## Abstract

The Casorati curvature of a submanifold M^{n} 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 (k^{2} _{1} + • •• + (k^{2} _{n}), whereby k_{1},..,k_{n}are 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 M̃^{n+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.

Original language | English |
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Pages (from-to) | 41-53 |

Number of pages | 13 |

Journal | Note di Matematica |

Volume | 29 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Dec 2009 |

Externally published | Yes |

## Keywords

- Casorati curvature
- Normal curvature
- Principal direction
- Squared length of the second fundamental form.