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Comparison Geometry for an Extension of Ricci Tensor

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Abstract

For a complete Riemannian manifold M with a (1,1)-elliptic Codazzi self-adjoint tensor field A, we use the divergence type operator \({L_A}(u): = div(A\nabla u)\) and an extension of the Ricci tensor to extend some major comparison theorems in Riemannian geometry. In fact we extend theorems such as mean curvature comparison theorem, Bishop–Gromov volume comparison theorem, Cheeger–Gromoll splitting theorem and some of their famous topological consequences. Also we get an upper bound for the end of manifolds by restrictions on the extended Ricci tensor. The results can be applied to some Riemannian hypersurfaces of Riemannian or Lorentzian space forms.

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Acknowledgements

The authors would like to gratefully thank the anonymous reviewer for his/her useful comments.

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Authors and Affiliations

  1. Department of pure Mathematics, Faculty of Sciences, Imam Khomeini International University, Qazvin, Iran

    Shahroud Azami

  2. Department of Mathematics, Tarbiat modares university, Tehran, Iran

    Seyyed Hamed Fatemi & Seyyed Mohammad Bagher Kashani

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  1. Shahroud Azami
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  2. Seyyed Hamed Fatemi
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  3. Seyyed Mohammad Bagher Kashani
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Correspondence to Seyyed Mohammad Bagher Kashani.

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Azami, S., Fatemi, S.H. & Kashani, S.M.B. Comparison Geometry for an Extension of Ricci Tensor. Results Math 76, 215 (2021). https://doi.org/10.1007/s00025-021-01521-3

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