[FoRK] Succinct Trig

Russell Turpin < deafbox at hotmail.com > on > Fri Jul 7 07:43:32 PDT 2006

James Tauber <jtauber at jtauber.com>:
>Only if you define a metric on the manifold. ..

Well, no. One point of differentiable manifolds is that you
get a geometry from them, with tangent spaces and geodesics,
and much of the other stuff needed for physics, without having
to bind yourself to a specific metric. You can put all sorts
of metrics on a differential manifold, as long as they preserve
local direction. (If I recall, correctly -- it's been many
years and gray hairs since I've studied this.) You're correct
that a differential topology has a lot of structure beyond
the underlying point-set topology, and that the latter isn't
enough for geometry.

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