On the extrinsic principal directions of Riemannian submanifolds

Stefan Haesen, Daniel Kowalczyk, Leopold Verstraelen

Research output: Contribution to journalArticleAcademicpeer-review

31 Citations (Scopus)

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 languageEnglish
Pages (from-to)41-53
Number of pages13
JournalNote di Matematica
Volume29
Issue number2
DOIs
Publication statusPublished - 1 Dec 2009
Externally publishedYes

Keywords

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

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