Some new obstructions to minimal and Lagrangian isometric immersions

Abstract

1. IntroductionThe main purpose of this paper is to introduce a new type of Riemannian curvature invariants and to show that these new invariants have interesting applications to several areas of mathematics; in particular, they provide new obstructions to minimal and Lagrangian isometric immersions. Moreover, these new invariants enable us to introduce and to study the notion of ideal immersions.One of the most fundamental problems in the theory of submanifolds is the immersibility (or non-immersibility) of a Riemannian manifold in a Euclidean space (or more generally, in a space form). According to a well-known theorem of J. F. Nash, every Riemannian manifold can be isometrically immersed in some Euclidean spaces with sufficiently high codimension.In order to study this fundamental problem, in view of Nash's theorem, it is natural to impose a suitable condition on the immersions. For instance, if one imposes the minimality condition on the immersions, it leads toPROBLEM 1. Given a Riemannian manifold M, what are the necessary conditions for M to admit a minimal isometric immersion in a Euclidean m-space Em?It is well-known that for a minimal submanifold in Em, the Ricci tensor satisfies Ric_??_0. For many years this was the only known general necessary Riemannian condition for a Riemannian manifold to admit a minimal isometric immersion in a Euclidean space.The main results of this article were presented at the 3rd Pacific Rim Geometry Conference held at Seoul, Korea in December 1996; also presented at the 922nd AMS meeting held at Detroit, Michigan in May 1997.