On geodesics of tangent bundle with fiberwise deformed Sasaki metric over Kählerian manifold (Q2912582)

Let \((M,g)\) be a Riemannian manifold. Denote by \(TM\) and \(T_1M\) the tangent bundle and the unit tangent bundle of \((M,g)\) with the Sasaki metric. Studying these manifolds some authors have shown that the Sasaki metric weakly inherits the base manifold properties. This is the reason to propose a deformation of the Sasaki metric to achieve some kind of ``flexibility'' of its properties. In this spirit the author proposes a fiber-wise deformation of the Sasaki metric on \(T_1M\) as well as on the slashed manifold \(TM_0:=TM\setminus M\) over the Kählerian manifold based on the Berger deformation of the metric on a unit sphere.NEWLINENEWLINEAssuming that \(M\) is a Kählerian locally symmetric manifold, \(\Gamma\) is a geodesic in the \(T_1M\) and \(\gamma=\pi\circ\Gamma\) its projection to the base \(M\), the author proves that all geodesic curvatures of \(\gamma\) are constant. Moreover, if \(M^{2n}\) \((n\geq3)\) is of constant holomorphic curvature then the geodesic curvatures of \(\gamma=\pi\circ\Gamma\) are all constant and \(k_6=\cdots=k_{n-1}=0\).NEWLINENEWLINEAnyhow, if \(\Gamma\) is a geodesic on the slashed tangent bundle \(TM_0\), then the projected curve \(\gamma=\pi\circ\Gamma\) does not posses this property.