Mathematics Colloquium: AMS: Applications of Nash's Embedding Theorem: Chen Invariants and New Inequalities for Curvature
4:00pm -- CUE 114
After Nash's Embedding Theorem was proved in the 1950s, there was some hope that if Riemannian manifolds could be regarded as Riemannian submanifolds of an Euclidean space, this will create a technical opportunity to use extrinsic help to better understand the manifolds' geometric properties. However, this hope had not been fully materialized, at least not in the first decades. The main reason for this was the lack of controls of the extrinsic properties of the submanifolds by the known intrinsic invariants. To overcome such a difficulty, as well as to provide answers to an open question on minimal immersions first suggested by S.-S. Chern in 1968, B.-Y. Chen introduced in the early 1990's a new class of Riemannian invariants. Since then, many results concerning these invariants, today known as Chen's curvature invariants, have been obtained by many geometers (see e.g. https://doi.org/10.1142/8003 ). We will explore the history of these ideas, their geometric motivation and present several recent results. An interesting question is whether there are other ways of using Nash's Embedding Theorem: this is an open quest.