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.

翻译：对于嵌入在内积空间中的流形，我们用全局坐标表达诸如{\it 哈密顿向量场、仿射联络和列维-齐维塔联络、曲率}等几何量。大多数量的全局公式不以坐标指标表示，而是通过使用{\it 仿射投影}到切丛上的{\it 算子值}表达式。对于嵌入流形的淹没像，我们引入了哈密顿向量场的{\it 提升}，从而允许在水平丛上使用嵌入坐标。我们推导了向量丛上仿射联络的{\it 高斯-科达齐方程}。该方法使得几何表达式可以在全局范围内求值，并可在现代数值框架中有效应用于实际问题。所考虑的实例包括刚体力学及格拉斯曼流形上的哈密顿力学。我们明确展示了对于具有反射器天线型代价函数的{\it Kim-McCann}度量，在固定秩半正定矩阵空间上的交叉曲率（MTW张量）具有非负交叉曲率，而相应的代价函数在格拉斯曼流形上（除射影空间外）可能具有负交叉曲率。