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

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32 views

Basic exercise differential forms

I have to show that the space of q-differential forms $\Omega^q(U)=\{0\}$ if and only if $q>n$ or $q<0$. Any ideas?
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1answer
51 views

What's going on with these identities involving $d$, $\mathcal L_X$, and $\iota_X$?

Let $\Omega^k$ denote the smooth $k$-forms on a given smooth manifold. Then we have the following operators: Exterior derivative: $d:\Omega^k\to\Omega^{k+1}$ (takes you to the right in the de Rham ...
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1answer
35 views

Stokes' theorem generalized the FTC part 2. Is there a known generalization for part 1?

Stokes' theorem generalizes the fundamental theorem of calculus (part 2) using differential forms. Is there a known generalization of part 1? edit In case anyone is unaware, The fundamental theorem ...
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2answers
77 views

would like proof of Theorem 4-10, from Spivak's Calculus on Manifolds

I'd like to understand a way to prove part 2 of this theorem. 2) If $\omega$ is a $k$-form and $\eta$ is an $l$-form, then $d(\omega\wedge\eta)=d\omega\wedge\eta+(-1)^k\omega\wedge d\eta$. I see ...
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0answers
22 views

Show that $d^2(w)=0$ for every $k$-form in $\mathbb{R}^n$

The question says to start showing this for 0-forms, and I did it. But how can I extend it for any k-form?
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3answers
36 views

Finding a $1$-form on a hyperboloid of one sheet which is closed but not exact

I want to find a closed 1-form $w$ on $$M=\{(x,y,z):x^2-y^2-z^2=-1\}\subset \mathbb R^3$$ which is not exact. I think that $$\frac{x\,\mathrm{d}x+y\,\mathrm{d}y+z\,\mathrm{d}z}{(x^2+y^2+z^2)^{3/2}}$$ ...
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1answer
60 views

Show that $f^*\omega = \det(df) \, dx_1\wedge\cdots\wedge dx_n$

Let $f:\Bbb{R}^n\to \Bbb{R}^n$ be a differentiable map given my $f(x_1,\ldots,x_n)=(y_1,\ldots,y_n)$, and let $$\omega=dy_1\wedge\cdots\wedge dy_n.$$ Show that $$f^*\omega = \det(df) \, ...
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1answer
38 views

Linear Maps from $V$ to $V^*$ defined by a 2-form

I came across this idea at the very beginning of a book and I don't quite seem to grasp it. It states given $\omega \in \bigwedge ^2 (V) $ you can define a linear map $ \omega^\#: V \to V^* $ by ...
3
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1answer
56 views

An exercise on differential forms

Let $\omega$ be a $1$-form in $U \subset \mathbb{R}^2$. A local integrating factor in $p$ for $\omega$ is a function $g: V \rightarrow \mathbb{R}$ defined in a neighbourhood $V$ of $p$ such that ...
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1answer
39 views

Closed form defines locally a function?

In my particular example, I have a volume form on a one-dimensional manifold, so i.e. this volume form $dx$ is closed. Now, I was wondering, does the integral function $f(y):=\int_{x_0}^{y} dx$ define ...
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3answers
51 views

The Euclidean Metric on $\mathbf R^3$ Induces an Index-Lowering Isomorphism $b:\mathfrak X(\mathbf R^3)\to \Omega^1(\mathbf R^3)$.

In Lee's Introduction to Smooth Manifolds, Second Edition, the line just before Equation 14.25 reads The Euclidean metric on $\mathbf R^3$ induces an index-lowering isomorphism $b:\mathfrak ...
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2answers
30 views

Closedness of a complex $1$-form defined by homogeneous functions

How can I prove this 1-form is a closed one within the specific subset $A\subset \mathbb C$ ? $$\omega=\frac{f(x,y)}{xf(x,y)+yg(x,y)}dx+\frac{g(x,y)}{xf(x,y)+yg(x,y)}dy$$ Where $f, g\in C^1$ are ...
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1answer
22 views

Notation for a projection of a differential form

Let $\omega = a_1 dx_1 + a_2 dx_2 + b_1 dy_1 + b_2 dy_2$. Is there any established notation to denote a mapping that "filters out" the $dy_i$-Terms? To be more precise, I invent my own one. Assume ...
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1answer
52 views

Can a volume form on a submanifold be extended to a parallel form in a neighbourhood?

