SUPREMUMS AND INFIMUMS OF RICCI AND SECTIONAL CURVATURES IN THE GEOMETRY OF COMPACT RIEMANNIAN MANIFOLDS

Abstract

In this paper, we investigate certain numerical characteristics of an $n$-dimensional connected, compact Riemannian manifold $(M,g)$, such as supremums and infimums of its Ricci curvature~$Ric$ and sectional curvature $sec$, and explore their applications. We present two illustrative results. First, if $(M,g)$ is a compact, connected Riemannian manifold of even dimension $n=2k\geq 4$ whose the supremum and infimum of its Ricci and sectional curvatures satisfy the strict inequality $2k\, sec_{\inf}>Ric_{\sup}$, then $M$ is diffeomorphic to either the Euclidean sphere~${\mathbb S}^{2k}$ of some radius $r>0$ or the real projective space~${\mathbb R\mathbb P}^{2k}$. Second, there exists no harmonic immersion of an $n$-dimensional compact, connected Riemannian manifold $(M,g)$  into the Euclidean sphere ${\mathbb S}^m$  of radius $r>0$ if the infimum of its Ricci curvature satisfies the strict inequality $Ric _{\inf}>n/2r^2$.