# Tagged Questions

For questions about differential forms which commonly arise in differential geometry, and sometimes in multivariable calculus.

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### What am I doing wrong when calculating this pullback?

Let $\omega = \sum_{j=1}^{n+1} x_j dy_j - y_j dx_j$ be a differential form on the sphere $S^{2n +1}$. Let $G = Z_2$ be the group acting on the sphere. I want to apply the following proposition to ...
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### Integral of a k-form

Just got to the culminating chapter in Munkres’s Analysis on Manifolds and I’ve been thrown for a loop. The author is in the process of defining the integral of a k-form η on A, an open set of R$^k$. ...
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### Exterior derivative on principal bundle

In Nakahara's Geometry, Topology and Physics on page 375, he constructs a Lie-algebra-valued one-form $\omega$ on a principal bundle $P$ by "lifting" a Lie-algebra-valued one-form $\mathcal A_i$ on an ...
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### Degree of smooth map of manifolds depends on orientation choice?

I'm a little to confused as to why it appears that the degree of a smooth map $f: M \to N$ between smooth manifolds appears to only be defined up to sign - I'm not sure where my mistake is. By ...
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### Connection planes associated to differential 1-forms

In textbook in differential geometry such an idea appears: there is a tangle bundle $\pi:TM\rightarrow M$ and we are actually looking at the trivialization $\pi^{-1}(U)=U\times \mathbb{R}^n$. We are ...
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### Showing de Rham cohomology $H^1(S^n)$ is zero

I'm trying to find an elementary way to see that the 1st de Rham cohomology of the n-sphere is zero for $n>1$, $H^1(S^n) = 0$. This is part of an attempt to find the de Rham cohomology of the n ...
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### Invariant forms on a manifold

Probably a silly question - still, it's been bugging me for some time now. Say that we have an invariant $1-$form $\omega$ on a smooth manifold $M$, acted on by a group $G$. Then \...
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### Does integration wrt to a differential form always come from a measure?

More precisely, is there an $n$-manifold $M$ with an $n$-form $\omega$ such that there is no measure $\nu$ on $M$ satisfying $$\int f \omega = \int f d\mu$$ for all compactly supported smooth ...
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### How do you compute a complex exterior derivative?

The context is deriving cauchy riemann equations using green's/stoke's theorem. The function is the complex function $f(x,y)=u(x,y)+iv(x,y)$ with associated one form $u(x,y)dx+iv(x,y)dy$. Here is my ...
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### Finding the expression of a one form in a chart.

Given a one form on a manifold the formula I was given for finding its expression in a given coordinate chart is very strange and I dont understand it. I would appreciate if someone could give me a ...
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### Solution to Cauchy-Riemann Differential Equation of Compact Support

I'm working through Forster's $\textit{Lectures on Riemann Surfaces}$ and am struggling with the following problem: Suppose $g \in \mathcal{E}(\mathbb{C})$ is of compact support. Prove there is a ...
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### Formula of a two form for a parallelizable manifold

Let $M^n$ be a parallelizable manifold with the nowhere dependent vector fields $X_1,\ldots, X_n$ forming a basis for the tangent space at each point of $M$. The Lie brackets of these fields are ...
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### Poincare's lemma from PMA Rudin

It's the so called Poincare's theorem from Rudin's book. I read this theorem fully but I have 2 questions: 1) How did he conclude that equations (120) holds? What did he use in his reasoning? This ...
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### Why is the $1$-form ${dz\over z}$ on $\mathbb{C}^*$ closed?

Why is the $1$-form $\displaystyle\frac{dz}{z}$ closed in $\mathbb{C}^*$? In general, how to compute a complex one form's derivative? Thank you!