Casorati curvatures

Abstract

The Casorati curvature of a submanifold Mⁿ of a Riemannian manifold M̃^n+m is known to be the normalized square of the length of the second fundamental form, C = 1/n∥h∥² , i.e., in particular, for hypersurfaces, C = 1/n (k ₁² + • • • + k_n² ), whereby k₁, . . . , k_n are the principal normal curvatures of these hypersurfaces. In this paper we define the Casorati curvature of a submanifold Mn in a Riemannian manifold M̃^n+m at any point p of Mⁿ for any unit tangent vector u of Mⁿ. The principal extrinsic (Casorati) directions of a submanifold at a point are defined as an extension of the principal directions of a hypersurface Mn at a point in M̃ⁿ⁺¹. A geometrical interpretation of the Casorati curvature of Mⁿ in M̃^n+m at p for any unit tangent vector u is given. A characterization of normally flat submanifolds in the Euclidean spaces is given in terms of a relation between the Casorati curvatures and the normal curvatures of these submanifolds.