  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 $\varepsilon$.

Differential Geometry