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

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2
votes
3answers
69 views

Generators of $H^1(T)$

Let $T$ denote the torus. I am working towards an understanding of de Rham cohomology. I previously worked on finding generators for $H^1(\mathbb R^2 - \{(0,0)\})$ but then realised that for better ...
2
votes
1answer
63 views

Constructing functions such that integral along any closed curve is non-zero

Consider smooth maps $f: \mathbb R^2 \setminus \{(0,0)\} \to \mathbb R$. How can I construct such an $f$ with the property that $$ \oint_C f \neq 0$$ for any closed curve $C$ around the origin? ...
1
vote
1answer
58 views

Using Poincaré lemma to find the generator $H^1$ of $\mathbb R^2 -\{(0,0)\}$?

I am working towards an undertanding of de Rham cohomology. For this reason I am trying to find generator(s) of $H^n_{dR}(\mathbb R^2 -\{(0,0)\})$ and currently I am working on the case $n=1$. I ...
4
votes
1answer
87 views

Understanding de Rham cohomology: geometrically speaking, when is a smooth function closed

On Wikipedia the de Rham cohomology groups are defined to be the cohomology groups of the de Rham cochain complex (equivalence classes of differential $k$-forms). By this definition the zeroth de ...
0
votes
0answers
40 views

Show that $\int_b d\omega=0$ where $b(s,t)=\Phi_s(c(t)) $

Let $c:[0,1]^k \rightarrow \mathbb{R}^n$; $t \mapsto c(t)$ be k-cell with $k < n$. Let $\mathbb{Y}$ denote a vector field on $\mathbb{R}^n$ with flow $\Psi_s$. Define a $(k+1)$-cell $b:[0,1]^{k+1} ...
2
votes
1answer
109 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 ...
1
vote
1answer
66 views

Show $D^2=0$ iff $D=e^{-f}de^{f}$ for some function $f$ , where $D\omega := d\omega+\alpha \wedge \omega $

Let $\alpha$ be a $1$-form on $\mathbb{R}^n$. Define the following which takes $k$-forms to $(k+1)$-forms. $$D\omega := d\omega+\alpha \wedge \omega $$ Show that $D^2=0$ iff $D=e^{-f}de^{f}$ for ...
1
vote
0answers
29 views

Proving that $\frac{d}{dt}\int \Phi_t^*\omega=\int_{\Phi_t \circ \partial c} i_{\mathbb{X}}\omega$

Let $\omega$ be a closed $k$-form on $\mathbb{R}^n$ and $c:I^k \rightarrow \mathbb{R}^n$ a $k$-cube on $\mathbb{R}^n$. Let $\mathbb{X}$ be a vector field on $\mathbb{R}^n$ with flow $\Phi_t$. Show ...
0
votes
1answer
51 views

Understanding the definition of a pullback of a differential $k$-form and applying it in $1-d$

I am having trouble understanding the definition of a pullback of a differential k-form in a basic course in differentiable geometry. This is the definition I am given. I believe it is easier to ...
1
vote
1answer
68 views

Show that $(\textbf{S}^*\textbf{B})(u,v)=\textbf{B}(\textbf{S}(u,v))\cdot \textbf{N}(u,v) \ du \wedge dv$

Let $\textbf{S}(u,v):[0,1]^2 \rightarrow \mathbb{R}^3$ be a singular $2$-cube which is smooth. Note that $0 \leq u,v \leq 1$. Let $B(\textbf{r})=B_x \ dy \wedge dz + B_y \ dz \wedge dx + B_z \ ...
2
votes
2answers
151 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 $\partial c=0$, ...
1
vote
1answer
53 views

$df\wedge \alpha=0$ implies $\alpha=g\,df$ for some $g$

Let $A \subset \mathbb{C}$ be an open set and $f\in {C}^{\infty}(A)$ with $df\neq0$. I consider $\alpha$ a $1$-form such that $df\wedge \alpha=0$ and I want to prove that exists a function $g$ such ...
1
vote
1answer
51 views

Reference to finite coverings causing injections on deRham cohomology

So, I've heard that if you have a finite degree covering of a compact connected manifold by another compact connected manifold of dimension $n$ (So $\pi :M \rightarrow N$) gives an injection on the ...
1
vote
1answer
136 views

The element of Volume of the Sphere and two formulas

Let $S^{n-1}$ be the unit sphere with the inner product $<.,.>$ that inherits from $\mathbb{R}^n$ and $V\in S^{n-1}$. Let $\{e_ i \}_{ i=1}^n $ be an orthonormal frame and let ...
2
votes
1answer
51 views

When are the exterior derivative and contraction of forms inverses?

I am trying to get a better feel for both the exterior derivative of a form and the contraction of a form by a vector field $X$. Basically, when are these inverses? If I have a one-form $\omega$ and ...
3
votes
0answers
139 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 ...
5
votes
2answers
79 views

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 ...
2
votes
1answer
91 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
58 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
101 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 ...
2
votes
1answer
97 views

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 &= \left(\frac{\partial}{\partial t}\hat{\Phi}_t \right) \hat{\Phi}_t^{-1} \\ &= \left(\frac{\partial}{\partial ...
0
votes
0answers
39 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
94 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
54 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
39 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
34 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
1answer
193 views

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

You are given the following statement of the Poincaré Lemma: If $\Phi_t$ is a one-parameter family of diffeomorphisms on $\mathbb R^n$ (not necessarily a subgroup) and $X_t$ the vector field ...
1
vote
1answer
79 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
80 views

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
32 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
52 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
35 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
27 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
47 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
23 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
58 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
104 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
210 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
33 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
87 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
43 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
122 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 ...
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vote
0answers
46 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
51 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
67 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
43 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
65 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
39 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
41 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
42 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 ...