Equivariant Embeddings of K\"alerian Symmetric Spaces

TL;DR

This paper studies equivariant embeddings of symmetric Kählerian manifolds, characterizing them in the case of complex projective spaces and identifying conditions for extrinsic symmetry based on curvature properties.

Contribution

It provides a characterization of equivariant embeddings of symmetric Kählerian manifolds, especially for ale9rian projective spaces, and links curvature conditions to extrinsic symmetry.

Findings

Characterization of equivariant embeddings of ale9rian symmetric spaces.

Identification of conditions under which embeddings are extrinsically symmetric.

Verification that parallel plurimean curvature implies extrinsic symmetry.

Abstract

In this article we investigate some properties of equivariant embeddings of a symmetric K\"ahlerian manifold. Motivated by a theorem of Cartan and Wallach on equivariant embeddings of symmetric spaces we characterize these embeddings in the special case of $\mathbb{CP}^n$. Further, we verify that if a equivariant embedding has parallel plurimean curvature then it is the extrinsically symmetric one.