension reduction using meridians "⇒" [8] Let X ⊂ V be extrinsic symmetric. The main idea is to reduce dim X while keeping a maximal torus T X .…”

Circles in self dual symmetric R-spaces

“…It is well-known that any maximal compact subgroup K of G (they are all conjugate with each other) acts transitively on M and there is a unique (up to homotheties) Riemannian metric g on M such the action of K is by isometries of g. By abuse of nomenclature we refer to it as the K-invariant Riemannian metric on M. Moreover, (M, g) is a compact, irreducible Riemannian symmetric space with cubic maximal tori (see [13], and also [3] for a geometrical proof of the rectangularity of maximal tori).…”