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The Bochner technique and modification of the Ricci tensor

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Abstract

In the well-known vanishing theorems of Bochner, the assumptions Ric ≥ 0 and Ric ≤ 0 are modified by using Hessian and Laplacian of a smooth positive function such that, when this function is constant, these assumptions return to Ric ≥ 0 and Ric ≤ 0. We prove that the assertions and results of Bochner’s vanishing theorems still hold under these modified assumptions. Additionally, the assumption Ric ≥ H > 0 given in the eigenvalue estimate theorem of Lichnerowicz is also modified in the same way, and we obtain estimates for the first positive eigenvalue of the Laplace operator.

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Authors and Affiliations

  1. Department of Mathematics, Anadolu University, 26470, Eskişehir, Turkey

    Murat Limoncu

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  1. Murat Limoncu
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Correspondence to Murat Limoncu.

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Limoncu, M. The Bochner technique and modification of the Ricci tensor. Ann Glob Anal Geom 36, 285–291 (2009). https://doi.org/10.1007/s10455-009-9163-y

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  • DOI: https://doi.org/10.1007/s10455-009-9163-y

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