Partially isometric immersions and free maps |
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Authors: | Giuseppina D’Ambra Roberto De Leo Andrea Loi |
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Institution: | 1.Dipartimento di Matematica,Università di Cagliari,Cagliari,Italy |
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Abstract: | In this paper we investigate the existence of “partially” isometric immersions. These are maps \({f:M\rightarrow \mathbb{R}^q}\) which, for a given Riemannian manifold M, are isometries on some sub-bundle \({\mathcal{H}\subset TM}\). The concept of free maps, which is essential in the Nash–Gromov theory of isometric immersions, is replaced here by that of \({\mathcal{H}}\) –free maps, i.e. maps whose restriction to \({\mathcal{H}}\) is free. We prove, under suitable conditions on the dimension q of the Euclidean space, that \({\mathcal{H}}\) –free maps are generic and we provide, for the smallest possible value of q, explicit expressions for \({\mathcal{H}}\) –free maps in the following three settings: 1–dimensional distributions in \({\mathbb{R}^2}\), Lagrangian distributions of completely integrable systems, Hamiltonian distributions of a particular kind of Poisson Bracket. |
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