Compatibility equations for isometric embeddings of Riemannian manifolds (Q1328234)

This paper considers local isometric immersions of Riemannian manifolds \(M\) into Euclidean spaces. Through a prolongation of the underlying PDE it constructs compatibility equations by a method due to A. Finzi, and discusses some of their geometric consequences. Some specific results: If a \(C^ 2\)-isometric embedding of \(M\) is \(C^ \infty\) except at a hypersurface \(H\) of \(M\), then \(H\) must be asymptotic (Corollary 11). Any two isometric embeddings contained in a \(C^ 3\)-neighborhood of an infinitesimally rigid elliptic (in the sense of N. Tanaka) isometric embedding are congruent (Theorem 13). Any two real analytic elliptic isometric embeddings of \(M\) coinciding along a hypersurface \(H\) of \(M\) must coincide on a neighborhood of \(H\) (Theorem 18).