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

learn more… | top users | synonyms

2
votes
0answers
55 views

Integration of forms on product manifolds

Let $M$ be a compact, connected and oriented smooth manifold of dimension $m$ and let $\pi_1,\pi_2:M\times M\rightarrow M$ be the projections to each factor. Given $\alpha\in H^m(M;\mathbb{R})$, I ...
0
votes
1answer
92 views

Curiosity about the wedge product and the Levi-Civita symbol

Let us use the definition of the wedge product of two vectors: $$\vec{u}\wedge\vec{v} = \vec{u}\otimes\vec{v} - \vec{v}\otimes\vec{u}$$ writing $\vec{u}$ and $\vec{v}$ in dyadic form as $\vec{u} = ...
1
vote
1answer
19 views

show that $K=E_2[\omega_{12}(E_1)]-E_1[\omega_{12}(E_2)]-\omega_{12}(E_1)^2-\omega_{12}(E_2)^2$?

$$K=E_2[\omega_{12}(E_1)]-E_1[\omega_{12}(E_2)]-\omega_{12}(E_1)^2-\omega_{12}(E_2)^2$$ where $K$ is Gaussian curvature, $E_i$'s are tangent frame field on surface $M$ in $R^3$, $v[\cdot ]$ is ...
5
votes
0answers
138 views

Is the de Rham complex a free (commutative?) differential graded algebra?

A differential graded algebra (dg-algebra) is a monoid object in the category of chain complexes with respect to the usual tensor product of complexes. A (graded) commutative dg-algebra is simply a ...
3
votes
0answers
321 views

push forward of differential form/ integration over fiber

It is elementary that differential forms can be pulled back via a smooth map between manifolds. However, I was reading a paper and came across a construction about push forward of a differential form ...
0
votes
1answer
37 views

Why is $\omega_1^2$ not semi-basic for $\pi : ASO(3) \rightarrow M \subset \mathbb{E}^3$?

In Cartan for Beginners, problem 2.4.3, the problem is that if $M$ is flat, show that there exist coordinates $x_1,x_2$ and an orthonormal adapted frame $(e_1,e_2,e_3)$ such that $\omega_1=dx_1, ...
4
votes
1answer
91 views

Local $\partial \bar{\partial}$-lemma..

I am trying to prove the local $\partial \bar{\partial}$ lemma. This says that for a polydisc in $\mathbb{C}^{n}$, a form in $A^{p,q}(U)$ being $d$-closed implies that it is $\partial ...
2
votes
2answers
125 views

Formula about time derivative of pushforward of family of forms: where is it from?

Proving Darboux's theorem, Hofer-Zehnder try to find, given $\omega$ a closed nondegenerate 2-form and $\omega_0$ the canonical symplectic form, a family of diffeomorphisms $\phi^t$ such that for all ...
4
votes
1answer
45 views

Why is the action of $SL(2, R)$ on holomorphic quadratic forms involve a square root?

Following this, given a quadratic differential $q$, it can be identified with the pair of real 1-forms $(\Im(q^{1/2}), \Re(q^{1/2}))$. Given a matrix $A \in SL(2, R)$, it acts on this pair by left ...
2
votes
1answer
56 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 ...
2
votes
0answers
65 views

Integration is only possible for forms and not for general tensors?

Integration is only possible for forms and not for general tensors? What is the true reason for this? Or can integration of $k$-forms be extended in some natural way to arbitrary $(k,l)$ tensors;if so ...
3
votes
1answer
147 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, ...
6
votes
0answers
120 views

What is the best way to learn Differential forms? [closed]

I'm taking a Multivariable Calculus class and my teacher has just started Differential forms. It is not making a lot of sense, though. I have tried reading "Geometric Approach to Differential forms" ...
3
votes
1answer
167 views

Volume form on $(n-1)$-sphere $S^{n-1}$

Let $\omega$ the (n-1) form on $\mathbb{R}^n$ $$\omega=\sum_{j=1}^{n}(-1)^{j-1}x_{j}dx_{1}\wedge\cdots\wedge \hat{dx_{j}}\wedge\cdots dx_{n}$$ Show that the restriction of $\omega$ to $S^{n-1}$ in ...
3
votes
1answer
63 views

Finding a form to solve a wedge product equation

We start out with $(\mathbb{R}^{2n},\omega_0)$, the "standard" symplectic manifold. We define $\Omega=\omega_0\wedge\dotso\wedge\omega_0$, i.e. $\Omega$ is the product of $\omega_0$ with itself $n$ ...
18
votes
0answers
405 views

Wanted: A purely algebraic proof of the Frobenius theorem on distributions

Is there a purely algebraic proof of the Frobenius theorem? Here's a rough sketch of what i'm looking for: Let $Der(R)$ denote the $R$-module of ($R$-valued) derivations of the algebra $R$ endowed ...
6
votes
1answer
49 views

Exterior 2-form, 1-form, Hodge star operator.

