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On the extrinsic principal directions of Riemannian submanifolds


Abstract


The Casorati curvature of a submanifold M^n of a Riemannianmanifold {\widetilde{M}^{n + m}} is known to be the normalized square of the lengthof the second fundamental form, C = \frac{1}{n}\|h\|^2, i.e., inparticular, for hypersurfaces, C = \frac{1}{n}(k_1^2 + \dots +k_n^2), whereby k_1,\dots,k_n are the principal normalcurvatures of these hypersurfaces. In this paper we in additiondefine the Casorati curvature of a submanifold M^n in aRiemannian manifold {\widetilde{M}^{n + m}} at any point p of M^n in any tangentdirection u of M^n. The principal extrinsic (Casorati)directions of a submanifold at a point are defined as an extensionof the principal directions of a hypersurface M^n at a point in{\widetilde{M}^{n + 1}}. A geometrical interpretation of the Casorati curvature ofM^n in {\widetilde{M}^{n + m}} at p in the direction u 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.

DOI Code: 10.1285/i15900932v29n2p41

Keywords:
Casorati curvature; principal direction; normal curvature; squared length of the second fundamental form

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