Abstract
Natural Riemann extensions are pseudo-Riemannian metrics (introduced by Sekizawa and studied then by Kowalski–Sekizawa), which generalize the classical Riemann extension defined by Patterson–Walker. Let \(M\) be a manifold with an affine connection and let \(T^{*}M\) be the total space of its cotangent bundle. On \(T^{*}M\) endowed with a natural Riemann extension, we study here the Laplacian and give necessary and sufficient conditions for the harmonicity of a certain family of (local) functions. We also prove a gradient formula for natural Riemann extensions.
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Acknowledgments
The first author is grateful to the Mathematical Institute of Charles University in Prague for the kind hospitality. The second author was supported by the grant GAČR 14-02476S.
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Dedicated to Ana-Maria Pastore to her 70s.
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Bejan, CL., Kowalski, O. On some differential operators on natural Riemann extensions. Ann Glob Anal Geom 48, 171–180 (2015). https://doi.org/10.1007/s10455-015-9463-3
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DOI: https://doi.org/10.1007/s10455-015-9463-3