Geodesicity and Isoclinity Properties for the Tangent Bundle of the Heisenberg Manifold with Sasaki Metric

TL;DR

This paper investigates the geometric properties of the tangent bundle of the Heisenberg manifold with the Sasaki metric, focusing on distributions, geodesics, and their isoclinic and geodesic characteristics.

Contribution

It establishes new results on the isocline and totally geodesic properties of distributions and characterizes geodesics in the tangent bundle with the Sasaki metric.

Findings

Horizontal and vertical distributions are isocline.

Certain distributions are totally geodesic but not isocline.

Lifts of geodesics from the base manifold are geodesics in the tangent bundle.

Abstract

We prove that the horizontal and vertical distributions of the tangent bundle with the Sasaki metric are isocline, the distributions given by the kernels of the horizontal and vertical lifts of the contact form $\omega$ from the Heisenberg manifold $(H_3,g)$ to $(TH_3,g^S)$ are not totally geodesic, and the distributions $F^H=L(E_1^H,E_2^H)$ and $F^V=L(E_1^V,E_2^V)$ are totally geodesic, but they are not isocline. We obtain that the horizontal and natural lifts of the curves from the Heisenberg manifold $(H_3,g)$, are geodesics in the tangent bundle endowed with the Sasaki metric $(TH_3,g^s)$, if and only if the curves considered on the base manifold are geodesics. Then, we get two particular examples of geodesics from $(TH_3,g^s)$, which are not horizontal or natural lifts of geodesics from the base manifold $(H_3,g)$.