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

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2
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1answer
89 views

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 ...
2
votes
2answers
49 views

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 ...
6
votes
1answer
60 views

Deciding whether a form in the exterior power $\bigwedge^k V$ is decomposable

Let $V$ be a vector space and $\bigwedge^kV$ be the $k$th exterior power. I'm trying to find a condition that characterizes when an element $\omega \in \bigwedge^kV$ is decomposable in the sense that $...
2
votes
1answer
39 views

Show that a k-form can be expressed as wedge product

I'm trying to show that given a $1$-form $\omega$ and a $k$-form $\alpha$ such that $\alpha \wedge \omega = 0$ then there exists a $(k-1)$-form $\beta$ such that $\alpha = \omega \wedge \beta$. I'm ...
3
votes
1answer
56 views

Differentiating the pull-back of a one-form

Let $\Omega$ be an open subset of a vector space $V$ and let $\alpha\colon \Omega\to V^*$ be a one-form on $\Omega$. Assuming that $\alpha$ is differentiable, then for any $x\in \Omega$, $D\alpha (x)\...
0
votes
1answer
62 views

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 ...
1
vote
1answer
39 views

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!
1
vote
1answer
46 views

Differential forms on a point

For the proof of Poincaré lemma, it's essential to evaluate $\Omega^p(*)$ where $*$ is zero dimensional manifold and $\Omega^p$ is a collection of all $p$-forms on given manifold. Clearly, $\Omega^0 (*...
1
vote
1answer
36 views

Intersection of kernels of linearly independent smooth 1-forms on $\mathbb R^n$

I'm trying to solve the following problem: Let $\omega^1,\dots,\omega^k$ be smooth $1$-forms on $\mathbb R^n$ that are linearly independent at each point of $\mathbb R^n$. For $p\in\mathbb R^n$, ...
0
votes
0answers
28 views

Definition of exact form

That's the definition of exact form in $E$. But if we look at the definition 10.18 we see that $\lambda \in C'$ but Rudin skip this condition. Can anyone please explain this moment
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0answers
34 views

Stokes' Theorem from PMA Rudin. Confusing moment with simplex

I am reading the proof of Stokes' theorem from PMA Rudin but one moment seems to very weird. Why Rudin considers the case when $\sigma=[0,\mathbf{e}_1,\dots, \mathbf{e}_k]$? After all $\sigma$ may ...
2
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0answers
39 views

Showing that $\|.\|$ is a norm of the space of 1-forms $\Omega^1(U)$, where $U\subset\mathbb{R}^n$.

Let $U\subset\mathbb{R}^n$ and let $\Omega^p(U)$ denote the vector space of $p$-forms ($p\in\mathbb{N}$). Define the isomorphism $\Phi:\Omega^{1}(U)\to\Omega^{n-1}(U)$ as $$\Phi\left(\omega=\sum_{i=1}^...
0
votes
2answers
67 views

$\mathbb{P}_{\mathbb{C}}^3$ is not isomorphic to $S^2 \times S^4$

I have been trying to solve this exercise given by my prof. The hint is to show that every $2$-form $w$ on $S^2 \times S^4$ is s.t. $w \wedge w = 0$, while this is not true in case of $\mathbb{P}_{\...
1
vote
1answer
33 views

Compact Poincaré dual of $S^{n-1}$ in $\mathbb{R}^n \backslash \{0\}$

I have been asked to find a sub manifold $S \subset M := \mathbb{R}^n \backslash \{0\} $ s.t. its compact Poincaré dual is a basis of the first cohomology group $H^1_c(M) = \mathbb{R}$. Now, $S$ must ...
0
votes
1answer
29 views

Symplectic form on $T^ ∗X$

If $\mu$ is any closed 2-form on a manifold $X$, then $d\alpha_X + π^∗\mu$ is also a symplectic form on $T^ ∗X$, where $\alpha_X$ is tautological one-form and $\pi:T^*X\to X$ is projection. In fact, ...
4
votes
0answers
69 views

Clifford, $p$-forms and spinors

I'm trying to understand the paper by Atiyah, Hitchin and Singer called: ''Self-duality in four dimensional Riemannian geometry", available here. I'm stuck at the point where it explains how the $p$-...
3
votes
2answers
97 views

How to understand the notion of a differential of a function

In elementary calculus (and often in courses beyond) we are taught that a differential of a function, $df$ quantifies an infinitesimal change in that function. However, the notion of an infinitesimal ...
4
votes
1answer
37 views

Finding all $2$-forms in the right half-plane that are invariant under glide transformations

I'm trying to find all 2-forms $\omega$ that are invariant under glide transformations in the right half-plane, $X = \{ (x,y) \in \mathbb{R}^2 : x > 0\}$. To do this, we can write the vector field ...
1
vote
1answer
55 views

Unit square as union of two simplexes

? If Rudin regarded $\Phi$ as a function of $2$-forms and suppose that $\omega=f(\mathbf{x})dx_{i_1}\land dx_{i_2}$ is $2$-form on $\mathbb{R}^m$ then $$I_{\Phi}(\omega)=\int \limits_{\Phi}\omega=\...
2
votes
0answers
31 views

