Abstract:

In 1956 John F. Nash, Jr. proved that a Riemannian manifold can be immersed isometrically into an Euclidean ambient space of dimension sufficiently large. In 1968, S.-S. Chern pointed out that a key technical element in applying Nash's Theorem effectively is finding useful relationships between intrinsic and extrinsic quantities characterizing immersions.  A turning point in the history of this question was an enlightening paper written by B.-Y. Chen in 1993, which paved the way for a deeper understanding of the meaning of the Riemannian inequalities between intrinsic and extrinsic quantities. One important development in the study of such geometric inequalities took place in 2007, with Zhiqin Lu's proof of the DDVV Conjecture. Pursuing this avenue, we present several new results related to the Riemannian study of the geometry of submanifolds.