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

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1answer
570 views

Pullback distributes over wedge product

I'm looking to prove that the pullback of a smooth function distributes over wedge product, i.e. $$\varphi^*(\omega \wedge \eta) = \varphi^* \omega \wedge \varphi^* \eta. $$ Here the question is ...
3
votes
1answer
54 views

1-form on $S^n$ with non-degenerate differential.

How it can be solved? Whether there is a $1$-differential form $w$ on $S^n$, such that $dw$ is non-degenerate on $T_aS^n$ for each $a \in S^n$?
3
votes
1answer
147 views

Independence of coordinate charts in the definition of the order of a pole of a meromorphic 1-form on a Riemann surface

I am currently reading Forster. Let $X$ be a Riemann suface, and $Y$ an open subset of $X$. Assume we have a meromorphic 1-form $\omega \in \Omega(Y \setminus \{a\})$ with a pole at $a \in Y$. One ...
3
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1answer
157 views

Why is Schouten-Nijnhuis bracket trivial on Poisson cohomology?

For a commutative algebra $A$, let a biderivation $P$ be called a Poisson structure if $[[P,P]]=0$ (the bracket is Schouten-Nijenhuis). Then one obtains a complex of multiderivations with $[[P,{}]]$ ...
3
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1answer
66 views

If $\alpha \wedge \omega = 0$ then $\alpha = f \omega$ for some $f$

Question: Let $\alpha$,$\omega$ be $1$-forms of class $C^1$ in $\mathbb R^3$. If $w(x) \neq 0$, for every $x \in \mathbb R^3$ and $\alpha \wedge \omega = 0$. Then $\alpha = f\omega$, where $f : ...
3
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0answers
51 views

Suggestion for reference book for differential forms, differentiable manifolds and other topics

I am currently taking a course on multivariable calculus and our professor is following the book by Do Carmo: Differential forms and applications. I feel the text is too rigorous, which I really ...
3
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1answer
134 views

Vectors, Forms, Multivectors, and Tensors

In researching some of the ways that vectors (and vector fields) generalize I find that there are apparently many different objects that generalize them -- matrices, differential forms/ covectors, ...
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1answer
65 views

An exercise on differential forms

Let $\omega$ be a $1$-form in $U \subset \mathbb{R}^2$. A local integrating factor in $p$ for $\omega$ is a function $g: V \rightarrow \mathbb{R}$ defined in a neighbourhood $V$ of $p$ such that ...
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0answers
31 views

The lie bracket as a limit

Let $M$ be a manifold and $f_t$ be the monoparametric group of local transformations generated by a vector field $X$. Suppose $w$ is a $k$-form and $x_1,\dots,x_k$ are vector fields. How can we see ...
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107 views

Differential Form Pullback Definition

I'm having some difficulty following how Spivak (Calculus on Manifolds) has induced the pullback on page 89. From reading elsewhere online it seems convention is to define the induced map of the ...
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0answers
139 views

Product of Two Orientable Manifolds is Orientable

I am trying to show that following: Let $M$ be an oriented smooth manifold of dimension $m$, and $N$ be an oriented smooth manifold of dimension $n$. Then $M\times N$ is orientable. Let ...
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1answer
55 views

Can We Write the Differential in Terms of Covectors?

Let $f:\mathbf R^n\to \mathbf R$ be a smooth map. We can write $df:T\mathbf R^n\to \mathbf R$ neatly as $$ df = \sum_{i=1}^n(\partial f/\partial x_i) dx_i $$ For a function $f:M\to \mathbf R$ defined ...
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0answers
147 views

Second structural equations in lorentzian space $\Bbb L^3$.

I'm rewriting O'Neill's "Elementary Differential Geometry"'s section on connection forms in Lorentz-Minkowski space $\Bbb L^3$, and I'm having trouble proving the second structural equations $${\rm ...
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0answers
136 views

Second fundamental form of a graph of a function using frame fields

I am trying to recover the second fundamental form $II$ for the graph $\Gamma(f)$ of a smooth real function $f$ in the Euclidean space $\mathbb{R}^3$ using first order frame fields. I have worked with ...
3
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1answer
86 views

Integration on $\mathbb{R}^n$ in terms of differential forms

One defines integration on a smooth manifold as follows: First define $\int_M \omega$ when $\omega$ is supported on a single coordinate chart by pulling back to $\mathbb{R}^n$ an integrating there, ...
3
votes
1answer
36 views

