3
votes
1answer
54 views

Differential Forms on submanifolds

Say I take an embedded submanifold of $\Bbb R^n$, like the sphere. Any differential form on $\Bbb R^n$ can be restricted to the sphere. My question is this: is any differential form on the sphere (or ...
1
vote
0answers
52 views

A 1-form on a smooth manifold is exact if and only if it integrates to zero on every closed curve

I am stuck on the following problem, which comes from a old qualifying exam. Prove that a 1-form $\phi$ on $M$ is exact if and only if for every closed curve c, $\int_{c} \phi =0$. One way is an ...
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 ...
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
84 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 ...
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)\\ ...
0
votes
0answers
54 views

Proving $d(f^*e_i')=0$

Let $f:M\to N$ be a differentiable map between two manifolds. $e_i$ is a basis vector of $N$ with respect to some chart and $e_i'$ its dual (i.e. $e_i'(e_j)=\delta_{ij}$). How do I prove the ...
0
votes
1answer
36 views

pullback on vector fields and 1-forrms

On a Riemannian manifold $M$ one can identify 1-forms and vector fields by $$ \alpha(p) = \langle X(p),\cdot\rangle_p $$ Since we can perform a pullback on both 1-forms and vector fields I expected ...
8
votes
1answer
236 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 ...
5
votes
1answer
88 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 ...
1
vote
0answers
56 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 ...
2
votes
2answers
126 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 - ...
1
vote
1answer
61 views

$df$ vanish in a compact manifold in at least 2 points

I need to prove that if $M$ is a compact manifold and $f$ is a smooth function in $M$, then $df$ vanish in at least 2 different points of $M$. I don't know where to start. Any suggestion will be ...
4
votes
0answers
127 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 ...
3
votes
1answer
45 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 $a \in S^n$?
6
votes
2answers
215 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 ...
2
votes
0answers
67 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?
1
vote
1answer
62 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 ...
4
votes
1answer
87 views

Integrating 2-form

In $\mathbb{R}^3$ I consider the compact 2-dimensional manifold $$ M=\left\{(x,y,z)\in\mathbb{R}^2: z=xy\right\} $$ which is orientated by the (global) map ...
4
votes
1answer
53 views

Smooth homotopic maps and closed forms..

does anyone have any idea for showing the following: Let $f_0, f_1:M\rightarrow N$ smooth homotopic maps between the manifolds $M$ and $N$. Suppose $M$ is compact with no boundary. Show that for every ...
1
vote
0answers
80 views

Is Cartan's magic formula applicable to time dependent vector fields?

Cartan's magic formula states: $$\mathcal{L}_v\omega = i_v\mathrm{d}\omega + \mathrm{d}i_v\omega$$ Is this also true for time dependent vector fields? If so: How can I prove it? If not: Is there a ...
4
votes
2answers
171 views

Orientations on Manifold

This is a very basic definition of orientable and very basic example 20.5 however ı could not understand definition in an good way so ı want you to explain my green writing please :) and my example ...
0
votes
1answer
69 views

What is a de Rham k-form?

I generally know what a differential k-form is. But what does it mean for a k-form to be a "de Rham" k-form? Thanks in advance!
8
votes
2answers
736 views

Inducing orientations on boundary manifolds

Given a $k$-manifold $M$, such that $\partial M$ is a $(k-1)$-manifold, there is a standard way in which $\partial M$ inherits the orientation of $M$. So if $M$ is oriented by the form field $\omega$, ...
1
vote
1answer
158 views

How general formulation of Stoke's theorem relate to Kelvin-Stokes theorem

I asked a similar question, but I realized the question is too vague and it's better to start a new one: We know that there are two usually used formulations of Stoke's theorem. One is vector ...
1
vote
1answer
49 views

Question about Interior product computation

I need to evaluate $\omega=i_X(dx\wedge dy)$ where $X$ is a vector field in $\mathbb{C}^2$ (which means $p=2)$. If I write $X(x,y)=(X_1(x,y),X_2(x,y))$, or simply $X=(X_1, X_2)$, then the interior ...
7
votes
2answers
170 views

Dimension of de Rham Cohomology groups?

