Abstract
In this paper, we study the geometry of the kernel of the Lichnerovicz Laplacian in the case of complete and, in particular, compact Riemannian manifolds, and also propose a lower estimate of its eigenvalues on a compact Riemannian manifold whose curvature operator is bounded from below and an upper estimate of its eigenvalues on a compact Riemannian manifold with the Ricci curvature bounded from below. We define the Lichnerovicz Laplacian on the space of smooth sections of the bundle of covariant tensors as is required by its original definition; this distinguishes our results from results obtained earlier.
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 169, Proceedings of the International Conference “Geometric Methods in the Control Theory and Mathematical Physics” Dedicated to the 70th Anniversary of Prof. S. L. Atanasyan, 70th Anniversary of Prof. I. S. Krasil’shchik, 70th Anniversary of Prof. A. V. Samokhin, and 80th Anniversary of Prof. V. T. Fomenko. Ryazan State University named for S. Yesenin, Ryazan, September 25–28, 2018. Part II, 2019.
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Stepanov, S.E., Tsyganok, I.I. On the Lichnerovicz Laplacian. J Math Sci 263, 415–422 (2022). https://doi.org/10.1007/s10958-022-05940-7
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DOI: https://doi.org/10.1007/s10958-022-05940-7
Keywords and phrases
- Riemannian manifold
- covariant tensor
- Lichnerovicz Laplacian
- kernel of the Laplacian
- eigenvalue of the Laplacian