7
votes
2answers
163 views

Interpretation of $p$-forms

Let $M$ be a smooth manifold, let $C^{\infty}(M)$ be set of all smooth functions from $M$ to $\mathbb R$ and let $Vec(M)$ denote the set of all vector fields on $M$. A $1$-form on $M$ is a ...
0
votes
1answer
30 views

What line does ω project vectors onto?

I have just started learn differential form from the bachman book (page 29)and I found some difficulties in the following problem in 2nd part. Let $ω(<dx,dy>) = −dx + 4dy$. 1. Compute $ω(<1, ...
1
vote
2answers
57 views

The property of an integral manifold of differential form

Let's define the $1$-form $\eta$ on $\mathbb{R}^3$ by formula: $$\eta = A(x,y,z)\;\mathrm{d}x + B(x,y,z)\;\mathrm{d}y + \mathrm{d}z \text{.}$$ And let's assume that $$\mathrm{d}\eta \wedge \eta = ...
1
vote
0answers
32 views

Stoke's theorem explanation

Firstly, Wikipedia informs me that Stoke's theorem might mean two different things so I'm pretty sure I'm talking about the general one. I'll just state what I understand from it and I'd like someone ...
1
vote
2answers
55 views

Wedge product and determinants

I am attempting to self-study differential forms this summer, but I ran into this definition for the wedge product in my book, and it doesn't make any sense to me. Note that $\Bbb R^3_p$ is the ...
1
vote
0answers
30 views

Wedge product of Lie algebra valued differential forms [duplicate]

Let $\mathfrak{g}$ be the Lie algebra of a matrix Lie group. Furthermore, let us consider the following $\mathfrak{g}$-valued $p$-form and $\mathfrak{g}$-valued $q$-form: \begin{equation} ...
1
vote
0answers
68 views

differential forms and orientation in Bott and Tu

I am reading Bott and Tu book on my own. If you have the text you can answer my question perhaps, if not it would be too long to explain it. I have gotten to page 29. I am confused about proposition ...
0
votes
1answer
55 views

Question about Alternating forms

So I understand the definition of an alternating form on $\mathbb{R}^m$, but I don't really understand the proof of the lemma. Could someone explain the first observation? Why is it so?
2
votes
1answer
69 views

Construct tensors from differential forms?

Let $(M,g)$ be a Riemannian manifold, differential forms are defined using tensors, could we define a tensor using a differential form? For example, if $\omega$ is a two-form on $M$ which is expressed ...
0
votes
0answers
49 views

Antipodal map commutes with antipodal map? [duplicate]

Suppose we have a closed form $d\omega$ on $S^{n}$, and antipodal map $i: S^{n} \to S$ n i.e $i:x \to −x$. How to see that the external differential commutes with antipodal map?
3
votes
1answer
39 views

Well-definedness of a coboundary map between a reduced $L^2$ de Rham cohomology group and a relative cohomology group

I'm working right now with this paper of Carron. And I think I'm stuck at a relatively simple question. On page 11 he is defining a coboundary map $b : H^k_{2, \text{reduced}}(M - K) \to H^{k+1}(K, ...
1
vote
0answers
25 views

prove that ''pullback'' maps forms to forms

Suppose we have $A: V_{1}\to V_{2}$ where $V_{1},V_{2}$ are real vector spaces. Then $A^{\star}:\mathcal{J}^{k}(V_{2})\to \mathcal{J}^{k}(V_{1})$ where $\mathcal{J}(V):=\{\text{space of all ...
1
vote
1answer
80 views

Change of Coordinate Formula for Differential Forms

Let $M$ be a manifold, $x$ local coordinates on an open set $U$, $y$ local coordinates on an open set $V$. In addition, let $(x, \alpha)$ and $(y, \beta)$ be two induced bases for the common part of ...
0
votes
1answer
26 views

the exact form in a manifold

Let $M$ be a compact manifold, $X$ is a vector field on $M$, $\alpha$ is a closed 2-form on $M$, $\phi: M\to M$ is a diffeomorphism such that $\phi^*\alpha=\alpha$, then I want to konw whether $$ ...
4
votes
0answers
62 views

A question on harmonic two-forms

Let $(M^4,g)$ be a closed Riemannian four-manifold with $b_2^+>0$ and $b_2^->0$, is it possible to find two harmonic two-forms $\alpha\in H^2_+(M)$ and $\beta\in H^2_-(M)$, such that ...
0
votes
1answer
43 views

When is an exact 2-form harmonic?

