IPHP transformations on tangent bundle of a Riemannian manifold with respect to a class of lift metrics

M. Zohrehvand

Eurasian mathematical journal, Tome 13 (2022) no. 2, pp. 82-92

Résumé

Let $(M_n, g)$ be an $n$-dimensional Riemannian manifold and $TM_n$ its tangent bundle. In this article, we study the infinitesimal paraholomorphically projective (IPHP) transformations on $TM_n$ with respect to the Levi-Civita connection of the pseudo-Riemannian metric $\tilde{g}=\alpha g^S+\beta g^C+\gamma g^V$, where $\alpha$, $\beta$ and $\gamma$ are real constants with $\alpha(\alpha+\gamma)-\beta^2\ne0$ and $g^S$, $g^C$ and $g^V$ are diagonal lift, complete lift and vertical lift of $g$, respectively. We determine this type of transformations and then prove that if $(TM_n,\tilde{g})$ has a non-affine infinitesimal paraholomorphically projective transformation, then $M_n$ and $TM_n$ are locally flat.