On the Gauss-Codazzi equations (Q919624)

It is a fundamental problem to find the minimum integer m such that an n- dimensional Riemannian manifold can be isometrically immersed into Euclidean m-space. The Gauss equation gives a useful information for this problem. The author regards an isometric immersion as a solution of a system of first order partial differential equations with respect to a differentiable map of a Riemannian manifold into Euclidean space. Then the author studies relations between the existence of solutions of the Gauss equation and the existence of isometric immersions. Some obstructions for existence of an isometric immersion are constructed and under certain conditions, a non-existence of an isometric immersion of \(PC^ 2\) into \(R^ 7\) is proved.