Global n-form on Calabi-Yau

I am now reading these lectures by Stefan Vandoren on complex geometry. Everything is fine in general, hiwever I am confused with how he defines 1-form on a Calabi-Yau 1-fold (or 2-form on CY$_2$). ...
3
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2answers
74 views

Using Stokes's theorem to calculate a value of integral

Use Stokes's theorem to calculate the integral $$I= \int_\Gamma (x^2+2y)dx+(y+z)dy+(z^2+x^2)dz$$ where $\Gamma$ is the boundary of $$\gamma=\left\{ (x,y,z):3x+y+3z=3,x\ge0,y\ge0,z\ge0\right\} $$ ...
3
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1answer
118 views

de Rham cohomology of $\mathbb{R}P^n$ via action by $SO(m+1)$

In lecture, my teacher proved the theorem that given a smooth $G$-action by a compact, connected Lie group on a manifold $M$, the de Rham cohomology of the $G$-invariant differential forms $H^p_G(M)$ ...
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0answers
117 views

Definition of differential forms

I am reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. The authors introduce some definitions about differential forms at the beginning of Section 2.2. These ...
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1answer
69 views

Cohomology of volume forms

If g and h are Riemannian metrics on the same manifold, say both of volume 1, then it follows (I guess from Poincaré duality) that their volume forms dvol_g and dvol_h are cohomologous. Question: is ...
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141 views

Intuitive meaning of $L_X\omega$

Suppose $\alpha$ is a one form and $\omega$ is two form such that $$\alpha= \omega(X,.)\text{ for some } X$$ Then what does intutive meaning of the following expression $$L_X\omega= \omega$$
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3answers
81 views

Integral depending on a path?

I need to check whether differential form $\omega$ has, in the domain $G$, such property that it's integral doesn't depend on path. In my exercise: $\omega = \frac{ydx -xdy}{x^2+xy+y^2}$ and $G= R^2 ...
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0answers
111 views

Is this a trivial Stokes exercise?

I'm working on this exercise from an old Spivak Differential Geometry book which I must be misunderstanding. The question reads: Let $M$ and $N$ be compact $n$-dimensional manifolds with boundary and ...
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0answers
58 views

Questions on a sub-bundle of $\mathbb R^3\setminus \{0\}$

Let us consider spherical coordinates $(r,\theta,\phi)$ on $\mathbb R^3$ and the manifold $M:=\mathbb R^3 \setminus \{0\}$. Let us consider the 1-form on $M$ $$ \omega = zdz ...
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0answers
100 views

Hodge decomposition on a manifold with a nontrivial connection

I am familiar with the notion of Hodge decomposition of an arbitrary differential form into an exact form, a co-exact form, and a harmonic form. Given a curved space with a connection, could you ...
3
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0answers
98 views

The canonical (1,1)-form on a compact Riemann surface gives locally a subharmonic function

Let $X$ be a compact connected Riemann surface of genus $g>0$. We have the so called canonical (1,1)-form $\mu$ on $X$ defined as follows. Choose an orthonormal basis $(\omega_1,\ldots, \omega_g)$ ...
2
votes
1answer
205 views

A non-vanishing one form on a manifold of arbitrary dimension

So the problem I have is: Let $\theta$ be a closed 1-form on a compact Manifold M without boundary. Further suppose that $\theta \neq 0$ at each point of M. Prove that $H^{1}_{dR}(M)\neq 0$. The ...
2
votes
2answers
180 views

Coordinate free definition of orientable manifold

Analysis and Algebra on Differentiable Manifolds, first edition, chapter 7.3.1, defines orientation on a vector space and orientable manifolds. There is a part of the definition that I do not ...
2
votes
2answers
163 views

Showing that $\int_{c} \omega =0$ when $\partial c =0$

Let $\omega$ be a $k$-form on $\mathbb{R}^n$ and suppose that $\omega=d\alpha$ for some $(k-1)$-form $\alpha$. Show that, for any singular $k$-cube $c$ on $\mathbb{R}^n$ with $\partial c=0$, ...
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4answers
934 views

Classic example of a non exact form

Let $\dfrac{xdy-ydx}{x^2+y^2}$ be a 1-form defined in $\mathbb{R}^2\backslash\{0\}$. Where can I find a detailed proof that it is not exact? I would prefer a proof that doesn't use results about ...
2
votes
1answer
78 views

If $d(f\omega)=0$, then $\omega \wedge d(\omega)=0$

Here's the question: Suppose that $\omega$ is a $k$-form on an open set $U$ of $\mathbb{R}^n$ and $f:U \to \mathbb{R}$ is a $C^\infty$ function such that $f(x) \neq 0$, for all $x \in U$, and ...
2
votes
2answers
55 views

