Some numerical characteristic inequalities of compact Riemannian manifolds

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/2r^2.