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

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3
votes
3answers
43 views
+50

How to find lagrangian submanifolds.

I am quite confused on the definition of a lagrangian submanifold $L$ of a symplectic manifold $(M,\omega)$. In particular, I read that $L \subset M$ is lagrangian iff the symplectic form field ...
1
vote
1answer
66 views

Proving $[L_X,i_Y]=[i_X,L_Y]=i_{[X,Y]}$

Let $X,Y$ be vector fields. $L_X$ is the Lie derivative and $i_X$ is the contraction of a $k$-form. I am really stuck on how you could prove the identity $[L_X,i_Y]=[i_X,L_Y]=i_{[X,Y]}$. Update: I ...
1
vote
1answer
49 views

Representation of $n$ form and $n-1$ form in local coordinates

Let $M$ denote a smooth $n$-dimensional manifold. (a) Let $\phi$ denote a smooth $n$ form which is nowhere zero. Show that every $x_{0} \in M$ has a neighborhood on which we can find smooth local ...
3
votes
2answers
81 views

Does $\omega \wedge \mathrm{d} \omega=0$ (where $\omega$ is a non-vanishing $1$-form) imply $\mathrm{d} \omega \in < \omega >$?

Let $\omega$ be a non-vanishing (for clarification: nowhere vanishing) smooth $1$-form on a smooth manifold $M$, if $\mathrm{d}\omega \wedge \omega =0$, do we already have $\mathrm{d}\omega= \sum a_i ...
0
votes
1answer
60 views
+50

Problem integrating in problem using the Poincaré Lemma

a) It is easy to show that $d\beta=0$. b) $\begin{align}\hat{\mathbb{X}}_t &= (\frac{\partial}{\partial t}\hat{\Phi}_t) \hat{\Phi}_t^{-1}) \\ &= (\frac{\partial}{\partial t}\hat{\Phi}_t) ...
0
votes
0answers
35 views

Pull-back of a one-form on a sphere.

Let $\iota: S^2 \to \mathbb{R}^3$ be the inclusion map and choose a chart $(U,f)$ on $S^2$, where $U=\{(x,y,z)\in \mathbb{R}^3: z>0\}$ and $$f: U \to \mathbb{R}^2,$$ $$ (x,y,z)\mapsto (x,y). $$ I ...
2
votes
1answer
83 views

Moving frame in a semi-Riemannian manifold

Can someone point me some reference for the moving frame theory in semi-Riemannian manifolds, using differential forms? In special, I'm looking for a version of Cartan's structural equations. I've ...
1
vote
0answers
38 views

Poincaré Lemma problems and computing contractions in an economical way

Let $x=(A,B,C,D)$ be coordinates on $\mathbb{R}^4$. $\displaystyle \beta = \frac{(AdB-BdA)\wedge(dC \wedge dD)+(dA \wedge dB)\wedge(CdD-DdC)}{(A^2+B^2+C^2+D^2)^2}$ I would like to compute ...
0
votes
1answer
32 views

Show that $d\beta=0 \iff p=n/2$

Let $\beta$ be the $(n-1)$-form on $\mathbb{R}^n \setminus \{0\}$ given by $\displaystyle \beta = \sum_{i=1}^{n}(-1)^{i-1}\frac{x^i dx^1 \wedge dx^2 \wedge \dots \wedge \hat{dx^i} \wedge \dots ...
3
votes
1answer
29 views

Wedge product descend to the cohomology

I found this statement in Raoul Bott "Differential Forms in Algebraic Topology": "Because the wedge product is an antiderivation, it descends to cohomology." Apparently this meant to be really obvious ...
4
votes
0answers
104 views
+50

Use of Poincare Lemma in solving $\nabla \times \textbf{A}(\textbf{r})=\frac{\textbf{r}}{r^3}$

Let $U = \mathbb{R}^3 \setminus \{(0,0,z) \}$ (ie $\mathbb{R}^3$ with the $z$-axis removed ) and consider $\beta$ on $U$ given by $\displaystyle \beta = \frac{x dy \wedge dz + y dz ...
0
votes
1answer
61 views

Identity concerning Lie derivative of $k$-form $\omega$

Let $X$ and $Y$ be vector fields on $\mathbb{R}^n$. Show that for $\omega$, a $k$-form on $\mathbb{R}^n$, $(L_XL_Y-L_YL_X)\omega=L_{[X,Y]}\omega $. I try using Cartan's magic formula and get that ...
1
vote
0answers
53 views
+50

Identity of the pushforward of a vector field using a Jacobi bracket.