Under what conditions can a general 2-form be written as a wedge product of two 1-form

Assume we have a 2-form $\omega \in \Lambda^2\mathbb{R}^n$. It is usually stated one can write $$\omega = \alpha \wedge \beta,$$ with $\alpha, \beta \in \Lambda^1\mathbb{R}^n$ only for $n < 4$. How ...
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3answers
275 views

Differential forms and double improper integral

Can someone suggest me a list of book or workbook with examples and solutions on differential forms and double improper integral?
2
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1answer
68 views

Complex differential forms on $CP^n$

Why Complex projective spaces don't admit some differential forms? To be more specific, I know that the space of complex forms is decomposed as direct sum of holomorphic and anti-holomorphic part; ...
2
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3answers
137 views

How to understand $d^2=0$ in differential form?

How to understand $d^2=0$ in differential form without a simple proof from the definition?
2
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1answer
85 views

Construct tensors from differential forms?

Let $(M,g)$ be a Riemannian manifold, differential forms are defined using tensors, could we define a tensor using a differential form? For example, if $\omega$ is a two-form on $M$ which is expressed ...
2
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1answer
192 views

diffeomorphism preserve a volume form

Let $\omega_1$, $\omega_2$ two volume form on a compact manifold $M$, we know that there exists a never-vanishing function $f$, s.t. $\omega_1=f\omega_2$. If $h$ is a diffeomorphism $M \to M$ ...
2
votes
3answers
253 views

Simple criteria for “closed $\Longrightarrow$ exact”

In determining whether a closed form is an exact form, there is a lot of differential geometry definitions etc. that come in. I'm interested: what is the dummy, Calc III version of when closed implies ...
2
votes
2answers
206 views

p-forms as multilinear maps

I'm studying differential geometry and am learning about differential forms. We have a very intuitive and simple way to understand 1-forms as linear maps on from the tangent space to the base field, ...
2
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1answer
348 views

Basic computations with differential forms

I've never had any experience with differential forms before, so I'm trying to work through a couple of examples to see if I understand what's going on. I think I understand what I've been doing so ...
2
votes
1answer
46 views

How do you integrate a $0$-form?

I am trying to learn about differential forms. I know that A $0$-form is just a scalar function $f$. My question is: How is the integral of a $0$-form defined? In particular, if $f$ is a function of ...
2
votes
1answer
41 views

Locally flat manifold from Frobenius, differential forms approach

It is a known fact that a Manifold $M$ is locally flat if and only if the Riemann curvature tensor (Levi-Civita connection) vanishes. To show the if part is easy, but I am trying to follow the details ...
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3answers
157 views

Divergence theorem in complex analysis

I am revisiting my understanding of integration by parts in several complex variables, but I have run into an apparent contradiction. This shows my understanding is flawed, which is somewhat ...
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1answer
214 views

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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3answers
74 views

Whats the connection between formss and vector fields?

I heard someone talking about how vector fields are the kernels of forms. Can someone give me a detailed explanation about how this works? Thanks.
2
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1answer
148 views

What does $\Omega^\bullet(M)$ mean?

What does $\Omega^\bullet(M)$ mean? I know that $\Omega^k(M)$ is the set of all differential k-forms. Thanks in advance!
2
votes
1answer
322 views

Wedge product with a non-degenerate form

Let $\alpha$ be a non-degenerate form in $\Lambda^k(V)$ for some vector space $V$, $\dim V = n$. (Here non-degenerate means that if $x\in V$ is nonzero, then $(y_1 , ... , y_{k-1}) \mapsto \alpha(x , ...
2
votes
1answer
67 views

Simplifying the Kahler form

In the link here, p.4, it says that, given a fundamental 2-from $\mathcal{K}$ $$\mathcal{K}=\frac{\sqrt{-1}}{2\pi}g_{i\bar{j}}dz^i\wedge d\bar{z}^{\bar{j}},$$ a manifold is said to be Kahler if this ...
2
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1answer
37 views

Volume Forms Induced by Embedding

Let $(M, g)$ be a Riemannian Manifold of dimension $d$, $g$ naturally gives rise to an invariant volume form $V_M \in \Omega^d(M)$. Let $\Sigma$ be a smooth embedded submanifold of dimension $d-1$ in ...
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1answer
38 views

Can every 2 form be represented as a linear combination of these specific two forms?

This question is Question 2 from Ilka's book on page 8. The first part is to prove that every $\omega^2\in \Lambda^2(V^{\ast})$ can be represented as \begin{equation*}\tag{1} ...