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TrickyDicky
TrickyDicky is offline
#147
Nov26-11, 12:52 PM
P: 2,892
Correction of post #136: It was quite obvious that the gaussian curvature had to be zero but if even the "experts" blunder I guess it's no big deal that I do (and quite a few times). My only excuse is my math ignorance and that I have been misled to some extent (about the core of the matter, certainly not about the Gaussian curvature)
The important thing is that the underlying theme of the thread which I stated in my second post (#4), that a topological sphere can have a flat metric in hyperbolic space and that a horosphere is a topological sphere in H^3 is still alive.
Now to the correction of #136, it should have said:

So since the horosphere is closed it has no boundary term (it is compact without boundary): We only need the integral of the gaussian curvature to obtain the Euler characteristic, and since it has infinite volume:

[tex]\begin{align}
2 \pi \, \chi(M) &= \int_Ʃ K \, dA = \\
&= \lim_{R \to \infty} \int_Ʃ \frac{1}{R^2} \, dA=4\pi \\
\chi(M) &= 2
\end{align}
[/tex]

Let me know if there's any problem with this.