Let $(M^{n+1},g)$ be a Riemannian manifold and let $\Sigma^n \hookrightarrow M$ be a smooth, closed, embedded submanifold. Let $\Omega$ be the volume form of $\Sigma$. It is well-known that a volume ...
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0answers
26 views

The lie bracket as a limit

Let $M$ be a manifold and $f_t$ be the monoparametric group of local transformations generated by a vector field $X$. Suppose $w$ is a $k$-form and $x_1,\dots,x_k$ are vector fields. How can we see ...
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1answer
48 views

Why does a differential form represent a vector field?

I'm trying to learn the Divergence/Stoke's theorem and I can't wrap my head around the meaning of a differential form in this context. What does it mean that a differential form represents a vector ...
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1answer
65 views

Computing the sheaf of 1-forms on a toric variety

Consider projective space $P^{2}$ and its corresponding fan. We have the affine opens defined by $U_{\sigma_{0}} = Spec(\mathbb{C}[x,y])$, $U_{\sigma_{1}} = Spec(\mathbb{C}[x^{-1},x^{-1}y])$ and ...
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0answers
57 views

Wedge product equality of 2n-forms in 2n+2 dimension

I have a question that seems to me clear but i couldn't prove it. I have a diffeomorphism between the cotangent bundle of the n-sphere and the complex quadric as $$T^*(S^n)\to (S^n)^{\mathbb{C}},\ ...
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1answer
37 views

Integration of $V$-valued differential form

When studying fibre bundles, connections and gauge theories it is usual to consider vector-valued differential forms, like the connection one-form, or it's pull back by a local trivialization known as ...
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1answer
42 views

Differential Forms on Surfaces. Show that $N\cdot (\nabla \times V)\eta=d\phi$ on $x(D)$.

Let $M$ be an orientable surface in $\Bbb R^3$ with a unit normal vector field $N$ and let $x: D\to M$ be a patch. Let $\eta$ be a differential 2-form on $x(D)$ defined by $\eta(x_u,x_v)=\pm\|x_u ...
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1answer
29 views

Exterior derivarive dependent only on point

For any one-form (a linear form on the tangent space of each point) we have its exterior derivative $d\omega$ which is a two-form defined by $d\omega(X,Y)=D_X(\omega(Y))-D_Y(\omega(X))-\omega([X,Y])$ ...
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1answer
41 views

How do you compute the pull-back of a complex differential (1,1)-form given its potential?

Let $F: X \to Y$ be a holomorphic map. Let $\omega$ be a complex differential $(1,1)$-form on $Y$, $\omega=\partial \bar \partial f$, where $f$ is a pluri-subharmonic function. How would one ...
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1answer
54 views

Symplectic Geometry of 2-sphere in stereographic projection

I am trying to put the symplectic form of the 2-sphere defined by $\omega_u(v,w) := \langle u,v\times w\rangle,$ where $u \in \mathbb{S}^2$ and $v, w \in T_u\mathbb{S}^2$ in stereographic coordinates ...
4
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2answers
196 views

Integrals of Pullbacks

This is a problem from Guillemin's Differential Topology: Suppose that $f_0, f_1: X \to Y$ are homotopic maps and that the compact boundaryless manifold $X$ has dimension $k$. Prove that for all ...
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0answers
20 views

why is $\omega^n \neq 0$ for a nondegenerate 2 form $\omega$. [duplicate]

Let $\omega$ be a nondegenerate alternating $2$-form on an $2n$-dimesional Vectorspace $V$, meaning that for all nonzero $v \in V$ the map $w \mapsto \omega(v,w)$ is not identically zero. Why is the ...
3
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2answers
30 views

Pushforward injective

Let $f : M \rightarrow N$ be a smooth surjective map between smooth manifolds. Now, consider a 2-form $\omega$ on $T_pN$. Does it now follow that the pullback satisfies? $f^* d \omega =0 \Rightarrow ...
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2answers
43 views

Exterior derivative of a form and $d(d\omega)=0$?

