# Tagged Questions

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### Question about a notation. Norm of the derivative of a function at a point

Given is an analytic function from $M$ to $N$, both equipped with conformal Riemannian metric, say $g$ and $h$ resp. What might the $h$ norm of the derivative of the function at a point mean? ...
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### universal covering of punctured plane and Poincaré metric

I want to prove the following result: Let $\Omega$ be the domain $\mathbb{C}\backslash \{a_1,a_2,a_3,a_4\}$. Its universal covering is the unit disk, and the standard Poincaré metric is pulled back to ...
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### Monodromy representation of Airy equation

Let $K=\Bbb{C}(z)$ with the usual derivation and consider the Airy dierential equation $y^{(2)}-zy$=0. How to determine the monodromy representration? Airy equation is not Fuchsian diferential ...
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### Orientation on Riemann surfaces

$\mathcal{X}$ is a Riemann surface and $\mathcal{E}^{(2)}(\mathcal{X})$ is the $\mathbb{C}$-Vector space of all differentiable $2$-forms on $\mathcal{X}$. I want to define the orientation of ...
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### Homotopy invariance of line integral on manifolds

Consider a 1-form: $\omega\in\Gamma(\mathrm{T}^*M)$ and two differentiable curves: $\gamma,\tilde{\gamma}:[a,b]\to M:\gamma(a)=\tilde{\gamma}(a),\gamma(b)=\tilde{\gamma}(b)$ together with a ...
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### How many differential forms on the complex plane?

I am puzzled by the fact that the two differential forms $$\begin{array}{cc} dz=dx+i dy , & d\overline{z}=dx-i dy \end{array}$$ are $\mathbb{C}$-linearly independent, even if the underlying ...
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### Geometric interpretation of $\partial/\partial z$

My understanding is that analytic derivative ,$\partial\phi/\partial z$, and anti-analytic derivative ,$\partial\phi/\partial\bar{z}$, are resp. tangent and normal to the curve $\phi$. Am i right?can ...
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### Closed but not exact one-form on $S^2$

I would like to know whether there is any nice prescription to give an example of a closed but not exact one-form on $S^2$ (not the $3$-ball). I assume to take some points out of this surface, e.g. 3. ...
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### Möbius transformation in the complex plane.

Assume that $U$ be a line in the complex plane. And assume a Möbius transformation $\phi$ sends $U$ again to a line. How can I classify all such $\phi$? I want to write my ideas. But, I ...
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### top journals in analysis

as an undergraduate I find analysis as my favorite.I want to read journals regarding that. give me top 5 journals in analysis(real,complex)? top 5 journals in differential geometry? and generally some ...
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### Describing co-ordinate systems in 3D for which Laplace's equation is separable

Laplace's Equation in 3 dimensions is given by $$\nabla^2f=\frac{ \partial^2f}{\partial x^2}+\frac{ \partial^2f}{\partial z^2}+\frac{ \partial^2f}{\partial y^2}=0$$ and is a very important PDE in ...
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### Showing holomorphic functions are preserved under pullback by a holomorphic map

Let $f: X\rightarrow Y$ be a holomorphic mapping of complex manifolds and assume for simplicity that $dim(X)=dim(Y)=1$. I want to show that it preserves holomorphic functions under pullback. We define ...
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### Definition of Holomorphic map of complex manifolds

Let $X,Y$ be complex manifolds and let $f: X \rightarrow Y$ be a continuous map. When exactly do we say that $f$ is "holomorphic"? I am interested in the basic definition (possibly using charts), not ...
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### Conjugate function reverse angle in a neighborhood

Can you please help me with this question. Prove that if conjugate of $F$ is analytic in a neighborhood of $z_0$ in $C$ then $F$ reverses angles at $z_0$. Can anyone please explain that for me? what ...
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### Specific homotopy between complex conjugation and the identity.

Consider the set $\mathcal{C} = C^{\infty}(\mathbb{C}^*, \mathbb{C}^*)$, where $\mathbb{C}^* = \mathbb{C}\backslash\{0\}$. Both $f(z) = z$ and $g(z) = \bar{z}$ can be seen as elements in ...
We consider the function $f(x,y)=x^2+y^2$ in $\omega = (0,1)^2.$ I am wondering about the existence of a $C^2-$extension $F$ of $f$ in $\Omega = (0,2)^2$ such that $F$ is harmonic in ...
### how to understand the tensor product canonical line bundle $\otimes$ dual bundle
Suppose we have a Riemann surface $M$ together with a holomorphic vector bundle $E \to M$ of rank n. let $K$ denote the canonical line bundle and let $E^*$ denote the dual bundle I am trying to ...