In $\mathbb{R}^{2n}$ with coordinates $x_1, x_2, \dots, x_{2n}$, consider an exterior 2-form$$\eta = \sum_{k=1}^n x_{2k-1} \wedge x_{2k}.$$Given a 1-form $\alpha = \sum_{i=1}^{2n} a_ix_i$, what is the ...
1
vote
0answers
39 views

On closed, symplectically embedded surfaces of ambient compact symplectic manifolds, how does one avoid pulling back a 2-form to an exact 2-form?

Obviously, if one pulls back an exact (with respect to the de Rham d) differential form by any map, then one obtains an exact form on the submanifold. But if one starts out with a form that isn't ...
0
votes
1answer
28 views

Expression for integral of a particular 1-form doesn't convince me

Hofer-Zehnder, Symplectic invariants and Hamiltonian dynamics, defines $\omega_0$ as the standard symplectic form, $\sum_1^ndy_j\wedge dx_j$, where $x_1,\dotsc,x_n,y_1,\dotsc,y_n$ are the coordinates ...
3
votes
1answer
107 views

How to visualize differential forms geometrically

I've been attempting to teach myself differential geometry and I have heard that one can visualise them geometrically and that this can sometimes be helpful for an intuitive understanding of them. For ...
2
votes
0answers
53 views

A compact $n$-manifold is orientable iff there is an everywhere nonzero $n$-form

Let $M$ be a compact differentiable manifold of dimension $n$ without boundary. Show that $M$ is orientable if and only if there exists a diffential $n$-form $\omega$ defined on $M$ and which is ...
1
vote
0answers
53 views

Identity regarding the components of a dual basis

This problem is from Robert Wald's "General Relativity." The problem is 4(b) from chapter 2. Let $Y_1\cdots Y_n$ be smooth vector fields on an $n$-dimensional manifold $M$ such that at each $p\in M$ ...
6
votes
2answers
131 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 ...
6
votes
1answer
449 views

Integral Definition of Exterior Derivative?

Is there a rigorous integral definition of the exterior derivative analogous to the way the gradient, divergence & curl in vector analysis can be defined in integral form? Furthermore can it be ...
1
vote
1answer
105 views

How to show $[\omega]=0$ implies $[\omega^n]=0$?

I'm trying to prove the following: If $(M, \omega)$ is a symplectic manifold and $[\omega]=0$ then $[\omega^n]=0$, where $[\omega]$ is the De Rham cohomology class of $\omega$. Well what I've done ...
2
votes
1answer
64 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
99 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
88 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
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 ...
15
votes
2answers
1k views

What do $dz$ and $|dz|$ mean?

I'm having a hard time understanding complex differentials. I know that when I have a field $\mathbb K$ and a $\mathbb K-$vector space $\mathbb K^n,$ then we define $dx_i\in \mathrm{Lin}(\mathbb ...
0
votes
2answers
56 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 ...
14
votes
3answers
645 views

The algebraic de Rham complex

Let $A$ be a commutative $R$-algebra (or more generally a morphism of ringed spaces). Then there is an "algebraic de Rham complex" of $R$-linear maps $A=\Omega^0_{A/R} \xrightarrow{d^0} \Omega^1_{A/R} ...
0
votes
0answers
43 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 $\omega$ ...
2
votes
1answer
152 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 ...
0
votes
1answer
35 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: ...
3
votes
1answer
34 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
204 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 ...
3
votes
2answers
49 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 ...
4
votes
1answer
383 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 ...
2
votes
1answer
59 views

Expressing a differential form in terms of a scalar function

We can express every k-form in the form $ \omega(x) = \sum_Id_Idx_I $ where $ I$ is k-tuple and $ d_I$ is just some scalar function of x. That's entirely understandable for me. But while reading ...
1
vote
0answers
44 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
21 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 $\omega_2$ ...
27
votes
4answers
625 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 ...
1
vote
1answer
108 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
1answer
57 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 $Z$. ...
4
votes
1answer
59 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 ...
0
votes
2answers
37 views

If $σ$ is an exact differential $1$-form on the plane, then the form $ω=σ+xdy$ is not exact

If $σ$ is an exact differential $1$-form on the plane, then prove that the form $ω=σ+xdy$ is not exact. In the previous part of the question we have calculated the integral of the differential ...
5
votes
3answers
98 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 ...
2
votes
1answer
60 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 ...
3
votes
1answer
83 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 ...