2022/02/25 by Ricardo A. E. Mendes, Mendes, Ricardo A. E.
Mathematics · Medicine · #Advanced Algebra and Geometry #Homotopy and Cohomology in Algebraic Topology #Ophthalmology and Eye Disorders
paper · pdf · doi:10.48550/arxiv.2202.12966
Given a compact Lie group G and an orthogonal G-representation V, we give a purely metric criterion for a closed subset of the orbit space V/G to have convex pre-image in V. In fact, this also holds with the natural quotient map V→ V/G replaced with an arbitrary submetry V→ X. In this context, we introduce a notion of "fat section" which generalizes polar representations, representations of non-trivial copolarity, and isoparametric foliations. We show that Kostant's Convexity Theorem partially generalizes from polar representations to submetries with a fat section, and give examples illustrating that it does not fully generalize to this situation.