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A Decomposition Theorem for Immersions of Product Manifolds
Published online by Cambridge University Press: 17 April 2015
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.
MSC classification
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- 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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