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Besse conjecture for compact manifolds with pinched curvature

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Abstract

On a compact n-dimensional manifold M, it has been conjectured that a critical point of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, is Einstein. In this paper, we prove the Besse conjecture for compact manifolds with pinched Weyl curvature. Moreover, we shall conclude that such a conjecture is true if its Weyl curvature tensor and the Kulkarni-Nomizu product of Ricci curvature are orthogonal.

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Acknowledgements

The author is partially supported by PPP/FAPEPI, Brazil, Grant 007/2018.

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  1. Departamento de Matemática, Universidade Federal do Piauí, Piauí, Teresina, 64049-550, Brazil

    H. Baltazar

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  1. H. Baltazar
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Correspondence to H. Baltazar.

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Baltazar, H. Besse conjecture for compact manifolds with pinched curvature. Arch. Math. 115, 229–239 (2020). https://doi.org/10.1007/s00013-020-01463-2

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