| Summary: | 碩士 === 國立中正大學 === 數學研究所 === 91 === In this report we fill in detailed computation for curvatures on the tangent bundle of a Riemannian manifold and its tangent sphere bundle with fixed radius r > 0.
The geometry of the tangent bundles of Riemannian manifolds has been studied thoroughly since the Sasaki metric premiered in 1958. Starting from the Sasaki metric, which is a natural extension for the metric of a Riemannian manifold to its tangent bundle, we are able to write down the corresponding
Christoffel symbols. Coupled with the notion of horizontal and vertical lifts of vector fields, the connection of lifted vector fields can be derived and then the Riemannian curvature tensor on the tangent bundle is at hand.
Together with submanifold theories, previous formulae can be applied to a special submanifold of the tangent bundle, called the tangent sphere bundle, and we shall compute some quantities on this manifold.
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