Comparative structure of Riemannian manifold and spin manifold

Abstract

We establish here connection between curvature of a generalized cylinder with geometric data on M with spinor metric structure by comparing the Dirac operators for two different metrics based on identification and existence of semi-Riemannian metric. Specific objective is to investigate properties of spinors on a manifold foliated by semi-Riemannian hypersurfaces using commutator expansion and its normal derivative. We derive algebraic properties of semi-Riemannian manifold initiated by H. Baunn by taking non-degenerate symmetric bilinear form. The two semi-Riemannian metrics on a manifold cannot always be joined by a continuous path of metrics even if they have the same signature. we show here that for a Codazzi tensor, the manifold can be embedded as a hypersurface into a Ricci flat manifold equipped with a parallel spinor which generalizes the case of Killing spinors. The classification of manifolds admitting Killing spinors that the cone over such a manifold possesses a parallel spinor.