Curvature groups of a hypersurface

Abstract

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