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Keywords:
semi-Riemannian manifold; cotangent bundle; natural Riemann extension; harmonic tensor field
Summary:
If $(M,\nabla )$ is a manifold with a symmetric linear connection, then $T^{*}M$ can be endowed with the natural Riemann extension $\bar {g}$ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to $\bar {g}$ initiated by C. L. Bejan and O. Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure $\mathcal {P}$ on $(T^{*}M,\bar {g})$ and prove that $\mathcal {P}$ is harmonic (in the sense of E. García-Río, L. Vanhecke and M. E. Vázquez-Abal (1997)) if and only if $\bar {g}$ reduces to the classical Riemann extension introduced by E. M. Patterson and A. G. Walker (1952).
References:
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