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

learn more… | top users | synonyms

0
votes
1answer
32 views

$\mathbb{P}_{\mathbb{C}}^3$ is not isomorphic to $S^2 \times S^4$

I have been trying to solve this exercise given by my prof. The hint is to show that every $2$-form $w$ on $S^2 \times S^4$ is s.t. $w \wedge w = 0$, while this is not true in case of ...
2
votes
2answers
78 views

How to understand the notion of a differential of a function

In elementary calculus (and often in courses beyond) we are taught that a differential of a function, $df$ quantifies an infinitesimal change in that function. However, the notion of an infinitesimal ...
1
vote
0answers
13 views

Showing that $\|.\|$ is a norm of the space of 1-forms $\Omega^1(U)$, where $U\subset\mathbb{R}^n$.

Let $U\subset\mathbb{R}^n$ and let $\Omega^p(U)$ denote the vector space of $p$-forms ($p\in\mathbb{N}$). Define the isomorphism $\Phi:\Omega^{1}(U)\to\Omega^{n-1}(U)$ as ...
4
votes
3answers
571 views

Differential Forms Help

I have a background in Analysis, specifically with Baby Rudin. However, as many people note, Rudin does not do a very good job discussing differential forms. Could someone please refer me to an ...
0
votes
0answers
34 views

Cut a hole in a Riemann surface

Consider $R$ a Riemann surface of genus $g\ge 2$ with no border component. Suppose we want to cut a hole in $R$: for example consider $J\subset R$ an embedding of the interval $[0,1]$ and then cut a ...
0
votes
0answers
15 views

Compact Poincaré dual of $S^{n-1}$ in $\mathbb{R}^n \backslash \{0\}$

I have been asked to find a sub manifold $S \subset M := \mathbb{R}^n \backslash \{0\} $ s.t. its compact Poincaré dual is a basis of the first cohomology group $H^1_c(M) = \mathbb{R}$. Now, $S$ must ...
0
votes
1answer
19 views

Symplectic form on $T^ ∗X$

If $\mu$ is any closed 2-form on a manifold $X$, then $d\alpha_X + π^∗\mu$ is also a symplectic form on $T^ ∗X$, where $\alpha_X$ is tautological one-form and $\pi:T^*X\to X$ is projection. In fact, ...
4
votes
3answers
591 views

Exterior derivative of a complicated differential form

Let $\omega$ be a $2$-form on $\mathbb{R}^3\setminus\{0\}$ defined by $$ \omega = \frac{x\,dy\wedge dz+y\,dz\wedge dx +z\,dx\wedge > dy}{(x^2+y^2+z^2)^{\frac{3}{2}}} $$ Show that $\omega$ is ...
4
votes
0answers
38 views

Clifford, $p$-forms and spinors

I'm trying to understand the paper by Atiyah, Hitchin and Singer called: ''Self-duality in four dimensional Riemannian geometry", available here. I'm stuck at the point where it explains how the ...
4
votes
1answer
33 views

Finding all $2$-forms in the right half-plane that are invariant under glide transformations

I'm trying to find all 2-forms $\omega$ that are invariant under glide transformations in the right half-plane, $X = \{ (x,y) \in \mathbb{R}^2 : x > 0\}$. To do this, we can write the vector field ...
1
vote
1answer
37 views

Unit square as union of two simplexes

? If Rudin regarded $\Phi$ as a function of $2$-forms and suppose that $\omega=f(\mathbf{x})dx_{i_1}\land dx_{i_2}$ is $2$-form on $\mathbb{R}^m$ then $$I_{\Phi}(\omega)=\int ...
1
vote
0answers
22 views

Examples of compactly supported exact differential forms [on hold]

I am having some trouble finding any examples of compactly supported exact differential forms on $\mathbb{R}^n$. I have found $e^{\frac{1}{x^2 -1}}$ when taken to be zero everywhere except on the ...
1
vote
0answers
29 views