Let $Z(u,v)$ be the vector field $Z(u,v)=(u^2+u,v^2+v)$, let $\Gamma_t$ denote its flow. I have shown that $[X,Z]=Z-X$. Show that $(\Gamma_t)_*X=e^{-t}X-(e^{-t}-1)Z$. Could someone please show me ...
1
vote
1answer
25 views

Green's operator, differential forms

In "Foundations of Differential Manifolds and Lie Groups" by Frank Warner on page 225 there is defined Green's operator: $G: E^p(M) \rightarrow (H^p)^{\perp}$ by setting $G(\alpha)$ to equal the ...
2
votes
0answers
39 views

Pullback of a 1 form on the circle

Q: Let $M$ be a smooth compact manifold, and suppose there is a smooth map $F:M \rightarrow S^{1}$ whose derivative is non-zero at every point. Prove that the de Rham cohomology space $H^{1}(M)$ is ...
1
vote
1answer
29 views

Finding the Lie derivative of a 2-form exercise

Let $\beta=-x dx \wedge dy + ydy \wedge dz$. The vector field is $X=(y,0,z)$.Find the Lie derivative. I try that $\begin{align}L_X \beta &=L_X (-x dx \wedge dy + ydy \wedge dz) = -x L_X(dx ...
0
votes
1answer
19 views

Identity about composition of the push forward of diffeomorphisms

I am able to do part a) and I believe it should be used in solving part b). I think that for part b) we should that $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$, $G: \mathbb{R}^n \rightarrow ...
1
vote
1answer
41 views

$d(\beta \wedge d\beta)=0$ if $k$ is even.

Let $\beta$ be a $k$-form. Show that $d(\beta \wedge d\beta)=0$ if $k$ is even. I get that $d(\beta \wedge d\beta)=d\beta \wedge d \beta + (-1)^k\beta \wedge d^2\beta=d\beta \wedge d \beta$. Why ...
0
votes
1answer
21 views

Exponential of a 2-form

What does $e^{\omega}$ means when $\omega$ is a $2$-form? Is it a $2$-form again? If it is a $2$-form, is its definition $\displaystyle e^{\omega}(u,v)=\sum_{n=0}^{\infty} \frac{\omega(u,v)^n}{n!}$?
3
votes
1answer
53 views

Inner product, differential forms and surfaces (Stokes' theorem)

I'm trying to understand how do you get the Kelvin-Stokes theorem \begin{equation} \int_{S} (\nabla\times \omega) \cdot \mathrm{d}S = \int_{\partial S} \omega \cdot \mathrm{d}r \end{equation} from the ...
2
votes
3answers
98 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?
12
votes
1answer
201 views

Proof of holomorphic Lefschetz fixed point formula using currents in Griffiths and Harris

I am trying to understand the proof of the Holomorphic Lefschetz fixed point formula on page 426 in Griffiths and Harris. However, I find their use of currents extremely confusing. They seem to go ...
0
votes
1answer
29 views

Can you switch the order of the determinants when changing variables using the Jacobian?

Let say we're changing the variables and we use the Jacobian to do this. Lets say we integrate in respect to $u$ and $v$, does it matter if we set up the integral like ...
2
votes
2answers
82 views

Differential forms on $S^1$

I'm reading this old question and there are some things I don't understand. For example, why in the case of $S^1$ can every $1$-form be written in the form $f(\theta)d\theta=c d\theta+dg(\theta)$ ...
2
votes
2answers
38 views

Wedge product computation

Let $\omega \in \Omega^{2}(\mathbb{R}^{2n})$ be the $2$-form $\omega=dx^{1} \wedge dx^{2} + dx^{3} \wedge dx^{4} + \dots + dx^{2n-1} \wedge dx^{2n}$. I want to compute the wedge product of $\omega$ ...
3
votes
1answer
114 views

Coordinate-free definition of integration of differential forms?

