A sharp scalar curvature inequality for submanifolds

2024/02/08 by Gururaja, H. A.
#53C40 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2402.05470

Abstract

Let Mⁿ, n≥ 3, be a complete Riemannian manifold of constant scalar curvature R and f: Mⁿ→ M^n+k(c) be an isometric immersion into a space form with flat normal bundle. Assume that f admits a principal normal vector field which has multiplicity n-1 at each point of Mⁿ. Our first result is global and states that (i) R≥ 0 if c=0; (ii) R> (n-1)(n-2)c if c> 0; and (iii) R≥ n(n-1)c if c< 0. These inequalities are optimal. Our second result states that if we further assume that the mean curvature field of f is parallel, then the sectional curvature of Mⁿ is bounded below by c. As a consequence, we classify submanifolds which satisfy the latter condition.

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