Boundary of oriented $k$-simplex from PMA Rudin

But paragraph which I marked by red line seems to me confusing. Let $k=3$ then $\sigma=[\mathbf{p}_0,\mathbf{p}_1,\mathbf{p}_2,\mathbf{p}_3]$ and $\partial ...
0
votes
1answer
55 views

Finding A 1-form on $R^2 - {\{(0,0)}\}$

I want to find a 1-form on $R^2 - {\{(0,0)}\}$ such that $w(Y) = 0$ and $w(X) = 1$. Here, $$X = -y\frac{\partial }{\partial x} + x\frac{\partial}{\partial y}\ \text{and}\ Y = x\frac{\partial ...
2
votes
1answer
49 views

A necessary and sufficient condition for the admittance of integrating factor

Let $\omega$ be a smooth 1-form on a smooth manifold $M$. A smooth positive function $\mu$ on some open subset $U\subset M$ is called an integrating factor for $\omega$ if $\mu\omega$ is exact on ...
1
vote
2answers
38 views

Specific example of integrating a 1-form over a curve

I was given the following definition in my course but no corresponding examples: Supppose $\gamma:[a,b]\rightarrow{M}$ is a smooth curve and $\omega$ a 1-form on $M$ (so $\omega:M\rightarrow{T^*M}$). ...
2
votes
1answer
57 views

Affine chains from PMA Rudin. Confusing examples

I understood the definition of affine $k$-chain and that he defines $\int \limits_{\Gamma} \omega$ as $(82)$. But I can't understand the last two above examples. What does they mean? Can anyone ...
0
votes
0answers
24 views

Equivalent conditions for $\mathfrak{F}$ to be a differential ideal

Heres the question: Let $\mathfrak{F}$ be an ideal of forms on a manifold $M$ locally generated by $r$ independent $1$-forms. Say $\mathfrak{F}$ is generated by $\omega_1,\ldots, \omega_r$ on $U$. ...
0
votes
0answers
28 views

Let $f : \mathbb{R}^n \to \mathbb{R}^n$, suppose that $\mu = dx^1\wedge\ldots\wedge dx^n$, then $f^{\ast}\mu = \det (df)\mu$

Let $f : \mathbb{R}^n \to \mathbb{R}^n$, suppose that $\mu = dx^1\wedge\ldots\wedge dx^n$, then $f^{\ast}\mu = \det (df)\mu$. I am trying to prove this. I made several low dimensional cases and ...
2
votes
1answer
54 views

Confusing moment in Theorem 10.27 from PMA Rudin

Theorem 10.27 If $\sigma$ is an oriented rectilinear $k$-simplex in an open set $E\subset \mathbb{R}^n$ then $$\int \limits_{\overline{\sigma}}\omega=\varepsilon\int \limits_{\sigma}\omega \qquad ...
2
votes
1answer
27 views

Prove the exterior derivative of the following (n-1) form is zero

Let $\omega(x)=\frac{1}{{\parallel x \parallel}^n}\displaystyle\sum_{i=1}^{n}(-1)^{i-1}x_{i} dx_{1} \wedge \dots \wedge \widehat{dx_{i}} \wedge \dots \wedge dx_{n}$ be a differential $(n-1)$ form on ...
2
votes
0answers
123 views

How to define the boundary operator using the exterior derivative?

I am looking for a way to define the boundary operator $\partial : M^n \to N^{n-1}$ from an $n$-dimensional manifold $M$ to its boundary $N$ using the the expression \begin{equation*} \int_M d \alpha ...
9
votes
3answers
196 views

What is the relation between dx in elementary calculus and dx in differential geometry?

