On the curvature of Riemannian submanifolds of codimension 2
Let M be an n-dimensional Riemannian manifold which is isometrically immersed into some Euclidean space \(E^{n+k}\) with codimension k, and let R denote the curvature tensor of M. Then the Gauss equation implies \[ (*)\quad rank R(x,y)\leq 2k \] for all tangent vectors \(x,y\in T_ pM\), \(p\in M\). The purpose of the paper under review is to determine polynomial relations of the curvature tensor R and to find new conditions on R which may serve as obstructions similar to (*). One of the given applications is: SU(3)/SO(3) cannot be isometrically immersed with codimension 2 in Euclidean space, even locally. The method used is representation theory of GL(n,\({\mathbb{R}})\). Since the required calculations are lengthy and complicated the author makes use of an algebraic programming system in many places of the paper.
scientific article; zbMATH DE number 3915154