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

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3
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0answers
81 views

Find the interior product of a basic p-form $\alpha = dx_1 \wedge dx_2 \wedge \ldots \wedge dx_p$ a and a vector field $X$

I'm reading Differentiable Manifolds by Nigel Hitchin, that is, his class notes for an Oxford course freely available here. In particular, I'm trying to understand the interior product on manifolds, ...
-4
votes
1answer
59 views

Pullback of differential form is zero [closed]

Let $ f:\Bbb R^m \to \Bbb R^n $ be differentiable map. Assume $ m<n$ and let $ w $ be a differential $k$-form in $\Bbb R^n $ , with $ k>m $. Show $ f^*w $ =0 Here $ f^* $ is the pullback of the ...
1
vote
1answer
50 views

Correspondence between Differential Forms and Vector Fields

I did not understand the highlighted text. Could anyone please explain it to me. There is a related post here- Differential Forms and Vector Fields correspondence. The first paragraph of the first ...
2
votes
1answer
28 views

Explanation of theorem on differential forms

This is text from Do Carmo's Differential forms and applications Page-10. Could anyone explain the highlighted step?If f* is applied to each of the term then how do we get the RHS of the highlighted ...
1
vote
2answers
44 views

Even dimensional real projective spaces cannot be combed

I have to prove that the even-dimensional real projective space cannot be combed, i.e. there isn't any non-vanishing smooth vector field. (I can't use Hopf theorem since those manifolds are not ...
1
vote
1answer
50 views

Euler number zero for odd dimensional compact manifolds

I need to prove that every compact manifold of odd dimension has Euler number zero. The Euler number of $M$ compact and oriented is $$ e(M):=\int_Ms_0^*\phi(TM) $$ where $s_0$ is the zero section of ...
0
votes
0answers
25 views

Basis of a linear map

A map, also known as function, is the set of ordered pairs $\left \{(a\in A;b \in B)| \forall a \exists!b(a\in A,b\in B) \right \}$ Manfredo and Do Carmo, in "Differental Forms and Applications" ...
1
vote
1answer
30 views

How the canonical symplectic form acts

I've read that the canonical symplectic form $\omega$ on $\mathbb R^{2n}$ is given by $$\omega=\sum_{i=1}^n dp_i\wedge dq_i,$$ where $(p_1,\dots,p_n,q_1,\dots,q_n)$ are the coordinates on $\mathbb ...
2
votes
1answer
24 views

Exterior Derivative in overlapping charts

Given a smooth manifold $M$, say of dimension $n$ and two charts $ (U,x)$ & $ (V,y)$, I want to prove that if $U \cap V \neq \emptyset $, then the exterior derivatives $d_x$ and $d_y$ coincide in ...
1
vote
1answer
56 views

What is an ordinary 2-form?

What is the difference between an ordinary 2-form and just a 2-form in general? Cant seem to find the definition for ordinary 2-form anywhere. Thanks in advance! Edit: If it helps, it was used in ...
4
votes
0answers
48 views

Composition of Lie Derivatives

I was wondering if anyone has any insight into what the composition of the lie derivative would look like. For example say you take the lie derivative of some p-form $ \omega $ with $X$ and then ...
0
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0answers
26 views

Geometrical interpretation of closed form

I'm studying Differential forms and my teacher generally gives geometrical interpretation of most things. So, I was wondering, what is the geometrical interpretation of the closed form and exact ...
1
vote
1answer
24 views

how (dx)^2 affect a formula

I am given α = xdx − ydy, β = zdxdy + xdydz, and required to compute αβ. After distributions and reorderings of differentials I get: $$xz(dx^2dy)+x^2(dxdydz)+yz(dy^2dx)-yx(dy^2dz)$$ Since I am not ...
3
votes
3answers
42 views

Where are the covectors in a line integral?

