On 4/22/13 4/22/13 2:10 AM, Shmuel (Seymour J.) Metz wrote: > In <ZPidnRFGArYb0O7MRVn_vwA@giganews.com>, on 04/20/2013 > at 09:22 PM, Tom Roberts <email@example.com> said: >> Displacements from a given point form a vector space only when the >> displacements are infinitesimal. > > I'm not sure what you mean by that.
Just that a set of all displacements from a given origin does not form a vector space unless there is an isomorphism to the tangent space at the origin.
>> Hmmmm. In physics, where we always have an ORIENTABLE manifold WITH >> METRIC, > > Certainly at an undergraduate level, but once you get into things like > gauge theories you need manifolds without a metric.