Some numerical characteristic inequalities of compact Riemannian manifolds

2025/04/15 by Stepanov, Sergey, Tsyganok, Irina
#53C20 #53C43 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.11041

Abstract

In this paper, we consider numerical characteristics of the connected compact Riemannian manifold (M, g) such as the supremum and infimum of the scalar curvature s, Ricci curvature Ric and sectional curvature sec, as well as their applications. Below are two examples of proven results. The first statement: If (M, g) be a connected, compact Riemannian manifold of even dimension n > 3 whose Ricci and sectional curvatures satisfy the strict inequality n Inf (sec) > Sup (Ric), then M is diffeomorphic to the Euclidean n-dimensional sphere of some radius r or the real projective n-dimensional space. The second statement: There is no harmonic immersion of an n-dimensional connected, complete Riemannian manifold (M, g) into the Euclidean n-sphere of radius r if there exists inf(Ric) such that Inf (Ric) > n/2r².

Citations

• On the theory of minimal submanifolds and harmonic maps from the point of view of the generalized Bochner technique
• Curvature of the Second kind and a conjecture of Nishikawa
• On higher order Codazzi tensors on complete Riemannian manifolds
• Riemannian Manifolds with positive sectional curvature
• Curvature, sphere theorems, and the Ricci flow
• Sphere Theorems in Geometry
• A general convergence result for the Ricci flow in higher dimensions
• A general convergence result for the Ricci flow in higher dimensions
• Another Report on Harmonic Maps
• A Report on Harmonic Maps
• Vanishing and Finiteness Results in Geometric Analysis
• Three-manifolds with positive Ricci curvature
• Harmonic Mappings of Riemannian Manifolds

Related

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• Some sharp differential sphere theorems for nonnegative scalar curvature manifolds
• The Boundary between Compact and Noncompact Complete Riemann Manifolds
• Compact harmonic RCD(K, N) spaces are harmonic manifolds
• The sphere theorems for manifolds with positive scalar curvature