Summary |
The Grassmann manifold inherits many canonical structures from the sur-rounding Euclidean space. Our purpose is to study all possible geometries defined by these structures in the sense of metric geometry. Both general met-ric and Finsler metric approaches are carried out, and it is discovered, then proved that general angular metric, given by composition of symmetric gauge function and canonical angle operator (∅οθ), defines the same length structure as the Finsler metric (∅οσ), where ∅ is the same and σ is the singular value operator well-defined on all the tangent spaces of Grassmann manifold. We have in fact considered all possible Finsler metrics, not even differentiable, on the Grassmann manifold, and find there counterparts as general metrics. Inspired by this result, we notice that being intrinsic is a distinctive quality for a general metric. How to characterize all intrinsic metrics on a nice space such as Grassmann manifold or even vector space is an appealing question to be considered. "My Calculus of Extension builds the abstract foundation of the theory of space; that is, it is free of all spatial intuition and is a pure mathematical science; only the special application to [physical] space constitutes geometry. However the theorems of the Calculus of Extension are not merely transla-tions of geometrical results into an abstract language; they have a much more general significance, for while the ordinary geometry remains bound to the three dimensions of [physical] space, the abstract science is free of this limitation." Hermann Günther Grassmann (1809-77), 1845 |