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

We propose a fiber-wise deformation of the Sasaki metric on slashed and unit tangent bundles over the Kälerian manifold based on the Berger deformation of metric on a unit sphere. The geodesics of this metric have different projections on a base manifold for the slashed and unit tangent bundles in contrast to usual Sasaki metric. Nevertheless, the projections of geodesics of the unit tangent bundle over the locally symmetric Kählerian manifold still preserve the property to have all geodesic curvatures constant.