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

learn more… | top users | synonyms

5
votes
2answers
96 views

Closed form on any submanifold closed?

Let $M$ be a manifold and $\omega$ a closed differential form, so i.e. $d \omega =0.$ If I now consider a submanifold $N$ of $M$. Does this mean that $\omega$ is still closed on $N$? This statement ...
2
votes
1answer
47 views

Computing Rham Cohomology

Suppose that we have a $C^{\infty}$ manifold $X$ with and atlas $\mathcal{A}=$($U_{\alpha},\varphi_{\alpha}$) such that for every two intersecting open sets $U,V \in \mathcal{A}$ the intersection is ...
1
vote
1answer
45 views

Symplectic form and wedge sum

The wedge sum of $k$ symplectic 2-forms is given by ( if $\omega = \sum_i e_i^* \wedge f_i^*$) $$ \omega^k = k! \sum_{1\le i_1 <...<i_k \le n} (e_{i_1}^* \wedge f_{i_1}^*) \wedge ... \wedge ...
3
votes
2answers
65 views

Definition of Cech-De Rham complex. Can't understand its definition!!!!

Let $M$ be a manifold and let $\mathcal{U}:=\{U_\alpha\}_{\alpha\in I}$ be an open covering, $I$ be a totally ordered set. For every $p$ and for every $\alpha_0<\dots<\alpha_p$, $$ ...
2
votes
1answer
36 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 ...
0
votes
2answers
43 views

2-form on a smooth manifold

Let $M$ be a smooth manifold, $f:M$ $\rightarrow \mathbb{R}$ differentiable and $p\in M$ with $df(p)=0$. I am trying to show that the application, $$\begin{matrix}\mathfrak{X}(M)\times ...
0
votes
0answers
31 views

Integrate symplectic two-form

I am supposed to show the following (maybe falsely stated) theorem Let $\Phi$ be a symplectic group action $\Phi: G \times M \rightarrow M$ on a manifold $M$. Assume that the symplectic form $\omega$ ...
0
votes
1answer
27 views

Analytic expression of a 1-form

Let $M$ be a differentiable manifold, $V\in\mathcal{X}(M)$ a vector field on $M$ and $\alpha\in\mathcal{X}^*(M)$ a 1-form. Let $L_{V\alpha}$ be another 1-form defined by: ...
2
votes
1answer
18 views

Restriction of differential $1$-forms to open subsets?

A vector field on a manifold $M$ is a linear map $X:C^\infty(M)\longrightarrow C^\infty(M)$ with an additional property. The set $\mathfrak{X}(M)$ of all vector fields on $M$ is a ...
2
votes
2answers
40 views

Volume forms and volume of a smooth manifold

Choose a volume form $\omega$ on $M$, oriented manifold. For every $F\in C^{\infty}_c(M)$, we define $$ \int_M F:=\int_M F\omega $$ where in the right hand term $M$ is taken wit positive orientation ...
2
votes
1answer
84 views

Manifold is not orientable

Let $M$ be a manifold of dimension $n$ such that there exist two charts $(U_a,\phi_a)$ and $(U_b,\phi_b)$ such that $U_a,U_b$ are connected and $U_a\cap U_b\ne\emptyset$. Moreover the ...
3
votes
2answers
32 views

2-form whose self-wedge does not vanish?

I know that any 2-form is decomposable if and only if its self-wedge vanishes. Is there an element $β ∈ A_2(R^n)$ such that $β ∧ β \neq 0$. Obviously, this $\beta $ must be indecomposable, but I ...
2
votes
1answer
56 views

Good book about differential forms

I'm a looking for a good book to self-study differential forms. Particularly, I'm looking for a book that is as similar as possible to Bert Mendelson's "Introduction to topology" (i.e. a book that ...
1
vote
0answers
34 views

Show that $1$-form has particular coordinate representation.

I want to show that for a nowhere-vanishing $1$-form $\alpha$ on an $n$-dimensional manifold $M$ we have that $\alpha \wedge d\alpha = 0$ if and only if around every point $p$ there exists a ...
0
votes
0answers
17 views

Order in integration of differential forms

Suppose $$G = dC + (b-1)\sum_{i=1}^{2} \delta_i \wedge \omega_i - \frac{b}{2\pi}d\phi \wedge \sum_{i=1}^{2}\omega_i$$ defines a 4-form in terms of a 3-form $C$, two forms $\omega_1$ and $\omega_2$ ...
5
votes
1answer
31 views

Relearning multivariable calculus through differential forms

While I learned multivariable calculus a few years ago, I have never felt I understand it well enough. Now I have time to go back and correct this. Since I have been through subjects like real ...
1
vote
1answer
45 views

Basis free way to show $\alpha(V,-)=\alpha(W,-)=0$ implies $\alpha([V,W],-)=0$?

