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Transactions of the American Mathematical Society

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Induced connections on $ S\sp 1$-bundles over Riemannian manifolds


Author: G. D’Ambra
Journal: Trans. Amer. Math. Soc. 338 (1993), 783-798
MSC: Primary 53C05; Secondary 58D15, 58E99
DOI: https://doi.org/10.1090/S0002-9947-1993-1106187-0
MathSciNet review: 1106187
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Abstract: Let $ (V,g)$ and $ (W,h)$ be Riemannian manifolds and consider two $ {S^1}$-bundles $ X \to V$ and $ Y \to W$ with connections $ \Gamma $ on $ X$ and $ \nabla $ on $ Y$ respectively. We study maps $ X \to Y$ which induce both connections and metrics. Our study relies on Nash's implicit function theorem for infinitesimally invertible differential operators. We show, for the case when $ Y \to W = {\mathbf{C}}{P^q}$ is the Hopf bundle, that if $ 2q \geq n(n + 1)/2 + 3n$ then there exists a nonempty open subset in the space of $ {C^\infty }$-pairs $ (g,\Gamma)$ on $ V$ which can be induced from $ (h,\nabla)$ on $ {\mathbf{C}}{P^Q}$.


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DOI: https://doi.org/10.1090/S0002-9947-1993-1106187-0
Article copyright: © Copyright 1993 American Mathematical Society

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