Abstract
The Casorati curvature of a submanifold of a Riemannianmanifold is known to be the normalized square of the lengthof the second fundamental form, , i.e., inparticular, for hypersurfaces, , whereby are the principal normalcurvatures of these hypersurfaces. In this paper we in additiondefine the Casorati curvature of a submanifold in aRiemannian manifold at any point of in any tangentdirection of . The principal extrinsic (Casorati)directions of a submanifold at a point are defined as an extensionof the principal directions of a hypersurface at a point in . A geometrical interpretation of the Casorati curvature of in at in the direction is given. Acharacterization of normally flat submanifolds in Euclidean spacesis given in terms of a relation between the Casorati curvaturesand the normal curvatures of these submanifolds.
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