# Tagged Questions

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### Two isomorphisms of the space of vector fields/forms

Suppose you have a diffeomorphism $$\phi: M \to N.$$ As far as I understand, this gives rise to two distinct isomorphisms $$a : \mathcal A^1(M) \cong \mathcal A^1(N) :b$$ between the space of ...
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### Interpretation of $p$-forms

Let $M$ be a smooth manifold, let $C^{\infty}(M)$ be set of all smooth functions from $M$ to $\mathbb R$ and let $Vec(M)$ denote the set of all vector fields on $M$. A $1$-form on $M$ is a ...
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### Generalizing the Hopf invariant to arbitrary manifolds

I recently ran across a qual question about a generalization the Hopf invariant to smooth maps $f: M^{4n-1} \to N^{2n}$ between arbitrary closed connected oriented manifolds of the indicated ...
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### Poincare dual of unit circle

I'm trying to self-study Differential Forms in Algebraic Topology by Bott and Tu. I've come across this exercise: Show that the closed Poincare dual of the unit circle in $R^2-\{0 \}$ is zero, ...
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### Change of Coordinate Formula for Differential Forms

Let $M$ be a manifold, $x$ local coordinates on an open set $U$, $y$ local coordinates on an open set $V$. In addition, let $(x, \alpha)$ and $(y, \beta)$ be two induced bases for the common part of ...
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### A question on harmonic two-forms

Let $(M^4,g)$ be a closed Riemannian four-manifold with $b_2^+>0$ and $b_2^->0$, is it possible to find two harmonic two-forms $\alpha\in H^2_+(M)$ and $\beta\in H^2_-(M)$, such that ...
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### When is an exact 2-form harmonic?

Let $\alpha$ be an exact two-form, $\alpha=d\beta$ for some one-form $\beta$, when is $\alpha$ harmonic? By uniqueness of harmonic forms in cohomology classes, it cannot be harmonic?
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### De Rham cohomology, and forms on manifolds

In String Theory and M-Theory by Becker, Becker and Schwarz, they introduce a group, $$C^{p}(M)$$ which they denote the group of all closed $p$-forms on the manifold $M$. Furthermore, they state ...
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### show that $\omega$ is exact if and only if the integral of $\omega$ over every p-cycle is 0

Let $M$ be an oriented smooth manifold and $\omega$ a closed $p$-form on $M$. Show that $\omega$ is exact if and only if the integral of $\omega$ over every $p$-cycle is $0$. In particular, how to ...
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### Proving $d(f^*e_i')=0$

Let $f:M\to N$ be a differentiable map between two manifolds. $e_i$ is a basis vector of $N$ with respect to some chart and $e_i'$ its dual (i.e. $e_i'(e_j)=\delta_{ij}$). How do I prove the ...
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### Why 'closed differential forms' are called 'closed'?

As is well known a differential form $\omega$ is called closed differential form if it satisfies $\mbox{d} \omega = 0 \, \, (\ast) \,$ where $\mbox {d}$ is the exterior derivative. I think the ...
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### The first proof for Poincare lemma in history

How can I get a reference about the first proof of Poincare lemma in history? I already know some methods of proof, but I do want to know the original approach. Thanks for your help!
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### Are these two definitions of exterior derivative equivalent?

I saw two definition of the exterior derivative of a $k$-form $\omega$. First definition: $$(d\omega)_p(v_0,\ldots,v_k)= \sum_{i=0}^k(-1)^id(\omega(v_0,\ldots,\hat{v_i},\ldots,v_k))_p(v_i)$$ Second ...
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### Diffeomorphic riemannian manifolds and volume forms

Maybe the question will be stupid, but I'm a beginner in riemannian geometry... We have two riemannian manifolds $(M,g)$, $(\overline M,\overline g)$ and a diffeomorphism $F:M\rightarrow\overline M$ ...
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### Closed not exact form on $\mathbb{R}^n\setminus\{0\}$

I'd like to construct a closed but not exact $n-1$-form $\omega$ on $\mathbb{R}^n\setminus\{0\}$ in analogy to the winding form: $$\frac{x~dy-y~dx}{x^2+y^2}$$ I think something like ...
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### Highest DeRahm Cohomology

Let $X$ be a $C^\infty$ manifold, compact oriented and connected of dimension $n$. How do you prove that the integration map $$\int_X: \omega \mapsto \int_X \omega$$ from $H^n_{DR}(X)$ to ...
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### Pullback of differential form on the double covering

On a double covering there is a differential form $\omega$ arises by the pullback of a differential form under the projection iff it is the pullback of $\omega$ under the map $i$, where $i$ is the map ...
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### Averaging differential forms

Let $M$ be a manifold with a circle action, i.e. with a 1-parameter group of diffeomorphisms $\phi_t:M\to M$ of period 1. I think of the average of a differential form $\omega \in \Omega^n(M)$ with ...
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### Poincaré Lemma Contractible Hypothesis

PoincarĂ©'s Lemma is often stated as saying that a closed differential form on a star-shaped domain is exact. More generally, it is true that a closed differential form on a contractible domain is ...