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The Sasaki metric (Riemannian geometry)

Gabe Khan

This diagram is meant to depict how the fibers distort as the base point (B) changes.

As the base point changes, the tangent planes of the Sasaki metric get distorted. To try to express this I drew the following picture. In terms of the Sasaki metric, the two vectors are actually orthonormal, which means that the space is warped from our viewpoint in 3D. However, that warping isn't obvious when you just look at the first picture. Furthermore, as the base point moves, the warping changes in some complicated ways. I'm not 100% sure that I did all the calculations correctly for this diagram, and there may be some small errors. Nonetheless, I feel like it gets across the point of how the tangent spaces "change shape" depending on the base point.