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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...