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Foldings of star manifolds

Published online by Cambridge University Press:  14 November 2011

H. R. Farran ,
E. El-Kholy  and
S. A. Robertson
H. R. Farran
Affiliation:
Mathematics Department, Kuwait University, P.O. Box 5969, 13060 Safat, Kuwait
E. El-Kholy
Affiliation:
Mathematics Department, Tanta University, Tanta, Egypt
S. A. Robertson
Affiliation:
Faculty of Mathematics, University of Southampton, Southampton SO9 5NH, U.K.
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Synopsis

This paper is a sequel to [4]. Its purpose is to show that the concept of isometric foldings of Riemannian manifolds can be extended to a much wider class of manifolds without losing the main structure theorem. We present here what we believe to be a definitive form of the folding concept for smooth manifolds.

The theory discussed here is based on the idea of a 1-spread [2], where the role of geodesies on a Riemannian manifold is assumed by smooth, unoriented and unparametrised curves on a smooth manifold. The absence of metrical structure forces a fresh approach to the basic definitions. A crucial feature of the Riemannian theory does survive, however, in this general setting: a 1-spread on a sufficiently smooth manifold M induces a 1-spread on sufficiently small spheres surrounding any point of M. With the help of this fact, we are able to construct an inductive definition of “star folding” f:M → N between smooth manifolds M and N, and to retain the theorem that the manifold M is stratified by the “folds”, each of which has the character of a “totally geodesic” submanifold with respect to the above-mentioned curves.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 6 , 1993 , pp. 1011 - 1016
Copyright
Copyright © Royal Society of Edinburgh 1993

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References

1Cartan, E.. Lecons sur la géométrie des espaces de Riemann Paris Gauthier-Villars, 1946.Google Scholar
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5Robertson, S. A.. Isometric folding of Riemannian manifolds, Proc. Roy. Soc. Edinburgh Sect A. 79 (1977), 275–284.Google Scholar