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A CLASSIFICATION OF HORIZONTALLY HOMOTHETIC SUBMERSIONS FROM SPACE FORMS OF NONNEGATIVE CURVATURE

Published online by Cambridge University Press:  31 May 2006

YE-LIN OU
Affiliation:
Department of Mathematics, The University of Oklahoma, Norman, OK 73019, USAylou@ou.edu, gerard@math.ou.edu
GERARD WALSCHAP
Affiliation:
Department of Mathematics, The University of Oklahoma, Norman, OK 73019, USAylou@ou.edu, gerard@math.ou.edu
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Abstract

In this paper, it is proved that for $n\geq 2$, any horizontally homothetic submersion $\varphi :\mathbb{R}^{n+1}\longrightarrow (N^{n}, h)$ is a Riemannian submersion up to a homothety. It is also shown that if $\varphi :\mathbb{S}^{n+1}\longrightarrow (N^{n}, h)$ is a horizontally homothetic submersion, then $n=2m$, $(N^{n}, h)$ is isometric to $\mathbb{C}P^{m}$ and, up to a homothety, $\varphi$ is a standard Hopf fibration $\mathbb{S}^{2m+1} \longrightarrow \mathbb{C}P^{m}$.

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Type
Papers
Copyright
© The London Mathematical Society 2006

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