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Classification of $(k,\mu )$-contact manifolds with divergence free Cotton tensor and vanishing Bach tensor

Volume 122 / 2019

Amalendu Ghosh, Ramesh Sharma Annales Polonici Mathematici 122 (2019), 153-163 MSC: Primary 53C15; Secondary 53C21, 52D10. DOI: 10.4064/ap180228-13-11 Published online: 8 March 2019

Abstract

We first prove that a $(k,\mu )$-contact manifold of dimension $2n+1$ with divergence free Cotton tensor is flat in dimension $ 3 $, and in higher dimensions, locally isometric to $S^n(4)\times E^{n+1}$. Finally, we show that a Bach flat non-Sasakian $(k,\mu )$-contact manifold is flat in dimension 3, and in each higher dimension, there is a unique $(k,\mu )$-contact manifold locally isometric, up to a $D$-homothetic deformation, to the unit tangent sphere bundle of a space of constant curvature $\not =1$. This result provides an example of a Bach flat metric that is neither Einstein nor conformally flat.

Authors

  • Amalendu GhoshDepartment of Mathematics
    Chandernagore College
    712 136, Chandanagar
    W.B., India
    e-mail
  • Ramesh SharmaDepartment of Mathematics and Physics
    University of New Haven
    West Haven, CT 06516, U.S.A.
    e-mail

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