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 of$M^n$ 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.