2019/04/17 by Masahiro Morimoto, Morimoto, Masahiro · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG
paper · pdf · doi:10.48550/arxiv.1904.08328
19 pages. Minor mistakes in the formulas (i) and (ii) of Remark 1 (p.8) in the first version have been corrected
openalex publication_date 2019/04/17 · arxiv created 2019/12/27 · arxiv updated 2019/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A weakly reflective submanifold is a minimal submanifold of a Riemannian manifold which has a certain symmetry at each point. In this paper we introduce this notion into a class of proper Fredholm (PF) submanifolds in Hilbert spaces and show that there exist so many infinite dimensional weakly reflective PF submanifolds in Hilbert spaces. In particular each fiber of the parallel transport map is shown to be weakly reflective. These imply that in infinite dimensional Hilbert spaces there exist so many homogeneous minimal submanifolds which are not totally geodesic, unlike in the finite dimensional Euclidean case.