Let $\alpha$ be an exact two-form, $\alpha=d\beta$ for some one-form $\beta$, when is $\alpha$ harmonic? By uniqueness of harmonic forms in cohomology classes, it cannot be harmonic?
2
votes
0answers
44 views

surface element of $S^{3}$

How does one show that the surface element of $S^{3}=\{x=(x_{1},...,x_{4})\in\mathbb{R}^{4}\;|x|^2=1\}$ is given by the following 3-form: $\omega=x_{1}dx_{2}\wedge dx_{3}\wedge ...
2
votes
1answer
56 views

what does it mean for a differential form to be well defined on a manifold?

What does it mean for a differential form to be well defined on some manifold. In particular, why the $2$-form $\omega=d\psi\wedge d\theta$ is well defined on $S^{2}$? Thank you in advance.
1
vote
0answers
36 views

integral of a 2-form over an oriented manifold

An old exam question: Let $M$ be oriented submanifold of $\mathbb{R}^{n}$ of dimension $k$, let $\omega$ be a $k$-form on $M$. 1) Define $\displaystyle\int_{M}\omega$ 2) let ...
2
votes
1answer
62 views

De Rham cohomology for $\mathbb{R^2}$

De Rham cohomology groups for $\mathbb{R^2}$. $H^{0}_{dR}(\mathbb{R}^{2})=\mathbb{R}$ since $Z^{0}(\mathbb{R}^{2})$ is the one dimensional space of locally constant functions on $\mathbb{R}^{2}$ and ...
1
vote
0answers
44 views

Commutative differential graded algebra

The situation is the following: Let $V$ be a Lie algebra of finite dimension, say n, and let be the graded commutative algebra $(\bigwedge^{\bullet}V)^*:=(\bigoplus_{k=0}^n\bigwedge^kV)^*$ and define ...
0
votes
0answers
50 views

Given a closed form, show it is exact

I came across an old exam problem and I wonder how to approach it: 1) Show that if $f$ is a $k$-form on $\mathbb{R}^{n}$ and $df=0$, $k>0$ then $f=dg$ for some $k-1$-form $g$. My first thoughts: ...
0
votes
1answer
56 views

Defining the integral of differential $1$-forms

Assume $M$ is a smooth manifold, $g:[0,1]\to M$ is a smooth curve on $M$, and $w$ is $1$-form on $M$. Definition: $$\int_gw=\lim_{\Delta\to 0}\sum_{i=1}^nw(x_i)$$ The tangent vectors $x_i$ are ...
2
votes
1answer
90 views

De Rham cohomology, and forms on manifolds

In String Theory and M-Theory by Becker, Becker and Schwarz, they introduce a group, $$C^{p}(M)$$ which they denote the group of all closed $p$-forms on the manifold $M$. Furthermore, they state ...
1
vote
1answer
71 views

diffeomorphism preserve a volume form

Let $\omega_1$, $\omega_2$ two volume form on a compact manifold $M$, we know that there exists a never-vanishing function $f$, s.t. $\omega_1=f\omega_2$. If $h$ is a diffeomorphism $M \to M$ ...
0
votes
1answer
33 views

How to show the identity $d(*df)=(\Delta f)\nu$ where $\nu=dx_1\wedge\ldots \wedge dx_n$?

