Geometric Properties of a Horizontally Deformed Cheeger-Gromoll Metric on the Tangent Bundle

We introduce a novel horizontally deformed Cheeger--Gromoll metric on the tangent bundle of a Riemannian manifold, thereby enlarging the spectrum of invariant metrics available in this context. A systematic analysis is performed on the geometric characteristics engendered by this metric, yielding explicit expressions for the Levi--Civita connection and the full array of curvature tensors. Moreover, we examined the geometry of a hypersurface embedded in the tangent bundle, providing a detailed analysis of its intrinsic structure, induced Levi--Civita connection, and associated curvature tensors.