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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...
Mar
31
comment Geometric interpretation of Gaussian curvature.
$\int_{\Sigma} |K| \ge \int_{\Sigma}K = \int_{\Sigma} dN_p dA_{\Sigma} = \int_{S^2} dA_{S^2} = 4 \pi $. Is it correct?
Mar
31
comment Geometric interpretation of Gaussian curvature.
@user86418 I can prove that the Gauss map is surjective, but I don't know how to continue....
Mar
31
asked Geometric interpretation of Gaussian curvature.