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The Tangent Bundle

Gabe Khan
The Sasaki metric is defined on the tangent bundle, which corresponds to points in the space along with the possible directions that one could travel. In this example, to specify a point in the tangent space, we pick a blue base point and its tangent space, which corresponds to the plane tangent to the sphere. We then pick a gray point within this tangent space.   For this example, the space is 4 dimensional, because there are two degrees of freedom for the blue point as well as two degrees of freedom for the gray point. However, this picture isn't that interesting, and doesn't make the intrinsic geometry at all apparent.