Riemannian DNA, Inequalities and Their Applications

The main purpose of this survey article is to present the new type of Riemannian curvature invariants (Riemannian DNA) and the sharp inequalities, involving these invariants and the squared mean curvature, originally introduced and established in [7,8]. These Riemannian DNA affect the behavior in general of the Riemannian manifold and they have several interesting connections to several areas of mathematics. For instance, they give rise to new obstructions to minimal and Lagrangian isometric immersions. Moreover, these invariants relate closely to the first nonzero eigenvalue of the Laplacian on a Riemannian manifold. These invariants together with the sharp inequalities gives rise naturally to the notion of “ideal immersions” or the notion of “the best ways of living”. We also explain the physical meaning of the notion of ideal immersions for Riemannian manifolds in a Riemannian space form based again on the sharp inequalities.