Geometry in global coordinates in mechanics and optimal transport
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For a manifold embedded in an inner product space, we express geometric quantities such as Hamilton vector fields, affine and Levi-Civita connections, curvature in global coordinates. Instead of coordinate indices, the global formulas for most quantities are expressed as operator-valued expressions, using an affine projection to the tangent bundle. For a submersion image of an embedded manifold, we introduce liftings of Hamilton vector fields, allowing us to use embedded coordinates on horizontal bundles. We derive a Gauss-Codazzi equation for affine connections on vector bundles. This approach allows us to evaluate geometric expressions globally, and could be used effectively with modern numerical frameworks in applications. Examples considered include rigid body mechanics and Hamilton mechanics on Grassmann manifolds. We show explicitly the cross-curvature (MTW-tensor) for the Kim-McCann metric with a reflector antenna-type cost function on the space of positive-semidefinite matrices of fixed rank has nonnegative cross-curvature, while the corresponding cost could have negative cross-curvature on Grassmann manifolds, except for projective spaces.
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