Embeddings from noncompact symmetric spaces to their compact duals

Every compact symmetric space $M$ admits a dual noncompact symmetric space $\breve{M}$. When $M$ is a generalized Grassmannian, we can view $\breve{M}$ as an open submanifold of it, consisting of space-like subspaces [4]. Motivated from this, we study the embeddings from noncompact symmetric spaces to their compact duals, including space-like embedding for generalized Grassmannians, Borel embedding for Hermitian symmetric spaces and the generalized embedding for symmetric $\mathrm{R}$‑spaces. We will compare these embeddings and describe their images using cut loci.

Keywords

symmetric space, Grassmannian, embedding

2010 Mathematics Subject Classification

53A40, 53C35

Full Text (PDF format)

We are grateful to J. J. Zhang for many useful comments and discussions. The first author is sponsored by Natural Science Foundation of Shanghai (No. 19ZR1411700). The second author is supported by National Natural Science Foundation of China (No. 11771455). The work of Leung described in this paper was substantially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 14301117) and a direct grant from The Chinese University of Hong Kong (Project No. 4053161).

Received 5 February 2018

Accepted 29 January 2020

Published 10 March 2021