1
vote
1answer
41 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
42 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
40 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
32 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
48 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
34 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
41 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
40 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
40 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 ...
4
votes
3answers
99 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
65 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
22 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
55 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
72 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
65 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
128 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
148 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
41 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
164 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
62 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
120 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
79 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
44 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
75 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 ...
1
vote
1answer
38 views

Proving Borsuk-Ulam with Stokes

What is the easiest way to deduce the Borsuk-Ulam theorem in the case $n=2$ by using integration on manifolds and Stokes theorem? So I want to prove the following: Given a map $f\colon S^2\rightarrow ...
0
votes
1answer
65 views

Condition for Differential Forms to Pass to the Quotient

everyone: I was reading this question : What do we mean when we say a differential form "descends to the quotient"? which is related to mine. But the reply given did not answer my question ...
1
vote
3answers
104 views

Prove $d(f\alpha)=d(f \wedge \alpha)$

I am reading the article http://en.wikipedia.org/wiki/Exterior_derivative and a definition of an exterior derivative from Axioms for the exterior derivative. How could I show that if $f$ is a function ...
5
votes
1answer
83 views

How do Chern classes behave under connected sums?

I often see people talking about Chern classes of manifolds like $\mathbb{CP}^2 \# \mathbb{CP}^2$, or $\mathbb{CP}^2\#\mathbb{CP}^2\#\overline{\mathbb{CP}^2}$, where $\#$ denotes connected sum and the ...
1
vote
1answer
123 views

What is the one form given its value for a vector field?

I read an article on vector fields. the author defined a 1-form on a manifold $M$ as $u(X)=\rho$ when $X$ is a given vector field and $\rho$ is a given real valued function defined on $M$. can we ...
2
votes
2answers
87 views

Integration of a 2-form

$\textit{What is}$ $\int_C{\omega}$ $\textit{where}$ $\omega=\frac{dx \wedge dy}{x^2+y^2}$ $\textit{and}$ $C(t_1,t_2)=(t_1+1)(\cos(2\pi t_2),\sin(2\pi t_2)) : I_2 \rightarrow \mathbb{R}^2 - ...
0
votes
0answers
56 views

Is a Differential 1-form ( on M^3)Dual to a Contact Vector Field a Symplectic Form?

say $w$ is a global contact 1-form on a 3-manifold $M^3$ , meaning $w \wedge dw \neq 0$ at any point in the manifold , and let $X$ be the vector field dual ( under, say, a choice of Riemannian metric ...
1
vote
2answers
106 views

Question about differential form

$\omega = y dx + dz$ is a differential form in $\mathbb{R}^3$, then what is ${\rm ker}(\omega)$? Is ${\rm ker}(\omega)$ integrable? Can you teach me about this question in details? Many thanks!
4
votes
0answers
100 views

Top de Rham cohomology

I just realized that I never really understood why $H_{dR}^n(M, \mathbb{R}) = \mathbb{R}$ if $M$ is compact and $H_{dR}^n(M, \mathbb{R}) = \{0\}$ if $M$ is not compact (provided that's true?). I'm ...
0
votes
0answers
64 views

differentiable equal r-forms

Let $\alpha$, $\beta$ be two $r$-forms continuous in $U\subseteq \mathbb{R}^n$ open. If $\int_M \alpha =\int_M \beta $ for all surface $M\subseteq U$   dimension $r$, compact, with boundary, then ...
3
votes
1answer
57 views

de Rham cohomology of $\mathbb{R}P^n$ via action by $SO(m+1)$

In lecture, my teacher proved the theorem that given a smooth $G$-action by a compact, connected Lie group on a manifold $M$, the de Rham cohomology of the $G$-invariant differential forms $H^p_G(M)$ ...
2
votes
1answer
77 views

Pullback distributes over wedge product

I'm looking to prove that the pullback of a smooth function distributes over wedge product, i.e. $$\varphi^*(\omega \wedge \eta) = \varphi^* \omega \wedge \varphi^* \eta. $$ Here the question is ...
2
votes
1answer
178 views

