Curvature groups of a hypersurface

Abstract:

A cochain complex associated with the vector $1$-form determined by the first and second fundamental tensors of a hypersurface $M$ in ${E^{n + 1}}$ is introduced. Its cohomology groups ${H^p}(M)$, called curvature groups, are isomorphic with the cohomology groups of $M$ with coefficients in a subsheaf ${\mathcal {S}_R}$ of the sheaf $\mathcal {S}$ of closed vector fields on $M$. If $M$ is a minimal variety, the same conclusion is valid with ${\mathcal {S}_R}$ replaced by a sheaf of harmonic vector fields. If the Ricci tensor is nondegenerate the ${H^p}(M)$ vanish. If ${\mathcal {S}_R} \ne \emptyset$, and there are no parallel vector fields, locally, the ${H^p}(M)$ are isomorphic with the corresponding de Rham groups.