Given a 2-form $\alpha$, I want to show that, if $X$ and $Y$ are vector fields such that $\alpha(X,Z)=\alpha(Y,Z)=0$ for all vector fields $Z$, then $\alpha([X,Y],Z)=0$ for all vector fields $Z$. ...
26
votes
4answers
526 views

Intuition behind an integral identity

A proof for the identity $$\int_{-\infty}^{\infty} f(x)\, dx=\int_{-\infty}^{\infty} f\left(x-\frac{1}{x}\right)\, dx,$$ has been asked before (for example, here), and one answer to that question ...
5
votes
3answers
74 views

Compute $\int_M \omega$

Let $M=\{(x,y,z): z=x^2+y^2, z<1\}$ be a smooth 2-manifold in $\Bbb{R}^3$. Let $\omega=xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\in \Omega^2(\Bbb{R}^3)$. Compute $$\int_M \omega.$$ I parametrised ...
3
votes
2answers
61 views

Non-integrability of distribution arising from 1-form and condition on 1-form

Suppose $M$ is a $(2k+1)$-dimensional manifold on which a 1-form $\alpha$ is defined. $M$ is termed as a contact manifold if the distribution arising from $\alpha$ is nowhere integrable, i.e. if: ...
1
vote
0answers
42 views

Differential forms defined by integration

Let $\omega_1,\omega_2$ be two n-forms on a $n$-dimensional manifold $M$. Now, imagine we have for every open $N \subset M$ that $$\int_{N}\omega_1 = \int_N \omega_2.$$ Can anybody show me how to ...
1
vote
1answer
44 views

The explicit expression for integral of forms

Could anyone please help me with the following three questions? They are simple questions, but I am confused. With a $2$-form $F=\frac{1}{2}F_{\mu\nu}dx^{\mu}\wedge dx^{\nu}$ in 4 dimension, what is ...
1
vote
1answer
67 views

Trouble understanding proof of this proposition on contact type hypersurfaces

I have finally started working on my thesis. I am reading the first reference book, which is Dusa McDuff, D. Salamon, Introduction to Symplectic Topology. It's a tough struggle, given my not-too-great ...
1
vote
2answers
40 views

Pullback of a linear map on a 2-form.

I am having a bit of trouble understanding a homework question and was seeking some clarification. Note, I have edited this question after I worked a couple of things out. Given a 2-form $v=dx_1 ...
2
votes
1answer
32 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} ...
0
votes
1answer
32 views

Prove that this $1$-differential form on $S^1$ is well-defined

Let $U_i:=\{p\in S^1:x_i\ne 0\}$, $i=1,2$, be two open sets of $S^1$. Define $$ \omega_p := \begin{cases} \Bigl(\bigl(-\frac{dx_2}{x_1}\bigr)|_{U_1}\Bigr)_p\,\,\,\;\text{if}\,\,p\in U_1\\ ...
1
vote
0answers
27 views

Zeroes of differential form embedded

Let us consider $S^1$ as a manifold embedded in $\mathbb{R}^2$. Let $dx_1\in\Omega^1(\mathbb{R^2})$. $$ Z_{\mathbb{R}^2}:=\{p\in\mathbb{R}^2:(dx_1)_p=0\}\\ ...
-3
votes
1answer
48 views

Differential Forms: Show that $d^2 = 0$ by explicit computation [closed]

Show that $d^2 = 0$ by explicitly computing $d^2\omega$ for $\omega$ a $1$-form in $\mathbb{R}^3$.
1
vote
2answers
38 views

Axioms for 2-forms, why should $\omega_x(\Delta x \wedge \Delta x) = 0$

I am having trouble understanding this piece about 2-forms from Terence Tao's "Differential Forms and Integration". I understand the bilinearity requirement in analogy to the one-dimensional case. ...
1
vote
1answer
36 views

Proving wedge product is associative

Fix a real vector space $V$ of finite dimension. Let's denote by $\Lambda^p(V)$ the vector space of $p$-forms on $V$ (i.e. alternating $p$-tensors). Then we have the product $\wedge : \Lambda^p(V) ...
2
votes
2answers
32 views

Exercise about differential forms and pull-back: $\omega_p=0$ if and only if $(\omega|_U)_p=0$ [duplicate]

Let $M$ be a manifold, $\omega\in\Omega^q(M)$. Let $U\subset M$ be an open subset embedded as manifold in $M$. Let $p\in U$. Show that $\omega_p=0$ if and only if $(\omega|_U)_p=0$. Notation: Let ...
3
votes
1answer
68 views

Integrating the Hopf invariant for $\pi:S^3\to S^2$

I've been working on the last part of problem 9., chapter 9 in Nakahara's Geometry, Topology and Physics all day, with no success, and am in need of some assistance. We are asked to compute the Hopf ...
2
votes
2answers
50 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 ...
4
votes
1answer
55 views

Choice of order in the Leibniz rule is arbitrary?

