  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.
