The structural consideration of the tangent bundleT(M n ) withg M over a riemannian manifoldM n
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A refinement of the geometry of tangent bundle is made by presenting the proposition((2.3)) on tensor fields of a tangent bundle and it is shown that the Riemann metric gM of a tangent bundle previously called the Sasaki lift is nothing but the direct sum of the vertical and horizontal lift of the Riemann metric defined on the base Riemann manifold. The geometric meaning of the unit tensor field and the almost complex structure is given on the basis of the proposition((2.3)). By means of B. O’Neill’s scheme and of Y. Muto’s notion the geometry of horizontal and vertical distribution is developed and it is shown that the fibre is totally geodesic while the horizontal distribution admits the second fundamental tensor field which is skew-symmetric. In Y. Muto’s sense the tangent bundle with gM is an isometric and parallel fibred space.
KeywordsVertical Distribution Tangent Bundle Geometric Meaning Horizontal Distribution Tensor Field
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