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Feb19-12, 09:35 AM
Sci Advisor
P: 1,716
Quote Quote by Mike706 View Post
Hello Everyone,

I am just wondering what the difference in these is. Could someone please give a brief example of non-coordinate based differential geometry vs the equivalent in coordinate based, or explain the difference (whichever is easier)? Also, what advantages does one have over the other?

Thanks for your help,

Edit - Also, are there specific names for these two types of DG? Thanks again.
Geometric structures can be described globally without reference to coordinates. However they can always be expressed locally in a coordinate system. In the 18'th century and maybe earlier I am not sure, geometers described curves and surfaces by parameterizations and then deduced the geometry from the parameters. This is a coordinate approach to geometry. But later is was realized that geometries exist as global structures across an entire surface and that the surface can not necessarily be described with a single coordinate system.

I can describe examples if you like.