My textbook defines the Hodge star operator as follows: Given $\omega\in \Omega^k(\mathbb R^n)$ we define $*\omega\in \Omega^{n-1}(\mathbb R^n)$ setting $$*(dx_{i_1}\wedge\ldots\wedge ...
1
vote
1answer
51 views

Flat Surfaces in $\mathbb{R}^3$ Can Be Bent Only Along Straight Lines

This is a problem out of Elementary Differential Geometry by Barrett O'Neill (Chapter 6 Section 3 Number 2). Let $M$ be a flat surface in $\mathbb{R}^3$ with principal curvatures $k_1$ and $k_2$, ...
2
votes
0answers
57 views

A few identities on differential forms

I just started studying differentiable manifolds, and I've encountered a few allegedly simple questions that aren't really simple to me. If possible, I'd be very happy to see full proofs (without ...
2
votes
2answers
80 views

surface area of a Torus using differential forms,

Im studing integration over manifolds. I want to compute the surface area of the torus. Im given the usual parametrization $f(u,v)= ((a+b\cos(v))\cos(u), (a+b\cos(v))\sin(u), b\sin(v))$ for $0\leq ...
1
vote
1answer
60 views

The kernel of a differential one-form

I'm thinking about the kernel of a differential one-form $\theta\in\Lambda^{1}(M)$: $$ Ker(\theta):=\left\{X\in\mathfrak{X}(M) \;|\; \theta(X)=0\right\} $$ Now suppose $X\in Ker(\theta)$, then is ...
3
votes
1answer
53 views

Recovering a frame field from its connection forms

I have a faced a research problem where I would need to recover a frame field given its connection forms. More precisely, I begin with an orthonormal frame field (given by data) in $\Re^3$ written as ...
1
vote
1answer
49 views

Winding number of a linear transformation?

I know that I am computing something incorrectly. I am trying to compute the index of a positive determinant linear bijection. The form I am using is $\omega = \frac{-y dx + x dy}{x^2 + y^2}$. I ...
1
vote
0answers
48 views

Does this Condition Force this Differential 2-form to be 0?

Let $dw$ be a 2-form; $\alpha$ be a 1-form, $g$ a function (i.e., a 0-form), $X$ be a fixed vector field , all living in a 3-manifold, so that $$dw(X,.)=g\alpha .$$ Does this condition force $dw$ to ...
1
vote
0answers
53 views

differential form and cylindrical coordinate

Problem. If $r, \theta, z$ are the cylindrical coordinate functions on $\mathbb > R^3$ , then $x = r\cos\theta, y = r\sin\theta, z = z$. Compute the volume element dx dy dz of $\mathbb R^3$ ...
1
vote
1answer
53 views

Cartan formalism calculation

Just to test out the Cartan formalism, I decided to apply it to the sphere. So, it admits a metric, $$\mathrm{d}s^2 = \mathrm{d}r^2 + r^2 \sin^2 \phi \mathrm{d}\theta^2 + r^2 \mathrm{d}\phi^2$$ from ...
6
votes
3answers
215 views

Interior product of differential forms

The interior product of a 2-form $\beta$ and vector field $X$ is defined by $(i_x\beta)(Y)=\beta(X,Y)$ where $Y$ is a vector field. This is the definition of a 2-form (and it's similar for a ...
4
votes
1answer
79 views

What's the mistake in this application of differential forms to vector calculus?

This is the first time I try to apply the calculus of differential forms to make some computation so sorry if I say something very silly. My try was the following: $M=\mathbb{R}^3$, and ...
1
vote
1answer
29 views

Verifying that for w a Contact Form , dw is not zero on Contact Planes.

I'm relatively new to contact forms, and to differential forms in generals; please forgive if this is too simple: I want to show that if $w$ is a contact form (say for a 3-manifold $M^3$), then $dw$ ...
0
votes
1answer
66 views

How to proceed this computation with differential forms?

