Geometry in global coordinates in mechanics and optimal transport

TL;DR

This paper develops global coordinate formulas for geometric quantities on manifolds embedded in inner product spaces, enabling effective numerical computation and analysis in mechanics and optimal transport, with applications to rigid body and Hamiltonian systems.

Contribution

It introduces operator-valued global formulas for geometric quantities and liftings of Hamilton vector fields, extending the toolkit for geometric analysis in mechanics and optimal transport.

Findings

Explicit formulas for curvature and connections in global coordinates.

Demonstrated nonnegative cross-curvature for Kim-McCann metric on positive-semidefinite matrices.

Identified cases where cross-curvature can be negative on Grassmann manifolds.

Abstract

For a manifold embedded in an inner product space, we express geometric quantities such as {\it 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 {\it operator-valued} expressions, using an {\it affine projection} to the tangent bundle. For a submersion image of an embedded manifold, we introduce {\it liftings} of Hamilton vector fields, allowing us to use embedded coordinates on horizontal bundles. We derive a {\it 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 {\it 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.