2
votes
2answers
38 views

Exercise on left invariant 2-forms

I'm trying to solve a problem about the Lie group of transformation over $\mathbb{R}$: \begin{equation} x\mapsto ax+b, \end{equation} where $a,b\in\mathbb{R}, a\neq 0.$ I'm asked to find the space of ...
1
vote
2answers
35 views

Given an $(n-1)$-form $\varphi$ on a smooth orientable $n$-manifold, there is a vector field $v$ such that $i_v\varphi = 0$.

I am working on the following problem. Let $M$ be a smooth orientable $n$-manifold, $n \geq 2$, and let $\varphi$ be a smooth $(n-1)$-form on $M$. Show that there is a vector field $v$ on $M$ such ...
2
votes
1answer
33 views

Bogus proof that the Liouville Form on the cotangent bundle is nondegenerate.

Suppose we have a manifold $M$ of dimension $n$ and its cotangent bundle $T^*M$. The Liouville form $\lambda$ on $T^*M$ is defined as $\lambda_{\omega_p} = \pi^*(\omega_p)$ where $\pi$ is the standard ...
0
votes
1answer
19 views

2-form corresponding to a contravariant vector and pseudo-forms

In Frankel's book he writes that in $R^{3}$ with cartesian coordinates, you can always associate to a vector $\vec{v}$ a 1-form $v^{1}dx^{1}+v^{2}dx^{2}+v^{3}dx^{3}$ and a two form $v^{1}dx^{2}\wedge ...
1
vote
0answers
24 views

Reference Request for differential ideals of Pfaffian forms on jet bundles

My setting is the following: Given two families of differential forms $\omega^i$ (with $i=1,...,m$) and $\mathrm d F^j$ (with $j=1...n-m$), define $$\eta^i := \sum_{j=1}^m a^i_j\omega^j + ...
2
votes
1answer
37 views

Differential Form Over $S^2$

I was checking problems on differential forms and I found the following one. Consider the sphere $S^2 \subseteq R^3$ and the map $\omega_p : T_pS^2 \times T_pS^2 \rightarrow \mathbb{R}$ given by ...
0
votes
0answers
31 views

$(1,0)$-forms on a complex manifold

I'm reading in a paper: "Let $\theta$ be a $(1,0)$-form with respect to $I$ ($I \theta = -i \theta$)". Here $I$ is the almost complex structure. Any idea why it says $I \theta = -i \theta$ and not $I ...
0
votes
1answer
39 views

Equality in de Rham cohomology

Let $U_1,U_2,...,U_r$ be open sets in $\mathbb{R}^n$ such that $U_i\cap U_j =\emptyset$ for all $i \neq j$. Then prove, $H^k_{dR}(\bigcup_{i=1}^{r} U_i)=\bigoplus_{i=1}^{r} H^k_{dR} (U_i)$
1
vote
1answer
30 views

De Rham cohomology group

We know $m$-th de Rham cohomology group on $U$ is defined to be, $H^{m}_{dR}(U)=ker(d^m)/im(d^{m-1})$ where $d^m:\Omega^m(U)\to \Omega^{m+1}(U)$'s are usual exterior derivative maps. Now its saying ...
3
votes
1answer
29 views

Calculating the pullback of a $2$-form

I have a $2$-form given by $\omega = dx \wedge dp + dy \wedge dq$ and a map $i : (u,v) \mapsto (u,v,f_u,-f_v)$ for a general smooth map $f : (u,v) \mapsto f(u,v)$. I want to calculate the pullback of ...
1
vote
0answers
35 views

differential form: a question in Novikov's book

I met a problem when reading the book of Novikov et al. "Modern geometry, vol 1". In page 200 of the book (GTM 93, English version), they say for 2-form $F$, $$ (F\wedge F)_{0123}=-{1\over ...
4
votes
3answers
90 views

Interior product between differential forms and vector fields

I don't understand what is meant when someone writes that forms (or form fields) "eat" vectors (or vector fields). For example when I have a one form field ω=3dx+5dy+3xdz and a vector field ...
1
vote
0answers
57 views

Integrability of 1-forms and Stokes' Theorem

Let $\alpha$ be a $1$-form defined on a manifold $M$ and $\Delta = ker (\alpha)$. The classical theorem of Frobenius says that $\Delta$ is integrable if $\alpha \wedge d\alpha =0$ i.e if $d\alpha$ is ...
0
votes
0answers
47 views

Differential forms and minor expansion, question about notation.

There are lectures by Theodore Shifrin on differential forms, and sadly one video ends suddendly where he explains some notation. I try to formulate it in my own words: When k=n, we have ...
4
votes
0answers
57 views

Differential forms and determinants

2-forms are defined as $du^{j} \wedge du^{k}(v,w) = v^{j}w^{k}-v^{k}w^{j} = \begin{vmatrix} du^{j}(v) & du^{j}(w) \\ du^{k}(v) & du^{k}(w) \end{vmatrix}$ But what if I have two concret ...
3
votes
0answers
37 views

Second fundamental form of a graph of a function using frame fields

I am trying to recover the second fundamental form $II$ for the graph $\Gamma(f)$ of a smooth real function $f$ in the Euclidean space $\mathbb{R}^3$ using first order frame fields. I have worked with ...
1
vote
2answers
135 views

Is a 1-form locally expressible as $dx$?

Given a 1-form $\alpha$ which is non-zero at every point of a manifold $M^n$, is it true that locally I can express the form as $dx$? (that is around each point there is a coordinate neighborhood such ...
3
votes
1answer
62 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 ...
3
votes
1answer
64 views

Integration on $\mathbb{R}^n$ in terms of differential forms

One defines integration on a smooth manifold as follows: First define $\int_M \omega$ when $\omega$ is supported on a single coordinate chart by pulling back to $\mathbb{R}^n$ an integrating there, ...
1
vote
0answers
36 views

Two isomorphisms of the space of vector fields/forms

Suppose you have a diffeomorphism $$ \phi: M \to N. $$ As far as I understand, this gives rise to two distinct isomorphisms $$ a : \mathcal A^1(M) \cong \mathcal A^1(N) :b $$ between the space of ...
7
votes
2answers
169 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
31 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
63 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
72 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
56 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
50 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
41 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
26 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
89 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
37 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
68 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
46 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
55 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
57 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
102 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
77 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
34 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
52 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
58 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
94 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
66 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 ...