Is there a simple way to prove that the de Rham cohomology groups of a compact manifold $M$ have finite dimension as $\mathbb{R}$-vector spaces?
0
votes
1answer
52 views

Restricting the domain of an integral on a manifold

I would like to prove the following: Guess. Suppose M is an orientable smooth manifold without boundary and W is an open set in M. If $\omega$ is a smooth n-form on M such that $\operatorname{supp} ...
3
votes
2answers
531 views

Chain rule and differential forms

It is easy to show that the differential forms of order $1$ obeys a form of chain rule. To be precise, $d(f(g(x)) = f^\prime(x) d(g(x))$. This can be for example proved by fixing a co-ordinate basis ...
5
votes
1answer
134 views

Diffeomorphic riemannian manifolds and volume forms

Maybe the question will be stupid, but I'm a beginner in riemannian geometry... We have two riemannian manifolds $(M,g)$, $(\overline M,\overline g)$ and a diffeomorphism $F:M\rightarrow\overline M$ ...
2
votes
0answers
66 views

Hodge decomposition on a manifold with a nontrivial connection

I am familiar with the notion of Hodge decomposition of an arbitrary differential form into an exact form, a co-exact form, and a harmonic form. Given a curved space with a connection, could you ...
3
votes
2answers
584 views

Pullback of differential form of degree 1

Good evening, In differential forms (in the proof of the naturality of the exterior derivative), I don't get why if $h\in \Lambda^0(U)$ and $f^*$ is the pullback then, $f^*dh=d(f^*h)$. I wrote ...
2
votes
1answer
390 views

non-vanishing k-form on a k-manifold in $\mathbb{R}^n$ implies orientability

I want to know how to prove the theorem: If M is a k-manifold in $\mathbb{R}^n$, then it is orientable if and only if there is a volume form defined globally on M. I'm currently stuck at this step: ...
5
votes
1answer
310 views

Line integral and integration of differential forms

The definition of integral of a $k$-form $\omega$ over a parametrized manifold $ Y_\alpha $ is $ \int_{Y_\alpha}\omega = \int_A\alpha^*\omega$ where $ \alpha\colon A \to R^n $. Let $ \gamma:(a, b) ...
0
votes
1answer
341 views

How can I find the winding number of a curve?

I need to find the winding number of the closed curve $c(t)=(a \cos(t),b \sin(t))^T $, where $a,b > 0$ and $c:[0,2\pi) \to\mathbb{R}^2\setminus\{0\}$. I don't understand how to do this.
2
votes
1answer
554 views

Covariant derivative of tensor in terms of local coordinates

Let $f:U \to \mathbb{R}^3$ be a surface with local coordinates $f_i=\frac{\partial f}{\partial u^i}$. Let $\omega$ be a one-form. I want to express $\nabla \omega$ in terms of local coordinates and ...
4
votes
3answers
148 views

elementary question regarding differential forms

Is it possible to give a high level explanation why changing the order of differentials will give rise to a minus sign ? I.e. why do we have $$ dx\,dt = - dt\,dx $$ (I am going to take a course on ...
4
votes
1answer
129 views

Highest DeRahm Cohomology

Let $X$ be a $C^\infty$ manifold, compact oriented and connected of dimension $n$. How do you prove that the integration map $$\int_X: \omega \mapsto \int_X \omega $$ from $H^n_{DR}(X)$ to ...
0
votes
0answers
156 views

differential form vanish somewhere on compact orientable without boundary manifold [duplicate]

(a)Let $ M =M^n$ a $n-$dimensional compact, orientable, differential manifold without boundary ( $\partial M = \emptyset$). Define $\omega$ a differential $(n-1)-$ form on $M$. Prove that there ...
5
votes
1answer
301 views

differential form

one form $\alpha$ over a smooth manifold is non vanishing means for every $p\in M$, $\alpha_p\neq 0$. But $\alpha_p$ is linear map $T_M\to \mathbb R$, hence $\alpha_p(0)=0$. So confusion arises ...
4
votes
1answer
252 views

$4$-form $ \omega \wedge \omega$ vanishes on $S^4$

If $\omega$ is a closed $2$-form on $S^4$, how can I show the $4$-form $ \omega \wedge \omega$ vanishes somewhere on $S^4$? I am guessing that the fact we're talking about the $2$-form being ...
5
votes
1answer
721 views

Relationship Between Differential Forms and Vector Fields

I am trying reach an understanding of precisely how the space of differential forms is related to the space of vector fields. These are the definitions that I understand and am using for these ...