There are a lot of structures on bundles (Tangent bundle, Cotangent bundle, Semi-Cotangent bundle, Tensor bundle, etc.) and n-dimensional differential manifolds Mn. The integrability of tensorial structures on a manifold and their extension to bundles such as tangent and cotangent bundles has been an active research topic for the last 60 years. Firstly, Japanese mathematician S.Sasaki (1912-1987) studies the differential geometry of tangent bundles of Riemannian manifolds in 1958. Later, the subject of lift and bundle constantly improved. Afterward, Ishihara and Yano (Ishihara and Yano,1973) obtained the integrability conditions of the F structure satisfying the condition of F3+F=0. By and by, a lot of structures on the manifold and bundles studies by valuable authors (Tachibana, 1960; Norden, 1960; Sato, 1968; Shirokov, 1966; Vishnevskii, 1970; Kruchkovich, 1972; Salimov, 1994). Differential geometric applications of tensor operators are a very fruitful area of research in the modern study of differential geometry. However, despite its importance, tensor operators, structures and related issues are not well known yet. In addition, there are very few reference books in this field that can be referenced. In this context, all structures on Mn and bundles from the beginning to the present combined in this book. We believe that this research book will provide a systematic description of tensor operators and structure theory and especially useful in master and doctoral education.