I've been studying Spivak's differential geometry book and he defines the exterior derivative of $\omega \in \Omega^k(M)$ in a coordinate system $(x,U)$ by $$d\omega = d\omega_{i_1\cdots i_k}\wedge ...
3
votes
1answer
88 views

Homework: calculation about differential form

Here is the question: Let $\omega = A dy\wedge dz + B dz \wedge dx + C dx \wedge dy$ in $\mathbf{R}^3$, and $d\omega = 0$. Denote \begin{eqnarray} \alpha = \int_0^1 tA(tx,ty,tz)dt\cdot(ydz-zdy)\\ ...
3
votes
1answer
86 views

What exactly are n-forms and how are they related to dual vectors?

I'm trying to get a hold of tensor analysis on manifolds and the idea of vectors and tangent spaces are just starting to click, but I don't really get how the differential of a function can be viewed ...
8
votes
1answer
221 views

Why are de Rham cohomology and Cech cohomology of the constant sheaf the same

I am comfortable with de Rham cohomology, sheaves, sheaf cohomology and Cech cohomology. I am looking to prove the following theorem: If $M$ is a smooth manifold of dimension $m$, then we have ...
6
votes
2answers
155 views

Why 'closed differential forms' are called 'closed'?

As is well known a differential form $ \omega $ is called closed differential form if it satisfies $ \mbox{d} \omega = 0 \, \, (\ast) \, $ where $ \mbox {d} $ is the exterior derivative. I think the ...
1
vote
1answer
44 views

If differential 1-forms agree on chains with integer coefficients, are they equal?

Let $M$ be a real, smooth manifold. Let $\omega_1$ and $\omega_2$ be differential 1-forms on $M$, and let $C_1(\mathbb Z,M)$ denote the set of 1-chains with integer coefficients. If \begin{align} ...
8
votes
1answer
194 views

When can a functional be written as the integral of a 1-form?

Let a real, smooth manifold $M$ be given. Let $\Gamma$ denote the set of all path segments on $M$, namely the set of all paths of the form $\gamma:[a,b]\to M$. Let $Q:\Gamma\to\mathbb R$ be a ...
5
votes
1answer
85 views

Pullback of a form using the Hopf fibration

I embed the 2-sphere and the 3-sphere in $\mathbb{R}^3$ and $\mathbb{R}^4$ respectively. Then denote by $\{x_1,x_2,x_3,x_4\}$ the coordinates on the 3-sphere and $\{y_1,y_2,y_3\}$ on the 2-sphere. So ...
2
votes
1answer
125 views

Exercice on a differential form

Let $\omega$ be a $q$-form on $\mathbb{R}^2$ and let $Z_{\mathbb{R}^2}(dx_1)=\{p \in \mathbb{R}^2 \colon (dx_1)_{|p}=0\}$ $Z_{S^1}(dx_1)=\{p \in S^1 \colon (dx1_{|S^1})_{|p}=0\}$ where ...
3
votes
2answers
119 views

Explanation for the integral of differential forms

In our course of differential geometry we defined the integral $\int_{U} \omega$ of a differential form $\omega=f dx_1\wedge \ldots \wedge dx_n: T^nU \rightarrow \mathbb R$ with $U\subseteq \mathbb ...
1
vote
0answers
54 views

Structure equations on the 3-sphere

On the 3-sphere I have found the vector fields $X_1=(-x_2,x_1,-x_4,x_3)$, $X_2=(-x_3,x_4,x_1,-x_2)$, $X_3=(-x_4,-x_3,x_2,x_1)$, in the basis $\left\{\frac{\partial}{\partial ...
3
votes
2answers
77 views

What is the value of the 2-form $X(u(Y))-Y(u(X))-u([X,Y])$, is it $2du(X,Y)$?

Let $X,Y$ be two vector fields on a Riemannian manifold $(M,g,\nabla)$ where $\nabla $ is the symmetric metric connection on $M$ and let $u$ be a 1-form. I want to find the value or a simple form or ...