On the extrinsic principal directions of Riemannian submanifolds

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 {norm of matrix}h{norm of matrix}² ,, i.e., in particular, for hypersurfaces, C = 1/n (k² ₁ + • •• + (k² _n), whereby k₁,..,k_nare the principal normal curvatures of these hypersurfaces. In this paper we in addition define the Casorati curvature of a submanifold M;ⁿ in a Riemannian manifold M̃^n+m at any point p of M;ⁿ in any tangent direction u of M;ⁿ. The principal extrinsic (Casorati) directions of a submanifold at a point are defined as an extension of the principal directions of a hypersurface Mn;ⁿ at a point in M̃^n+m. A geometrical interpretation of the Casorati curvature of M;ⁿ in M̃ⁿ⁺¹ at p in the direction u is given. A characterization of normally flat submanifolds in Euclidean spaces is given in terms of a relation between the Casorati curvatures and the normal curvatures of these submanifolds.