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- Title
Holomorphic Retractions of Bounded Symmetric Domains onto Totally Geodesic Complex Submanifolds.
- Authors
Mok, Ngaiming
- Abstract
Given a bounded symmetric domain Ω the author considers the geometry of its totally geodesic complex submanifolds S ⊂ Ω. In terms of the Harish-Chandra realization Ω ⋐ ℂn and taking S to pass through the origin 0 ∈ Ω, so that S = E ⋂ Ω for some complex vector subspace of ℂn, the author shows that the orthogonal projection ρ: Ω → E maps Ω onto S, and deduces that S ⊂ Ω is a holomorphic isometry with respect to the Carathéodory metric. His first theorem gives a new derivation of a result of Yeung's deduced from the classification theory by Satake and Ihara in the special case of totally geodesic complex submanifolds of rank 1 and of complex dimension ≥ 2 in the Siegel upper half plane ℋ g , a result which was crucial for proving the nonexistence of totally geodesic complex suborbifolds of dimension ≥ 2 on the open Torelli locus of the Siegel modular variety A g by the same author. The proof relies on the characterization of totally geodesic submanifolds of Riemannian symmetric spaces in terms of Lie triple systems and a variant of the Hermann Convexity Theorem giving a new characterization of the Harish-Chandra realization in terms of bisectional curvatures.
- Subjects
SYMMETRIC domains; SUBMANIFOLDS; GEODESICS; SYMMETRIC spaces; RIEMANNIAN manifolds; ORTHOGRAPHIC projection
- Publication
Chinese Annals of Mathematics, 2022, Vol 43, Issue 6, p1125
- ISSN
0252-9599
- Publication type
Article
- DOI
10.1007/s11401-022-0380-z