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

learn more… | top users | synonyms

0
votes
0answers
10 views

Obtain first-order system

Question: Let $G(x,t)=g''(x-a(t))$, where $g$ and $a$ are smooth. Combine equations: $$\frac{\partial u}{\partial x}=v;\frac{\partial u}{\partial t}=w$$$$\frac{\partial v}{\partial t}=\frac{\partial ...
0
votes
1answer
13 views

Derive necessary and sufficient conditions meaning

Question: Let $\mathbb{X}$ and $\mathbb{Y}$ be vector fields on $\mathbb{R}^3$ given by $$\mathbb{X}(x,y,z)=(1,0,p(x,y)r(z))$$ $$\mathbb{Y}(x,y,z)=(0,1,q(x,y)r(z))$$ where $p,q$ and $r$ are smooth, ...
1
vote
2answers
76 views

Exactness of $dx,dy$

Let $dx,dy$ denote differential $1$-forms. It is easy to verify that they are closed. My question is: Does there exist a space such that either $dx$ or $dy$ or both are exact? (A ...
2
votes
3answers
94 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 ...
0
votes
1answer
71 views

Deriving generators for $H^1(T)$: what are $dx$ and $dy$?

By trial and error I found that $dx,dy$ are generators of $H^1_{dR}$ of $T=S^1\times S^1$. Verifying that they generate the first cohomology group is not difficult. My problem is: I found them by ...
4
votes
1answer
57 views

Find the Poincare Dual of a ray in $\mathbb{R}^2-\{0\}$

This is example-exercise 5.16 in Bott and Tu (which I'm independently reading through.) The problem states: Let $M=\mathbb{R}^2-\{0\}$, and $X\subseteq M$ be the closed submanifold ...
1
vote
1answer
53 views

Generators of $H^1 (T)$: take two

Previously, I asked about how to prove that $dx + dy$ is a generator of the de Rham cohomology group of the torus. Now it occurred to me that $dx$ and $dy$ are both also generators of $H^1(T)$. ...
1
vote
2answers
81 views

Relation between volume form and cross product

Euclidean three-dimensional space (it's simpler). Defining $\eta={e^*}^1 \wedge {e^*}^2 \wedge {e^*}^3$, with $\{{e^*}^1,{e^*}^2,{e^*}^3\}$ dual of the orthonormal basis, and indicating the classic ...
0
votes
1answer
39 views

Find a one-form $\alpha_0$ such that $\lim_{t\to 0}\hat\Phi_t^*\beta=d\alpha_0$

Given before question: $\hat{\mathbb{X}}_t\circ\hat\Phi_t=\frac{\partial}{\partial t}\hat\Phi_t$ and $$\beta=\frac{(x^2-y^2)dy\wedge dz+2xydz\wedge dx}{(x^2+y^2)^2}$$ I have shown that $d\beta=0$ ...
1
vote
0answers
81 views

Frobenius theorem method

Question: Let $A(z)$ be smooth with nonvanishing derivative $A'(z)$, and consider the system with first-order PDE's, $$\frac{\partial u}{\partial x}=\frac{\partial K}{\partial ...
3
votes
1answer
42 views

Finding the Lie derivative of a complex valued function

Question: Let vector field $\mathbb{X}$ be given by $\mathbb{X}(x,y)=(-y,x)$ Let $f(x,y)$ be the complex-valued function given by $f(x,y)=(x+iy)^m$ where $m>0$ Show that $L_\mathbb{X}f=imf$ My ...
1
vote
1answer
34 views

Poles of Complex Functions or One-Forms?

The function $f(z) = \frac{1}{\sqrt{z}\sqrt{1-z}}$ with branch cuts chosen so that $f$ is analytic on $\mathbb C-[0,1]$ has a pole at infinity according to this walkthrough of a branch cut contour ...
1
vote
2answers
67 views

Compute the contraction of a 1-form with a vector field

Question: Let $\alpha$ be the $1$-form on $\mathbb{R}^3$ given by $\alpha=zdy-ydz$ and let $\mathbb{X}$ be the vector field on $\mathbb{R}^3$ given by $\mathbb{X}=(0,y,-z)$. Compute ...
5
votes
1answer
59 views

What does $\alpha$ be the $1$-form and $\beta$ the $2$-form on $\mathbb{R}^3$ mean?

In an earlier post to math.stackexchange I asked a question beginning with: Let $\alpha$ be the $1$-form and $\beta$ the $2$-form on $\mathbb{R}^3$ given by $$\alpha=(x+y)\,dy+(x^2-y^2)\,dz$$ ...
2
votes
0answers
83 views

Compute the wedge product

This is my first time computing the wedge product, I am not sure if I have done it correctly as I do not have solutions, if I have gotten the answer right or am doing the right method please say. ...
3
votes
1answer
65 views

How to determine whether a differential $1$-form is globally welldefined?

This is a question that occurred after working on finding a generator of the first de Rham cohomology group of the torus. It was pointed out to me that the differential $1$-form $$ dx + dy$$ was ...
3
votes
2answers
120 views

Applying the Frobenius theorem to a decomposable 2-form

So I have the following problem: Suppose $\omega=\phi \wedge \theta$ is a closed decomposable 2-form on $M$ a manifold (decomposable just means it can be written as a wedge of 1-forms). Suppose $p\in ...
2
votes
3answers
72 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
70 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
62 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
95 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
44 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
130 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
70 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
31 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
64 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
76 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
157 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
56 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
61 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
144 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
2answers
64 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
140 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
86 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
101 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
69 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
145 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
102 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
48 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
109 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
59 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
40 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
52 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
197 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
85 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
92 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
38 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
62 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
41 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
48 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 ...