Boundary of oriented $k$-simplex from PMA Rudin

But paragraph which I marked by red line seems to me confusing. Let $k=3$ then $\sigma=[\mathbf{p}_0,\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}_3]$ and $\partial \sigma=[\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}...
0
votes
1answer
62 views

Finding A 1-form on $R^2 - {\{(0,0)}\}$

I want to find a 1-form on $R^2 - {\{(0,0)}\}$ such that $w(Y) = 0$ and $w(X) = 1$. Here, $$X = -y\frac{\partial }{\partial x} + x\frac{\partial}{\partial y}\ \text{and}\ Y = x\frac{\partial }{\...
2
votes
1answer
56 views

A necessary and sufficient condition for the admittance of integrating factor

Let $\omega$ be a smooth 1-form on a smooth manifold $M$. A smooth positive function $\mu$ on some open subset $U\subset M$ is called an integrating factor for $\omega$ if $\mu\omega$ is exact on U. ...
1
vote
2answers
42 views

Specific example of integrating a 1-form over a curve

I was given the following definition in my course but no corresponding examples: Supppose $\gamma:[a,b]\rightarrow{M}$ is a smooth curve and $\omega$ a 1-form on $M$ (so $\omega:M\rightarrow{T^*M}$). ...
1
vote
1answer
58 views

Affine chains from PMA Rudin. Confusing examples

I understood the definition of affine $k$-chain and that he defines $\int \limits_{\Gamma} \omega$ as $(82)$. But I can't understand the last two above examples. What does they mean? Can anyone ...
0
votes
0answers
26 views

Equivalent conditions for $\mathfrak{F}$ to be a differential ideal

Heres the question: Let $\mathfrak{F}$ be an ideal of forms on a manifold $M$ locally generated by $r$ independent $1$-forms. Say $\mathfrak{F}$ is generated by $\omega_1,\ldots, \omega_r$ on $U$. ...
0
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0answers
28 views

Let $f : \mathbb{R}^n \to \mathbb{R}^n$, suppose that $\mu = dx^1\wedge\ldots\wedge dx^n$, then $f^{\ast}\mu = \det (df)\mu$

Let $f : \mathbb{R}^n \to \mathbb{R}^n$, suppose that $\mu = dx^1\wedge\ldots\wedge dx^n$, then $f^{\ast}\mu = \det (df)\mu$. I am trying to prove this. I made several low dimensional cases and ...
2
votes
1answer
39 views

Prove the exterior derivative of the following (n-1) form is zero

Let $\omega(x)=\frac{1}{{\parallel x \parallel}^n}\displaystyle\sum_{i=1}^{n}(-1)^{i-1}x_{i} dx_{1} \wedge \dots \wedge \widehat{dx_{i}} \wedge \dots \wedge dx_{n}$ be a differential $(n-1)$ form on $\...
1
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1answer
55 views

Confusing moment in Theorem 10.27 from PMA Rudin

Theorem 10.27 If $\sigma$ is an oriented rectilinear $k$-simplex in an open set $E\subset \mathbb{R}^n$ then $$\int \limits_{\overline{\sigma}}\omega=\varepsilon\int \limits_{\sigma}\omega \qquad (81)$...
0
votes
2answers
58 views

Example of exact form

Consider the differential 1-form $\omega = ydx+dy$. I need to show that this is not exact, and find an example of a function $G(x,y)$ such that $G\omega=G(x,y)(ydx+dy)$ is an exact form. I have done ...
0
votes
1answer
43 views

Theorem 10.22 from PMA RUdin

We know that $(dy_I)_T=dt_{i_1}\land \dots \land dt_{i_k}$ and using definition 10.18 we get $$d((dy_{I})_{T})=d1\land dt_{i_1}\land \dots \land dt_{i_k}=0$$ since $dc=0$ for any $c\in \mathbb{R}^1$. ...
1
vote
1answer
36 views

Derivative of $0$-form

Rudin states that $1$-form $xdy$ is not the derivative of any $0$-form. By contradiction, suppose that that exists $0$-form whose derivative is $xdy$. Then $f\in C'$ and $df=xdy$. But $d^2f=dx\land ...
4
votes
1answer
40 views

Noncanonical isomorphism of spaces of differential forms

Let $\pi: V \to M$ be a smooth $n$-dimensional vector bundle over $M$. Are the spaces of differential forms $\Omega^i(V)$, $\Omega^i(V^*)$ noncanonically isomorphic? If so, how do I see this? Is there ...
2
votes
2answers
71 views

Frank Warner's definition of the Hodge star

Frank Warner's book, chapter 2, excercise 13 states the following: If $V$ is an oriented inner product space ($n$ dimensional) there is a linear map $\ast \colon \Lambda (V) \to \Lambda (V)$, ...
1
vote
1answer
85 views

Does such property exist for differential forms?

