2024/06/17 by Christos-Raent Onti, Onti, C. -R., Kleanthis Polymerakis +3
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2406.11692
openalex publication_date 2024/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate specific intrinsic curvatures ρk (where 1≤ k≤ n) that interpolate between the minimum Ricci curvature ρ1 and the normalized scalar curvature ρn=ρ of n-dimensional Riemannian manifolds. For n-dimensional submanifolds in space forms, these curvatures satisfy an inequality involving the mean curvature H and the normal scalar curvature ρ^⊥, which reduces to the well-known DDVV inequality when k=n. We derive topological obstructions for compact n-dimensional submanifolds based on universal lower bounds of the Ln/2-norms of certain functions involving ρk,H and ρ^⊥. These obstructions are expressed in terms of the Betti numbers. Our main result applies for any 1≤ k ≤ n-1, but it generally fails for k=n, where the involved norm vanishes precisely for Wintgen ideal submanifolds. We demonstrate this by providing a method of constructing new compact 3-dimensional minimal Wintgen ideal submanifolds in even-dimensional spheres. Specifically, we prove that such submanifolds exist in \mathbbS6 with arbitrarily large first Betti number.