We know that in differential geometry, $d^2\omega=0$, where $\omega $ is a form and $d$ is the exterior derivative. However if this form happens to be the exterior derivative of another form ...
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1answer
42 views

(Co)Tangent bundle of Cone manifold

Given a Riemannian manifold $(M,\bar{g})$, we can construct the Riemannian cone manifold $(C(M), g )$ as follows. Topologically, $C(M)$ is $M \times \mathbb{R}_{>0}$. We equip this with the ...
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1answer
27 views

exact and closed differential forms

This exercise is taken from the Meyer-Hall-Offin book on Hamiltonian systems. Let $Q(p,q)$ and $P(p,q)$ be smooth functions defined on an open set in $\mathbb{R}^2$. Consider the four differential ...
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1answer
38 views

$e^{xy}dx \wedge dy$: determine the $1$-form that it induces on $S^1$ and check if the obtained $1$-form respects or not the induced orientation

Consider the $2$-form $e^{xy}dx \wedge dy$ on $\mathbb{R}^2$. Determine the $1$-form that it induces on $S^1$, viewed as the boundary of $B_2$. Check if the obtained $1$-form respects or ...
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1answer
53 views

Construct a $k$-form on $S^k$ with nonzero integral

How to construct a $k$-form on $S^k$ with nonzero integral ? I think this can be done by Bump function $\rho$ on $R^k$ and define $w = \rho\, dx_1\,dx_2 \cdots dx_k$ on $R^k$. Now pull it back by the ...
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2answers
15 views

If $σ$ is an exact differential $1$-form on the plane, then the form $ω=σ+xdy$ is not exact

If $σ$ is an exact differential $1$-form on the plane, then prove that the form $ω=σ+xdy$ is not exact. In the previous part of the question we have calculated the integral of the differential ...
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1answer
48 views

Differential calculation in multiple variables function (cannot understand 2nd order differential form)

This question is somehow related to this question. Consider a multiple variables function $G(u, v) = \left(\matrix{x(u,v) &=& G_x(u,v)\\y(u,v) &=& G_y(u,v)\\z(u,v) &=& ...
3
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1answer
52 views

Proving that the pullback map commutes with the exterior derivative

I'm trying to prove that the pullback map $\phi^{\ast}$ induced by a map $\phi:M\rightarrow N$ commutes with the exterior derivative. Here is my attempt so far: Let $\omega\;\in\Omega^{r}(N)$ and let ...
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1answer
33 views

Standard integral Kähler form on $\mathbf{CP}^1 \times \mathbf{CP}^2$

$\newcommand{\Proj}{\mathbf{CP}}$What is the "standard integral Kähler form" on $\Proj^1 \times \Proj^2$? Does that mean Fubini-Study form on $\Proj^1$ and $\Proj^2$?
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1answer
13 views

show that $K=E_2[\omega_{12}(E_1)]-E_1[\omega_{12}(E_2)]-\omega_{12}(E_1)^2-\omega_{12}(E_2)^2$?

$$K=E_2[\omega_{12}(E_1)]-E1[\omega_{12}(E_2)]-\omega_{12}(E_1)^2-\omega_{12}(E_2)^2$$ where $K$ is Gaussian curvature, $E_i$'s are tangent frame field on surface $M$ in $R^3$, $v[.]$ is directional ...
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1answer
52 views

Show $H_{dR}^1(S^n)=0$ for $n>1$ without de-Rham Theorem, and some similar questions.

