On the Classifying of the Tangent Sphere Bundle with Almost Contact B-Metric Structure

One of the classical fundamental motifs in differential geometry of manifolds is the notion of the almost contact structure. As a counterpart of the almost contact metric structure, the notion of the almost contact B-metric structure has been an interesting research field for many mathematicians in differential geometry of manifolds, and the geometry of such structures has been studied frequently. There is a classification for the almost contact B-metric structures, named the relevant classification, with respect to the covariant derivative of the fundamental tensor of type (1, 1). In this paper, we basically use this classification to achieve our goals. On the other hand, many of mathematicians have widely considered the concept of lifted metric on the tangent bundle and tangent sphere bundle of a Riemannian manifold (M, g). The idea of constructing a lifted metric on the tangent bundle was a strong inspiration for many of mathematicians and finally, the notion of g-natural metric as the most general type of lifted metrics on tangent bundle TM of a Riemannian manifold (M, g) was introduced in 2005. In this paper, we consider a pair of associated g-natural metrics on the unit tangent sphere bundle T1M with B-metric, and we classify this structure with respect to the relevant classification of almost contact manifold with B-metric. tensor../files/site1/files/%D9%BE%DB%8C%D8%BA%D8%A7%D9%86.pdf