Is it true that if I take two surfaces that are topologically equivalent, I can find a conformal mapping between them?

I often hit this problem : Consider a surface defined by the equation $z = f(x, y)$, the differentials of this function are $\frac{\partial f}{\partial x}\mathrm{d}x$ and $\frac{\partial f}{\partial y}\mathrm{d}y$. But suppose that the surface is rotated by any space rotation $R$ : the points of the surface are now $\left(\matrix{x’\\y’\\z’}\right) = R\left(\matrix{x\\ y\\ f(x, […]

Problem Prove that if $\mathbf{y}:E\to M$ is a proper patch, then $\mathbf{y}$ carries open sets in $E$ to open sets in $M$. Deduce that if $\mathbf{x}:D \to M$ is an arbitrary patch, then the image $\mathbf{x}(D)$ is an open set in $M$. (Hint: To prove the latter assertion, use Cor 3.3.) Finally, prove that every […]

The two surfaces $S_1$, $S_2$ parameterised respectively by $$\sigma_1(u,v) = (u\cos v,u\sin v, \ln u)$$ $$\sigma_2(u,v) = (u\cos v, u\sin v, v),$$ are, as I understand, not locally isometric. How can this be proven? They have the same Gaussian curvature, so that’s not enough. The coefficients of the first fundamental form in each are not […]

A Cylinder is such a common surface. But is there a parametrization for an isometrically $ R^2 $ bent cylinder whose major and minor dimensions are along x, y axes? I used an approximation to parametrize, that is why the generators are bent between boundaries that seem to have elliptic profiles. Most probably there is […]

I don’t understand this How $ dS = \sqrt{ \left ( \partial g \over \partial x\right )^2 + \left ( \partial g \over \partial y\right )^2 + 1 } \; dA \; \; $ ?? Is $ dA = dx\times dy$??

When dealing with differentiable surfaces one defines a function $f:S\rightarrow \mathbb{R}$ as being differentiable if its expression in local coordinates is differentiable. But one could also define it to be differentiable if there exists differentiable function $F: V\subset \mathbb{R}^3 \rightarrow \mathbb{R}$ from an open set $V$ of $\mathbb{R}^3$ such that $S\subset V$ and $F|_{S} = […]

What’s the K-group of a surface? I also want to know how to calculate such group and if there is a explicit characterization of the generators.

Let $\partial M$ be $C^2$ closed surface in $\mathbb{R}^3$, $M$ is open. Show that $$ f(x) = \frac{\int_{\partial M} \left| \frac{1}{||y-x||} n_y \cdot \nabla_y \frac{1}{||y-x||} \right| dS_y}{\left| \int_{\partial M} \frac{1}{||y-x||} n_y \cdot \nabla_y \frac{1}{||y-x||} dS_y\right|} $$ is bounded in $\overline{M}$ I already know few things about this kind of integral, I already had two questions […]

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to prove Proposition 2.7. The only resource where I could find something is Farb and Margalit: A primer on MCGs. […]

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