Closed but not exact one-form on $S^2$

I would like to know whether there is any nice prescription to give an example of a closed but not exact one-form on $S^2$ (not the $3$-ball). I assume to take some points out of this surface, e.g. 3. ...
2
votes
0answers
92 views

Volume form on $\mathbb{S}^2$

Let $\omega = x_1 \,dx_2 \wedge dx_3 + x_2 \,dx_3 \wedge dx_1 + x_3\, dx_1 \wedge dx_2$, with $(x_1,x_2,x_3) \in \mathbb{S}^2$, be a volume form on $\mathbb{S}^2$, and let $f : \mathbb{S}^1 \times ...
3
votes
1answer
43 views

1-form on $S^n$ with non-degenerate differential.

How it can be solved? Whether there is a $1$-differential form $w$ on $S^n$, such that $dw$ is non-degenerate on $T_aS^n$ for each $a \in S^n$?
2
votes
0answers
50 views

Help with local coordinates computations-differential forms

I am reading a lecture notes on differential forms and tangent bundles on complex manifolds and I got stuck on one line where the author does not explain how he did the computation: Let ...
2
votes
3answers
82 views

Simple criteria for “closed $\Longrightarrow$ exact”

In determining whether a closed form is an exact form, there is a lot of differential geometry definitions etc. that come in. I'm interested: what is the dummy, Calc III version of when closed implies ...
2
votes
3answers
98 views

On the scalar product of a vector field and the exterior derivative of a smooth map on a surface

I am reading Roger Penrose's wonderful book 'The Road to Reality: A Complete Guide to the Laws of the Universe'. I need some help in understanding the scalar product of a vector field and the ...
4
votes
1answer
390 views

How to calculate the pullback of a $k$-form explicitly

I'm having trouble doing actual computations of the pullback of a $k$-form. I know that a given differentiable map $\alpha: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ induces a map $\alpha^{*}: ...
8
votes
0answers
140 views

algebraic $1$-forms vs analytic $1$-forms

First let's fix some definitions: Definitions: Complex manifold (of dimension n): Is a locally ringed space $(X,\mathscr F)$, where there is an open cover $\bigcup_{i\in I} U_i=X$ such that ...
2
votes
2answers
94 views

Show that $a \wedge * b = g(a,b) \operatorname{vol}$

$\newcommand{\vol}{\operatorname{vol}}$ Let $\omega$ be a $p$-form on a Riemannian manifold $M^n$ with metric $g$ and let $\vol_{i_1,\ldots,i_n}=\sqrt{\lvert g\rvert} \epsilon_{i_1,\ldots,i_n}$ be a ...
6
votes
2answers
161 views

Why are differential forms more important than symmetric tensors?

In differential geometry, differential forms are totally anti-symmetric tensors and play an important role. I am led to wonder why do we not study totally symmetric tensors as much as forms. What ...
1
vote
0answers
37 views

invariance of 2-form under $SO(3)$

I'm trying to understand how to derive forms that invariant under action of some group. For example 2-form on $S^2$ and on $\mathbb{R}^3$ is very interesting for me (because I have troubles with it). ...
2
votes
0answers
53 views

differential of a differential form

Given a differential form $w$ on a manifold, I know how to calculate $dw$ in local coordinates. But is there any way to define $dw$ independent of local coordinates?
2
votes
2answers
82 views

p-forms as multilinear maps

I'm studying differential geometry and am learning about differential forms. We have a very intuitive and simple way to understand 1-forms as linear maps on from the tangent space to the base field, ...
1
vote
1answer
85 views

A (not so?) simple question about differential forms

Let $M^n$ be a compact orientable manifold and let $\omega$ be a $(n-1)$-form in $M^n$. I want to show that there is $p\in M$ such that $(d\omega)_p=0$. Can somebody help me, please ? Thanks :)