One of the rules which characterizes the exterior derivative is that, for $\varphi$ a real-valued function and $\omega$ a $k$-form, we have $$d(\varphi \cdot \omega) = d\varphi \wedge \omega + ...
2
votes
2answers
32 views

Line integrals in differential form

I'm a bit confused as to the format of line integrals in differential form (i.e. the form in which Green's theorem is often presented). For example: $$ \oint\limits_\mathcal{C} \left( y^2 \mathrm{d}x ...
0
votes
0answers
42 views

Is there a deRham (co)homology for vector-valued differential forms?

Is there a deRham (co)homology for vector-valued differential forms? The deRham (co)homology of differential forms has been well-discussed and well-founded, along with the fact that for the exterior ...
0
votes
0answers
69 views

Differential forms- riddle me this.

I stumbled over this answer on math.stackexchange Let $x$ be a point in $\rm M$. Then because $\omega$ is non degenerate at $x$, the antisymmetric matrix $\omega_x$ has full rank on the tangent ...
3
votes
3answers
76 views

Exterior derivative

Let $$\omega = \frac{1}{2} \sum_{i,j} \omega_{i,j} dx_i \wedge dx_j$$ be an antisymmetric form, so i.e. $\omega_{i,j} = - \omega_{j,i}.$ Now, in some lecture notes I found that $$d \omega ...
2
votes
2answers
40 views

Dimension of the $0^{\text{th}}$ de Rham cohomology group of $U$

Let $U\subset\mathbb{R}^n$ be an open set. I proved that $$ H^0_{DR}(U):=\frac{\text{closed forms}}{\text{exact forms}}=\{f\in C^{\infty}(U):\,f\,\,\text{is locally constant} \} $$ I have to show ...
1
vote
2answers
36 views

Finite universal covering induces injective maps on cohomology

I am trying to prove the following: Suppose $M$ is a smooth, connected manifold with finite fundamental group and $f : \widetilde{M} \rightarrow M$ is its (smooth) universal cover. Show that $f^* : ...
0
votes
1answer
27 views

Hamiltonian vector field and symplectic geometry

I want to show the following theorem: For any Hamilton function $H : M \rightarrow \mathbb{R}$ on some symplectic manifold $M$ and symplectomorphism $f : M \rightarrow M$ we have $X_{H \circ f} = ...
0
votes
0answers
30 views

Integration by parts in the $x_i$ direction in the integral which uses differential forms

Let $\Omega \subset \mathbb{R}^n$ be open and bounded and let $K \subset \Omega $ be compact. Let $\xi \in C_0^\infty (\Omega )$ with $0\leq \xi \leq 1$ in $\Omega $ and $\xi \equiv 1 $ on $K$. For ...
3
votes
1answer
58 views

Stokes' theorem and symplectic geometry

Let $V = \mathbb{R}^2,$ as a vector space then the Poincaré invariant is an integral $\int_{\gamma} \theta$ where $\theta = p dx $ is the symplectic 1-form and $\gamma$ a closed curve. Now, it is ...
1
vote
0answers
26 views

Is there a treatment/development of the Stokes' Theorem using differential forms and the Henstock-Kurzweil integral i.e. the gauge integral?

I'm working through an analysis text independently to prepare for grad school, and the author has discussed the limitations of both the Riemann and Lebesgue integrals and only hinted at the power of ...
2
votes
2answers
34 views

Pullback of $1$-form in coordinates

Let $$\theta(p) = \sum_{i=1}^n f_i(p) \, dx_i$$ be a $1$-form in local coordinates. then we define $F^*(\omega(p))(X_1,\ldots,X_n) = \omega(F(p))(DF(p)(X_1),\ldots,DF(p)(X_n))$ as the pullback of a ...
1
vote
1answer
35 views

Exercise about wedge product and multilinear forms

I'm considering $\omega\in \Lambda^{2q+1}(V^\ast)$, i.e. a multilinear skew-symmetric form. I want to prove that $\omega\wedge\omega=0$. How shall I proceed? Any suggestions? Do I have to write ...
0
votes
0answers
30 views

Line integral and differential forms

Let $df = \frac{\partial f}{\partial x_1 } dx_1 + \frac{\partial f }{\partial x_2 } dx_2$ be a $1-form.$ I know that that the line integral (along a curve $\gamma:[0,t] \rightarrow \mathbb{R}^2$) is ...
0
votes
0answers
31 views

Basic exercise differential forms

I have to show that the space of q-differential forms $\Omega^q(U)=\{0\}$ if and only if $q>n$ or $q<0$. Any ideas?
4
votes
1answer
51 views

What's going on with these identities involving $d$, $\mathcal L_X$, and $\iota_X$?

Let $\Omega^k$ denote the smooth $k$-forms on a given smooth manifold. Then we have the following operators: Exterior derivative: $d:\Omega^k\to\Omega^{k+1}$ (takes you to the right in the de Rham ...
1
vote
1answer
34 views

Stokes' theorem generalized the FTC part 2. Is there a known generalization for part 1?

Stokes' theorem generalizes the fundamental theorem of calculus (part 2) using differential forms. Is there a known generalization of part 1? edit In case anyone is unaware, The fundamental theorem ...