Abstract
The Casorati curvature of a submanifold Mn of a Riemannian manifold M̃n+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,..,knare 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 M̃n+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 M̃n+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.
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.