I am studying differential forms from Rudin's PMA. Here's the definition of $k$-form Also he proves the anticommutative relation: $dx_1 \land dx_2=-dx_2\land dx_1$ Does the following expressions ...
1
vote
1answer
51 views

Method to calculate the de Rham cohomology of $\mathbb{R}\mathrm{P}^n$

I'm trying to follow through a method to calculate the de Rham cohomology groups of $\mathbb{R}\mathrm{P}^n$ from the de Rham cohomology groups of $S^n$. I'm trying to show that differential k-forms ...
0
votes
1answer
29 views

Are curvature forms in complex line bundles symplectic

I know that the curvature form $F_\nabla$ of a connection $\nabla$ in a complex line bundle $L \to B$ is presymplectic (i.e. antisymmetric and closed). Does it also have to be non-degenerate, i.e ...
0
votes
0answers
15 views

How to show that a tensorial field,$t$, $k$-times covariant over a differential variety $M$ is differentiable by its components

How can i show that a tensorial field,$t$, $k$-times covariant over a differential variety $M$ is differentiable if and only if the following functions (The components of t) are differentiable.: $$t_{...
0
votes
0answers
46 views

vanishing of differential form on connected compact manifold

Let $M$ be a $n$ dimensional compact connected manifold. Let $\alpha$ be a differential form of degree $n$ such that $$ \int_{M} \alpha = 0 $$ then I would like to show that $M$ vanishes at at least ...
2
votes
1answer
45 views

What is the skew-symmetric part of the covariant derivative of a one-form?

This is a followup question to here. Let $E \to M$ be a vector bundle with connection $D := \nabla$. Extend $D$ to $E^*$ and $\text{Hom}(E, E)$. Let $E = TM$ here, and suppose that the torsion is $0$:...
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votes
0answers
38 views

Integration of forms on manifolds

If you have a $n$-form $\omega$ on $\mathbb{R}^n$, then $\omega = f \mathop{}\!\mathrm{d}x_1 \wedge \dots \wedge \mathop{}\!\mathrm{d}x_n$ locally. Integrating $\omega$ is easy now - let's assume $\...
3
votes
1answer
61 views

Zeroes of exact differential forms on compact manifold

Let $M$ be a $n$ dimensional compact differentiable manifold. I would like to show that any exact differential form of degree $n$ vanishes at at least one point. I think it is a generalization of the ...
2
votes
1answer
31 views

Prove that $p-$form $\omega$ on $M\times N$ is $\delta \pi(\alpha)$ iff $i(X)\omega=0$ and $L_X\omega=0$

Let $M$ be connected and let $\pi:M\times N \rightarrow N$ be the natural projection. Prove that $p-$form $\omega$ on $M\times N$ is $\delta \pi(\alpha)$ for some $p-$form $\alpha$ on $N$ if and only ...
2
votes
2answers
52 views

Integrate ${xdy+ydx} \over {x^2+y^2}$ on the circle $(x-1)^2+(y-1)^2=1$

I tried to integrate ${xdy+ydx} \over {x^2+y^2}$ on the circle $(x-1)^2+(y-1)^2=1$ counterclockwise. Used Grin's theorem, then went to polar cordinates but can't integrate the expression I got. So I ...
0
votes
1answer
30 views

Showing that the exterior derivative of a 1-form is 0.

The question is $$ Let\quad f : \Bbb R \to \Bbb R$$ $$\omega = f(||\mathbf x||)(\sum_{i=1}^n x_{i}dx_{i}) \in \mathcal A^1(\Bbb R^n) $$ $ (a) $ Assuming f is differentiable, prove that $d\omega = 0$ ...
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0answers
31 views

Elementary properties of diff. form from PMA Rudin

How he got $d\omega$ in the RHS of $(39)$? Or maybe it's a typo?
-1
votes
1answer
41 views

How am I supposed to answer this question?

I've got the following question: The first part of this question I can do. I've deduced that the DE has oscillatory solutions for all $\lambda > 16$; that the Eigenvalues are given by $$ \...
0
votes
3answers
48 views

Reference request: integration of *one*-forms along curves on a differentiable manifold.

Could somebody please direct me to a book/lecture notes with an introduction to integration of one-forms along curves in a differentiable/Riemannian manifold -- preferably leaning more towards ...
1
vote
1answer
31 views

The value of the integral of the curvature of a complex line bundle

I am trying to show that if $L \to M$ is a complex line bundle endowed with a connection $\nabla$, $F$ is the curvature form, and $S \in M$ a closed surface, then $\int \limits _S F \in 2 \pi \textrm ...
0
votes
1answer
43 views

Prove the pullback of the wedge product is the wedge product of the pullbacks.

Let $F:V \rightarrow W$ be a linear map. Show that $F^{\ast}(\omega \wedge \eta)=(F^{\ast}\omega) \wedge (F^{\ast}\eta)$ for all $\omega \in \Lambda^{p}(W) , \eta \in \Lambda^{q}(W)$. Where $F^{\ast}...
0
votes
1answer
38 views

Integral of Differential 1-form

Let $\omega$ be the closed $1$-form $\omega = \frac{xdy - ydx}{x^2 + y^2}$ and let $S$ be the unit circle and let $C = \{(x,y) : (x-3)^2 +y^2 = 1\}$. I'm trying to find $\int _S \omega$ and $\int ...