Question Without using de Rham's theorem, prove: (1) Show $H_{dR}^1(S^n)=0$ for $n>1$. (2) Use (1) to show $H_{dR}^1(RP^n)=0$ (3) a n-form $\Omega$ is exact on $S^n$ if and only if $$ ...
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1answer
51 views

Notation for the set of holomorphic differential forms

I used to write $\Omega^p(M)$ for the set of $p$-forms on $M$ and $\Omega^{p,q}(M)$ for the set of $(p,q)$-forms on a complex manifold $M$. Now some authors use $\Omega^p(M)$ to denote the set ...
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0answers
49 views

Cohomology classes of the DeRham cohomology

May be $TM$ a tangent bundle of the manifold $M$ and $\wedge^n TM$ the set of all $n$-forms. The map $d: \bigwedge^n TM \rightarrow \bigwedge^{n+1}TM$ is called the exterior derivative and it holds ...
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0answers
34 views

Integral Over the N-Sphere in the framework of chains:

Integration over manifolds is commonly defined with object called chains. What about if I want to integrate the exterior derivative of a $k-form$ over the n-sphere and use Stokes theorem: ...
4
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1answer
69 views

Poincaré Lemma, differential forms and I do have troubles

I think I need some hints about a proof I am currently reading in order to understand it. This question is similar to the construction used in Lemma 17.9 in the book "Introduction to smooth manifolds" ...
2
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1answer
39 views

Pullback Solid Angle, Stereographic projection

I have an issue with a differential geometry task. Given is the solid angle form: $$\omega = \frac{\epsilon_{ijk} x^i dx^j \wedge dx^k}{2 [ (x^1)^2 + (x^2)^2 +(x^3)^2]^{3/2}}$$ The aim of the task ...
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1answer
62 views

The adjoint of left exterior multiplication by $\xi$ for hodge star operator

As we know, for $V$ vectoral space and a orientation $\mathcal{O}$ on $V$ and $e_{1},...,e_{n}$, the hodge star operator $\ast:\wedge V^*\rightarrow\wedge V^*$ is defined for ...
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1answer
45 views

Lie derivative for a wedge product $\omega_{1}\wedge\omega_{2}$

I have to prove that $L_X\omega_{1}\wedge\omega_{2}=(L_X\omega_{1})\wedge\omega_{2}+\omega_{1}\wedge(L_X\omega_{2})$ using the definition ...
2
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0answers
53 views

Volume form on $(n-1)$-sphere $S^{n-1}$

Let $\omega$ the (n-1) form on $\mathbb{R}^n$ $$\omega=\sum_{j=1}^{n}(-1)^{j-1}x_{j}dx_{1}\wedge\cdots\wedge \hat{dx_{j}}\wedge\cdots dx_{n}$$ Show that the restriction of $\omega$ to $S^{n-1}$ in ...
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1answer
65 views

For $(n-1)$-form $\omega$ on $M^{n}$ compact, orientable without boundary, then $d\omega$ vanish for some point

Let $M^{n}$ manifold compact, orientable without boundary and $\omega$ $(n-1)$-form then there is $p\in M$ such that $d\omega(p)=0$. This is for my homework of integration on manifolds & Stokes ...
4
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1answer
99 views

Is $\omega=\sin\varphi\,\mathrm{d}\theta\wedge\mathrm{d}\varphi$ an exact form?

Let $\omega=\sin\varphi\,\mathrm{d}\theta\wedge\mathrm{d}\varphi$ be a $2$-form on $\mathbb{R}^3\setminus\{0\}$. Then ...
0
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0answers
60 views

Is this differential 2-form closed

Consider a unit sphere $S^2 \subset R^3$ and a map $\omega_p : T_pS^2 \times T_pS^2 \to \mathbb{R}$ defined by $$\omega_p(u,v) = (u \times v) \cdot p$$ How do I know is this 2-form (on $S^2$) closed ...
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1answer
41 views

Poincare: Change of form to a primitive 1-form

Given a 2-form w on $R^3$, such that dw=0; how does one find all 1-forms k such that dk=w? Provided a homotopy H(t) one can pull back w and apply the Poincare operator on it. But what if one does not ...
0
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1answer
22 views

Deduce that there is a cup product that is well-defined

I have showed that if $\alpha$ and $\beta$ are closed forms on a smooth manifold $M$, then $\alpha \wedge \beta$ also be closed. Further, if one of $\alpha$ or $\beta$ is exact, than $\alpha \wedge ...