Let $\omega$ be an $n$-form on an oriented $n$-manifold $M$. To integrate $\omega$, we choose an atlas $(O_\alpha, (x^1_\alpha,\dots, x^n_\alpha))_\alpha$ for $M$ and a partition of unity ...
1
vote
0answers
43 views

How to Interpret Exterior Derivative as Infinitesimal

In Riemann Integral, one can intuitively interpret $dx$ as infinitesimal, and it makes sense, but in differential forms, it lost this interpretation, is there a way to make connection between these ...
2
votes
2answers
48 views

Arc Length and Differential Forms

Suppose $\gamma$ is circle in $\mathbb{R}^3$ defined by coordinates $\begin{pmatrix}r\cos\theta\\r\sin\theta\\0\end{pmatrix}$, and function $F: \gamma \rightarrow \mathbb{R}^3$ is defined by ...
1
vote
0answers
56 views

double Hodge star operator

$*$ is the Hodge star operator acting on differential $k$-forms on $\mathbb{R}^{n}$ as below: $$*:Λ^k(\mathbb{R}^{n}) \to Λ^{n-k}(\mathbb{R}^{n})$$ $$\alpha \wedge (\star \beta) = \langle ...
1
vote
0answers
35 views

Wedge product of differential forms

I'm trying to grasp the notation and concept of wedge products(, and tensors as well). In my lecture notes, the following expansion/notation for a $(n,r)$-tensor is used: In a basis $\left\{ ...
0
votes
1answer
57 views

Canonical bundle of a fibered product

Let $f: X \to Z$ and $g: Y \to Z$ be smooth morphisms of smooth projective varieties. Consider the fibered product \begin{array}{ccc} X \times_Z Y &\stackrel{\tilde{f}}{\longrightarrow}& Y\\ ...
2
votes
1answer
34 views

exterior product of forms is exact.

I don't know what to do to prove the following statement: Let $U \subset \mathbb R^n$ be an open set and let $\alpha$ be a $k$-form on $U$ and $\beta$ be an $l$-form on $U$. Suppose both $\alpha, ...
1
vote
0answers
40 views

Zeros of $f$ in a disk

If $f$ holomorphic in a domain $U$ and $f(z)\neq 0$ for all $z\in U$ then every zero of $f$ is such that $f(q)=0$ and $\det(Df_{p})>0$. Using that I have to prove that if $f$ keeps that conditions ...
0
votes
1answer
41 views

Differential forms theorem reference request

Let $f: A\subset\mathbb{R}^n\rightarrow A$ be smooth ($A$ not necessarily open) and homotopic to the identity map ${\rm id}_A$. If $s^k$ is a singular $k$-chain with image set $A$ and such that ...
1
vote
0answers
34 views

Pull back of a vector representing a 2-form in $\mathbb R^3$

Let $\omega$ be a $2$-form defined in $\mathbb R^3$. I know that it can be represented by a vector field $\xi$ in such a way: $$ \omega_x (v,w) = \xi(x)\cdot (v\times w) $$ ($x,v,w \in \mathbb R^3$ ...
1
vote
0answers
45 views

Closed 1-forms implies exact 1-forms

I have two problems, the first one I think I´ve prove it, but I have problems on the second one. Let $\omega$ a closed $1$-form defined on a open $U\subset \mathbb{R^2}$ and let ...
0
votes
1answer
25 views

Integration of an equation in terms of differential forms

Suppose we have the equation in terms of differential forms $$ d\mathcal{Y}=f(C)dC,$$ here $d\mathcal{Y}(x^i)$ is one form, $C(x^i)$ is a scalar, $f(C)$ is a function of $C$ only. Can it be ...
0
votes
1answer
42 views

Wedge product is zero

Suppose $C$ is a scalar, $\mathcal{Y}$ is one-form and there is the equation $$ dC\wedge \mathcal{Y}=0 \quad (1)$$ What is the most general solution of this equation? Using the geometrical ...
0
votes
2answers
60 views

Relationship between divergence operators defined with respect to two different volume forms.