I've recently started studying differential geometry and was really hoping that in doing so I'd finally have an answer to something that's been bugging me since I first learnt calculus - what is ...
-1
votes
2answers
17 views

How can I convert this second order equation into a first order equation? [closed]

In a previous exercise sheet, we were asked to transform the second order differential equation $$ x'' = -x + \alpha x^{3} $$ Into a first order equation. The solutions have since been released, but ...
0
votes
2answers
48 views

Example of exact form

Consider the differential 1-form $\omega = ydx+dy$. I need to show that this is not exact, and find an example of a function $G(x,y)$ such that $G\omega=G(x,y)(ydx+dy)$ is an exact form. I have done ...
0
votes
1answer
40 views

Theorem 10.22 from PMA RUdin

We know that $(dy_I)_T=dt_{i_1}\land \dots \land dt_{i_k}$ and using definition 10.18 we get $$d((dy_{I})_{T})=d1\land dt_{i_1}\land \dots \land dt_{i_k}=0$$ since $dc=0$ for any $c\in ...
1
vote
1answer
34 views

Derivative of $0$-form

Rudin states that $1$-form $xdy$ is not the derivative of any $0$-form. By contradiction, suppose that that exists $0$-form whose derivative is $xdy$. Then $f\in C'$ and $df=xdy$. But ...
0
votes
1answer
21 views

Are curvature forms in complex line bundles symplectic

I know that the curvature form $F_\nabla$ of a connection $\nabla$ in a complex line bundle $L \to B$ is presymplectic (i.e. antisymmetric and closed). Does it also have to be non-degenerate, i.e ...
4
votes
1answer
35 views

Noncanonical isomorphism of spaces of differential forms

Let $\pi: V \to M$ be a smooth $n$-dimensional vector bundle over $M$. Are the spaces of differential forms $\Omega^i(V)$, $\Omega^i(V^*)$ noncanonically isomorphic? If so, how do I see this? Is there ...
2
votes
2answers
65 views

Frank Warner's definition of the Hodge star

Frank Warner's book, chapter 2, excercise 13 states the following: If $V$ is an oriented inner product space ($n$ dimensional) there is a linear map $\ast \colon \Lambda (V) \to \Lambda (V)$, ...
1
vote
1answer
78 views

Does such property exist for differential forms?

I am studying differential forms from Rudin's PMA. Here's the definition of $k$-form Also he proves the anticommutative relation: $dx_1 \land dx_2=-dx_2\land dx_1$ Does the following expressions ...
1
vote
1answer
41 views

Method to calculate the de Rham cohomology of $\mathbb{R}\mathrm{P}^n$

I'm trying to follow through a method to calculate the de Rham cohomology groups of $\mathbb{R}\mathrm{P}^n$ from the de Rham cohomology groups of $S^n$. I'm trying to show that differential k-forms ...
10
votes
1answer
130 views

Dimension of diffeomorphism groups preserving some $2$-tensor.

For a finite-dimensional smooth manifold $M$, let $\mathrm{Diff}(M)$ be its diffeomorphism group. Suppose we are given a $2$-tensor $\mathcal{K}$ on $M$, and let $$\mathrm{Diff}_{~\mathcal{K}}(M) = ...
0
votes
0answers
15 views

How to show that a tensorial field,$t$, $k$-times covariant over a differential variety $M$ is differentiable by its components

How can i show that a tensorial field,$t$, $k$-times covariant over a differential variety $M$ is differentiable if and only if the following functions (The components of t) are differentiable.: ...
2
votes
1answer
41 views

What is the skew-symmetric part of the covariant derivative of a one-form?

This is a followup question to here. Let $E \to M$ be a vector bundle with connection $D := \nabla$. Extend $D$ to $E^*$ and $\text{Hom}(E, E)$. Let $E = TM$ here, and suppose that the torsion is ...
0
votes
0answers
39 views

vanishing of differential form on connected compact manifold

Let $M$ be a $n$ dimensional compact connected manifold. Let $\alpha$ be a differential form of degree $n$ such that $$ \int_{M} \alpha = 0 $$ then I would like to show that $M$ vanishes at at least ...
3
votes
1answer
55 views

Zeroes of exact differential forms on compact manifold

Let $M$ be a $n$ dimensional compact differentiable manifold. I would like to show that any exact differential form of degree $n$ vanishes at at least one point. I think it is a generalization of the ...
0
votes
0answers
32 views

Integration of forms on manifolds

If you have a $n$-form $\omega$ on $\mathbb{R}^n$, then $\omega = f \mathop{}\!\mathrm{d}x_1 \wedge \dots \wedge \mathop{}\!\mathrm{d}x_n$ locally. Integrating $\omega$ is easy now - let's assume ...
2
votes
1answer
26 views

