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Normal and osculating maps for submanifolds of RN

Published online by Cambridge University Press:  14 November 2011

I. Cattaneo Gasparini  and
G. Romani
I. Cattaneo Gasparini
Affiliation:
Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Via A. Scarpa 10, 00161 Rome, Italy
G. Romani
Affiliation:
Dipartimento di Matematica “G. Castelnuovo”, P.1-Aldo Moro 5, 00185 Rome, Italy
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Synopsis

Let Mn be a manifold supposed “nicely curved” isometrically immersed in ℝn+p. Starting from a generalised Gauss map associated to the splitting of the normal bundle defined from the values of the fundamental forms of M of order k (k ≧ 0), we give necessary and sufficient conditions for the map to be totally geodesic and harmonic . For k = 0 is the classical Gauss map and our formula reduces to Ruh–Vilm's formula with a more precise formulation due to the consideration of the splitting of the normal bundle.

We also give necessary conditions for M, supposed complete, to admit an isometric immersion with . This theorem generalises a theorem of Vilms on the manifolds with second fundamental forms parallel (case k = 0). The result is interesting as the class of manifolds satisfying the condition is larger than the class of manifolds satisfying .

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 114 , Issue 1-2 , 1990 , pp. 39 - 55
Copyright
Copyright © Royal Society of Edinburgh 1990

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