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A Decomposition Theorem for Immersions of Product Manifolds

Part of: Local differential geometry Global differential geometry

Published online by Cambridge University Press:  17 April 2015

Ruy Tojeiro
Ruy Tojeiro*
Affiliation:
Universidade Federal de São Carlos, Via Washington Luiz, Km 235, 13565-905 São Carlos, Brazil (tojeiro@dm.ufscar.br)
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Abstract

We introduce polar metrics on a product manifold, which have product and warped product metrics as special cases. We prove a de Rham-type theorem characterizing Riemannian manifolds that can be locally or globally decomposed as a product manifold endowed with a polar metric. For such a product manifold, our main result gives a complete description of all its isometric immersions into a space form whose second fundamental forms are adapted to its product structure in the sense that the tangent spaces to each factor are preserved by all shape operators. This is a far-reaching generalization of a basic decomposition theorem for isometric immersions of Riemannian products due to Moore as well as of its extension by Nölker to isometric immersions of warped products.

Keywords

product manifoldspartial tubesdecomposition of isometric immersions

MSC classification

Secondary: 53B25: Local submanifolds 53C40: Global submanifolds
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 1 , February 2016 , pp. 247 - 269
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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References

1. Carter, S. and Dursun, U., Partial tubes and Chen submanifolds, J. Geom. 63 (1998), 30–38.CrossRefGoogle Scholar
2. Carter, S. and West, A., Partial tubes about immersed manifolds, Geom. Dedicata 54 (1995), 145–169.Google Scholar
3. Dillen, F. and Nölker, S., Semi-parallelity, multi-rotation surfaces and the helix-property, J. Reine Angew. Math. 435 (1993), 33–63.Google Scholar
4. Hiepko, S., Eine innere Kennzeichnung der verzerrten Produkte, Math. Annalen 241 (1979), 209–215.CrossRefGoogle Scholar
5. Moore, J. D., Isometric immersions of Riemannian products, J. Diff. Geom. 6 (1971), 159–168.Google Scholar
6. Nölker, S., Isometric immersions of warped products, Diff. Geom. Applic. 6 (1996), 31–50.Google Scholar
7. Reckziegel, H. and Schaaf, M., De Rham decomposition of netted manifolds, Results Math. 35 (1999), 175–191.Google Scholar
8. Tojeiro, R., Conformal de Rham decomposition of Riemannian manifolds, Houston J. Math. 32 (2006), 725–743.Google Scholar