what's with that transparency

[Michal Frackowiak]

On Fri, 2002-11-15 at 05:26, szajtan odwieczny wrote:
> ok....at first thanks - you gave me quite homework to do, it will take a few days until I manage to read that book but I just can't sit like that after what you wrote.
> firstly I want to remark that I don't know what that damn word manifold mean - I cant find it anywhere - I can just suppose.

a manifold is: rozmaitosc rozniczkowa.
let me put some words here which should clear something:

1. manifold is only a "imaginary" structure, you can think of it as of
something smooth without peak, with or without edges
2. manifold itself is not very useful in GR (OTW)
3. riemann manifold is a manifold with given metric. only with the
metric you get the recipe for calculating distances.
4. the shape of the metric is not important at all - the condition is
that there are mappings (for n-dim manifold) which LOCALLY map M^n into
R^n with given coordinate system (there is no coordinate system on the
manifold itself). set of mappings should cover the whole manifold.
5. in fact the metric is also not described on the manifold, but on R^n,
into whih the manifold can be at every its point mapped.

keeping this in mind, a 2-torus can be mapped EVERYWHERE (but locally)
into R^2, and assuming metric ds^2 = dx^2 + dy^2 its flat. thats all.
simple and easy.

how to get curvature from the metric?
1. calculate connection coeficients (wspolczynniki koneksji)
\Gamma^\mu_{\nu\rho}
2. calculate Riemann tensor

if all the terms in the Riemann tensor =0, space is flat. for ds^2 =
dx^2 + dy^2 is flat for sure, because even connection =0.

regards - michal
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