On the Structure of Submanifolds with Degenerate Gauss Maps

Document Type

Article

Publication Date

12-1-2001

Abstract

An n-dimensional submanifold X of a projective space PN(C) is called tangentially degenerate if the rank of its Gauss mapping γ: X → G(n, N) satisfies 0 < rank γ < n. The authors systematically study the geometry of tangentially degenerate submanifolds of a projective space PN(C). By means of the focal images, three basic types of submanifolds are discovered: cones, tangentially degenerate hypersurfaces, and torsal submanifolds. Moreover, for tangentially degenerate submanifolds, a structural theorem is proven. By this theorem, tangentially degenerate submanifolds that do not belong to one of the basic types are foliated into submanifolds of basic types. In the proof the authors introduce irreducible, reducible, and completely reducible tangentially degenerate submanifolds. It is found that cones and tangentially degenerate hypersurfaces are irreducible, and torsal submanifolds are completely reducible while all other tangentially degenerate submanifolds not belonging to basic types are reducible.

Identifier

0007581286 (Scopus)

Publication Title

Geometriae Dedicata

External Full Text Location

https://doi.org/10.1023/A:1011908210052

ISSN

00465755

First Page

205

Last Page

226

Issue

1-3

Volume

86

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