On Sun, 23 Dec 2007, Chris McCormick wrote:
I am definately no expert in this area, but this guy and his ideas
always fascinated me as an alternative to Euclidean geometry:
<http://en.wikipedia.org/wiki/Riemann>
<http://en.wikipedia.org/wiki/Riemannian_manifold>
<http://en.wikipedia.org/wiki/Riemannian_geometry>
The "Riemann curvature tensor" is useful for describing curvature of a
space. However, the notation is hard to get used to. For 2-D spaces, you
can avoid that notation, but for 3-D and 4-D spaces, as used in
relativistic physics, you need the full arsenal. I've learned the 1-D and
2-D special case. The amount of info you need to handle depends on whether
you are endo or exo, that is, whether you are inside the space and don't
care about the outside, or whether you are looking at the space from a
greater space that contains it.
For a sphere (the surface of a ball), the curvature is the same
everywhere. This is a special case of elliptic spaces, in which the
curvature is positive everywhere. In this case you can simplify a lot of
things, and you don't need the whole theory, so you can actually put a
63-D spherical space in a 64-D euclidean space with relative ease.
_ _ __ ___ _____ ________ _____________ _____________________ ...
| Mathieu Bouchard - tél:+1.514.383.3801, Montréal QC Canada
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