In this paper, we consider the tangent bundle of a Rie- mannian manifold (M, g) with g-natural metrics and among all of these metrics, we specify those with respect to which the unit tangent sphere bundle with induced g-natural metric is totally geodesic. Also, we equip the unit tangent sphere bundle T 1 M with g-natural contact (paracon- tact) metric structures, and we show that such structures are totally geodesic K-contact (K-paracontact) submanifolds of TM, if and only if the base manifold (M, g) has positive (negative) constant sectional cur- vature. Moreover, we establish a condition for g-natural almost contact B-metric structures on T 1 M such that these structures be totally geodesic submanifolds of TM.
