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 on 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 on 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 on (TH_3,g^s), which are not horizontal or natural lifts of geodesics from the base manifold (H_3,g).
DRUTA, SIMONA LUIZA and PIU, PAOLA
"Geodesicity and isoclinity properties for the tangent bundle of the Heisenberg manifold with Sasaki metric,"
Turkish Journal of Mathematics: Vol. 36:
2, Article 14.
Available at: https://journals.tubitak.gov.tr/math/vol36/iss2/14