Prove that $p-$form $\omega$ on $M\times N$ is $\delta \pi(\alpha)$ iff $i(X)\omega=0$ and $L_X\omega=0$

Let $M$ be connected and let $\pi:M\times N \rightarrow N$ be the natural projection. Prove that $p-$form $\omega$ on $M\times N$ is $\delta \pi(\alpha)$ for some $p-$form $\alpha$ on $N$ if and ...
3
votes
1answer
124 views

How to derive the standard symplectic form on a 2-sphere in cylindrical polar coordinates?

I have already know that at a point $p\in S^2$ with $u,v$ the tangent vectors at $p$. Then the standard symplectic form is $\omega_p(u,v) :=\langle p,\,u\times v\rangle$ where $u\times v$ is the outer ...
2
votes
0answers
41 views

What is an integral differential form and how do we recognize it as such?

I am reading about embedding theorems of various types of manifolds (Kodaira's embedding theorem being one of them), and one condition that repeats in all of them is that the manifolds should be ...
0
votes
1answer
29 views

Showing that the exterior derivative of a 1-form is 0.

The question is $$ Let\quad f : \Bbb R \to \Bbb R$$ $$\omega = f(||\mathbf x||)(\sum_{i=1}^n x_{i}dx_{i}) \in \mathcal A^1(\Bbb R^n) $$ $ (a) $ Assuming f is differentiable, prove that $d\omega = 0$ ...
2
votes
2answers
46 views

Integrate ${xdy+ydx} \over {x^2+y^2}$ on the circle $(x-1)^2+(y-1)^2=1$

I tried to integrate ${xdy+ydx} \over {x^2+y^2}$ on the circle $(x-1)^2+(y-1)^2=1$ counterclockwise. Used Grin's theorem, then went to polar cordinates but can't integrate the expression I got. So I ...
-1
votes
1answer
41 views

How am I supposed to answer this question?

I've got the following question: The first part of this question I can do. I've deduced that the DE has oscillatory solutions for all $\lambda > 16$; that the Eigenvalues are given by $$ ...
0
votes
0answers
31 views

Elementary properties of diff. form from PMA Rudin

How he got $d\omega$ in the RHS of $(39)$? Or maybe it's a typo?
0
votes
3answers
45 views

Reference request: integration of *one*-forms along curves on a differentiable manifold.

Could somebody please direct me to a book/lecture notes with an introduction to integration of one-forms along curves in a differentiable/Riemannian manifold -- preferably leaning more towards ...
0
votes
1answer
24 views

Proof Verification: A differentiable aplication $\psi : M\to N$ is differentiable if and only if: $\psi^{*}f\in C^{\infty}(M)$

Show that a differentiable aplication $\psi$ over $M$ to a differentiable variety $N$ is differentiable if and only if: $$\psi^{*}f\in C^{\infty}(M)$$ For: $f\in C^{\infty}(N)$ Where $\psi^{*}f$ is ...
1
vote
1answer
28 views

The value of the integral of the curvature of a complex line bundle

I am trying to show that if $L \to M$ is a complex line bundle endowed with a connection $\nabla$, $F$ is the curvature form, and $S \in M$ a closed surface, then $\int \limits _S F \in 2 \pi \textrm ...
0
votes
1answer
29 views

Prove the pullback of the wedge product is the wedge product of the pullbacks.

Let $F:V \rightarrow W$ be a linear map. Show that $F^{\ast}(\omega \wedge \eta)=(F^{\ast}\omega) \wedge (F^{\ast}\eta)$ for all $\omega \in \Lambda^{p}(W) , \eta \in \Lambda^{q}(W)$. Where ...
0
votes
1answer
36 views

Integral of Differential 1-form

Let $\omega$ be the closed $1$-form $\omega = \frac{xdy - ydx}{x^2 + y^2}$ and let $S$ be the unit circle and let $C = \{(x,y) : (x-3)^2 +y^2 = 1\}$. I'm trying to find $\int _S \omega$ and $\int ...