ArthurStuart
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 Mar 31 asked Closure of algebraic groups Jan 16 awarded Yearling Dec 22 awarded Popular Question Apr 30 awarded Taxonomist Apr 14 asked Total curvature of an ovaloid. Apr 14 comment Triangulations of surface. I'm sorry, what is a gon? Apr 14 asked Asymptotic geodesic on hyperboloid. Apr 13 asked Frenet frame and tangent space. Apr 7 comment Maximal circles and sphere. $p$ and $v$ are ortogonal, and maximum circle is a great circle. Apr 6 asked Maximal circles and sphere. Mar 31 comment Characterize the sphere using mean curvature. But the connected sum of two $T^2$ has hyperbolic points... right? (Ok $T^2$ has only a hole) Mar 31 comment Characterize the sphere using mean curvature. @HeeKnowLee Only if $\Sigma$ is a sphere... true? Mar 31 comment Characterize the sphere using mean curvature. $\int |K|= 2 \pi \chi(T^2)= -4 \pi$ Mar 31 comment Characterize the sphere using mean curvature. I'm sorry. If $\Sigma$ is a compact generic surface and $\int_{\Sigma} |K|= 4 \pi$, can I say that $\Sigma$ has no hyperbolic points? Mar 31 accepted Characterize the sphere using mean curvature. Mar 31 comment Characterize the sphere using mean curvature. Ok thank you! I have a little curiosity. We know that $H^2 \ge K$. If $\int_{\Sigma}K= 4 \pi$ can I deduce that $\Sigma$ has not hyperbolic points? Mar 31 comment Characterize the sphere using mean curvature. @aGer Let's suppose that $\int_{\Sigma}H^2= 4 \pi$ but $\Sigma$ not a sphere. We know that $H^2 \ge K$, so $\int_{\Sigma}H^2 \ge \int_{\Sigma}K = 2 \pi \chi(\Sigma)$... Gauss-Bonnet? Mar 31 asked Characterize the sphere using mean curvature. Mar 31 comment Geometric interpretation of Gaussian curvature. Ok, thank you! The last thing: can I argue something about the elliptic or hyperbolic nature of the points of $\Sigma$? Is it possible to say that $\Sigma$ has an elliptic point? Moreover, is it possible to say that if $\int_{\Sigma}|K|=4 \pi$ than $\Sigma$ has not hyperbolic points? Mar 31 comment Geometric interpretation of Gaussian curvature. $4 \pi = \mathcal{A}(S^2) = \int_{S^2} 1 \, d\mathcal{A}_{S^2} = \int_{\Sigma}|Jac(N)(p)| .$ We have by definition $|Jac(N)(p)|= |(dN)_p(e_1) \wedge (dN)_p(e_2)| ,$ where $\{e_1, e_2\}$ is an o.n. basis of $T_p\Sigma$. We choose $e_1$ and $e_2$ as the principal direction $\Sigma$ in $p$. We obtain $$|Jac(N)(p)|=|k_1(p)k_2(p)||e_1 \wedge e_2|= K(p) .$$ So to finish: $$4 \pi = \int_{\Sigma} K \le \int_{\Sigma} |K| .$$ I have a problem with the module also in this case...