I'm trying to connect the idea of covectors to integration. From what I understand, given a basis $\{e_1, \dots, e_n\}$ of a vector space $V$, there exists a dual basis of covectors $\{f^1, \dots, ...
0
votes
1answer
43 views

2-forms as wedge product of 1-forms

Why can all $2$-forms on $\operatorname{T}_p (\mathbb{R}^3)$ be written as product of $1$-forms? What is a counter-example to show this isn't true in a space other that $\operatorname{T}_p ...
0
votes
1answer
34 views

Integrals of non-compactly supported forms

Let's recall the definition of integral of a compactly supported smooth $m$-form $\omega$ over an orientable $m$-differential manifold $M$: if $\{(U_\alpha,\phi_\alpha )\}$ is a differentiable atlas, ...
0
votes
1answer
45 views

De Rham cohomology disjoint union of cylinders

Could someone explain me (sorry for the, maybe, trivial question :-) ) how to prove that $H^1_{DR}((S^1\times\mathbb{R}) \sqcup(S^1\times\mathbb{R}))=\mathbb{R}^2$? I'm talking about the de Rham ...
1
vote
0answers
23 views

Change of the sequence of differentation in physics?

Assume to have a quantity A which is calculated from the formula $A=\frac{dB}{dC}$. dC can be written as dC=dEdF so $A=\frac{dB}{dEdF}$. I assume that the differential of A is also ...
2
votes
1answer
81 views

Lie derivative w.r.t. time-dependent field

Some time ago, I asked this question. CvZ answered, and with my additional answer I thought I had solved the problem. Yesterday evening, I copied those calculations into my thesis, and having $t$ and ...
2
votes
1answer
74 views

Exterior derivative commutes with postcomposition by symmetric multilinear functionals?

Let $\frak{g}$ be a finite-dimensional real Lie algebra, $\varphi: \bigotimes^l \frak{g} \to \mathbb{R}$ a symmetric multilinear functional, and $\psi \in \Omega^k(M; \bigotimes^l \frak{g})$ a ...
0
votes
0answers
58 views

The use of a partition of unity in the construction of integrals on manifolds

While reading the usual construction of the integral on a manifold (as in Isaac Chavel, "Riemannian Geometry", page 147) I encountered the following point that I do not understand: Chavel considers an ...
3
votes
1answer
65 views

Different definitions for exterior derivative? [duplicate]

Let $\omega \in \Omega^k(M)$ and $X_i \in \Gamma(TM)$. In Spivak Volume I page 213 the exterior derivative is given invariantly by the formula \begin{align*}d \omega(X_0, \ldots, X_k) = & \sum_i ...
2
votes
1answer
40 views

Deriving formula for formal adjoint

My question is in relation to the derivation of the formal adjoint for a connection $D:\Omega^{p-1}(\text{Ad}E)\rightarrow \Omega^p(\text{Ad}E)$ - I am reading through this derivation in Jost's ...
1
vote
1answer
31 views

With indices and without indices.

Let $S = (S^{\mu\nu})$ be a two form on a four space and $\tilde S$ is the (Hodge) dual of $S$. On what condition can we have $$\partial_{\mu}(S^{\mu\nu}+\tilde{S}^{\mu\nu}) = 0\Leftrightarrow d(S+ ...
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 ...
2
votes
0answers
54 views

Are those two equivalent mathematically speaking?

Starting from the action $$S=-\frac{1}{4}\int{F_{\mu\nu}F^{\mu\nu}d^4x}$$ (until here this is physics but my question is about the math) that is: I got the following equation of motion solving for ...
0
votes
1answer
93 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} = ...
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
323 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 ...
4
votes
1answer
94 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
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 ...
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, ...
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
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 $n$ ...
5
votes
3answers
136 views

Coordinate Free Proof of $ d^2 = 0$

We all know that when the exterior derivative is applied twice to a $k$ form, it always yields the null $k+2$ form (i.e. $d^2 = 0$). However, the only proofs I've seen of this does it in some chart ...
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 ...
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
2answers
126 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 ...
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 ...
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 ...
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 ...
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
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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
89 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
43 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 ...