Let us assume that you have a volume form $\mu$ defined on a manifold $\mathcal{M}$. Then you can define the divergence operator with respect to this metric, such that the following relationship holds ...
1
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0answers
23 views

closed and bounded form

I have this problem, Let $\omega$ a closed $1$ form in $\mathbb{R^{2}}\setminus {0} $ such that $\omega$ restricted to the set $D$ is bounded with $D=\left \{ x\in\mathbb{R} \text{ such that }\left | ...
6
votes
3answers
110 views

Wedge product = set intersection?

In a research article [1] I found the following formulation: The wedge product may be considered as set intersection. For example, surfaces of constant $f(x,y,z)$ and surface of constant ...
1
vote
0answers
35 views

differentiable curve

I´m a little stuck with this problem, I think is false but I can´t find a counter example, here is the problem Let $\omega$ a 1-form defined in $U\subset \mathbb{R^{2}}$(it can be $\mathbb{R^{n}}$, ...
1
vote
4answers
68 views

For a differentiable map $f: \mathbb{R^n}\to \mathbb{R^n}$, Show that $f^*({dy_1 \wedge\cdots \wedge dy_n})=\det(df)dx_1\wedge \cdots\wedge dx_n$

Let $f: \mathbb{R^n}\to \mathbb{R^n}$ be a differentiable map given by $f(x_1,\cdots, x_n) = (y_1,\cdots,y_n)$. Show that $f^*({dy_1 \wedge\cdots \wedge dy_n})=\det(df)dx_1\wedge \cdots\wedge dx_n$ ...
0
votes
2answers
46 views

A question on wedge product of differential forms

Let $\omega$ be a $k$-form, is it true that $$\omega\wedge\omega=0?$$ is it true that $$d\omega\wedge d\omega=0?$$
0
votes
0answers
36 views

Is there a way to use this interpretation of differential forms on manifolds?

I read Rudin's "Principles of Mathematical Analysis". In the part of Differential Forms, he defined them formally. I particularly enjoyed the formal viewpoint, since everywhere else it seems that the ...
4
votes
1answer
74 views

What exactly does “differential forms are coordinate free” mean?

Most introductory texts on differential forms praise their property of allowing for a "coordinate free formulation". What exactly does this mean? What would be a concrete example for which a ...
0
votes
0answers
22 views

differential forms and index

The other question i can´t solve is this, If $\varphi$ a differential transformation such that $\varphi (x,y)=(f(x,y),g(x,y))$ and define $i(\varphi ,D)=\frac{1}{2\pi }\int _{\gamma }\theta _{0}$. ...
1
vote
0answers
43 views

1-forms and zero simple

Let $\varphi$ a differential transformation such that $\varphi (x,y)=(f(x,y),g(x,y))$ and $D\subset U$ such that $\varphi$ restricted to $\partial D=\gamma$ be distinct zero and we define $i(\varphi ...
1
vote
0answers
21 views

Pullback of area form of manifold by local chart map

Let $M$ be an orientable $2$-manifold in $\mathbb{R}^n$, with area form $dA$. Let $\phi: U \to M$ be a local coordinate system for $M$, with $U \subseteq \mathbb{R}_{uv}^2$. It is stated that ...
1
vote
1answer
43 views

A question on self-dual differential 2-forms

This question is from Lemma 2 in Derdzinski's paper. Let $$\omega=e_1\wedge e_2+e_3\wedge e_4, \eta=e_1\wedge e_3+e_4\wedge e_2, \theta=e_1\wedge e_4+e_2\wedge e_3$$ be a basis for self-dual ...