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Harmonic manifolds with some specific volume densities

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Abstract

We show that a noncompact, complete, simply connected harmonic manifold (M d, g) with volume densityθ m(r)=sinhd-1 r is isometric to the real hyperbolic space and a noncompact, complete, simply connected Kähler harmonic manifold (M 2d, g) with volume densityθ m(r)=sinh2d-1 r coshr is isometric to the complex hyperbolic space. A similar result is also proved for quaternionic Kähler manifolds. Using our methods we get an alternative proof, without appealing to the powerful Cheeger-Gromoll splitting theorem, of the fact that every Ricci flat harmonic manifold is flat. Finally a rigidity result for real hyperbolic space is presented.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology, Powai, 400 076, Mumbai, India

    K Ramachandran & A Ranjan

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  1. K Ramachandran
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  2. A Ranjan
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Correspondence to K Ramachandran.

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Ramachandran, K., Ranjan, A. Harmonic manifolds with some specific volume densities. Proc. Indian Acad. Sci. (Math. Sci.) 107, 251–261 (1997). https://doi.org/10.1007/BF02867256

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  • DOI: https://doi.org/10.1007/BF02867256

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