On tangent sphere bundles with contact pseudo-metric structures

In this paper, we introduce a contact pseudo-metric structure on a tangent sphere bundle $T_\varepsilon M$. we prove that the tangent sphere bundle $T_{\varepsilon}M$ is $(\kappa, \mu)$-contact pseudo-metric manifold if and only if the manifold $M$ is of constant sectional curvature. Also, we prove that this structure on the tangent sphere bundle is $K$-contact iff the base manifold has constant curvature $\varepsilon$.