Questions tagged [differential-forms]
For questions about differential forms, a class of objects in differential geometry and multivariable calculus that can be integrated.
3,888 questions
3
votes
1
answer
109
views
When is the Simple Tensor Product of Non-Zero Alternating Multilinear Forms Alternating?
I am exploring the relationship between the tensor product ($\otimes$) and the exterior product ($\wedge$) of multilinear alternating forms (differential forms/covectors).
Let $V$ be a vector space ...
- 75
0
votes
0
answers
33
views
Wedge product in characteristic r [closed]
I very much prefer in characteristic zero the Kobayashi-Numizu definition of wedge product in terms of the Alternator Alt without the coefficient (p+q)!/(p!q!), but reading Warner I realized that the ...
- 75
4
votes
1
answer
131
views
Question on a proof by Spivak
I am studying the proof of the following theorem from Spivak's book “A comprehensive introduction to differential geometry"
In the last passage of the proof (I think), Spivak uses the change of ...
- 921
1
vote
0
answers
88
views
Difference between the Levi-Civita symbol and the volume form
Let $B = \{e_1, \dots, e_n \}$ be an orthonormal basis of $T_pM$ and $B^*=\{\theta^1,
\dots , \theta ^n\}$ the basis of $T_pM^*$. My professor defined the volume form (or the Levi-Civita tensor) as:
$...
- 833
0
votes
1
answer
69
views
Where does this partition-of-unity argument not work in general?
I'm trying to solve problem 15-1 from Lee's Introduction to Smooth Manifolds, 2nd ed which states
Suppose $M$ is a smooth manifold that is the union of two orientable open
submanifolds with connected ...
2
votes
0
answers
47
views
Splitting of the Chiral de Rham differential for affine space
I am currently reading through Malikov, Schechtman and Vaintrob's paper Chiral de Rham Complex. In the proof of Theorem 2.4, i.e. that the chiral de Rham complex extends the usual de Rham complex for ...
3
votes
1
answer
119
views
Why isn't $\mathbb{RP}^n$ always orientable?
If $G$ is a Lie group and $H$ is a closed subgroup, the homogeneous space $G/H$ admits a $G$-invariant volume form if and only if ${\Delta_G}_{|H} = \Delta_H$ (where $\Delta_G$ and $\Delta_H$ are the ...
0
votes
0
answers
38
views
Question if a lemma from Miranda is classical u-sub
In Miranda’s Algebraic Curves & Riemann surfaces, in chapter IV. Integration on manifolds, Lemma 3.9(f) reads:
If $F:X\to Y$ is a holomorphic map between Riemann surfaces, then the operation (push ...
- 5,320
2
votes
1
answer
187
views
Problem out of Miranda on Zero mean theorem
Let $X$ be a compact Riemann surface and suppose the zero mean theorem holds:
Zero mean Theorem (p.318 Miranda):If $X$ is an algebraic curve and $\eta$ is a $C^\infty(X)$ $2-$form on it,
then there ...
- 5,320
1
vote
0
answers
139
views
Miranda X.$2$.F.
Let $X$ be a compact Riemann surface. Let Bar: $\Omega^1(X)\to H^{(0,1)}_{\bar{\partial}}(X)$
by sending $\omega$ to the equivalence class of $\bar{\omega}$ is $\Bbb{C}-$linear, and $1-1$.
Attempt:
So ...
- 5,320
1
vote
0
answers
140
views
IV.I.$1$ out of Miranda Algebraic curves & Riemann Surfaces.
Let $L$ be a lattice in $\Bbb{C}$, and let $\pi$ be the natural protection. Show that $dz,d\bar{z}$ are well-defined holomorphic $1-$forms on $X=\Bbb{C}/L$.
Attempt:
I used charts because 2 are enough ...
- 5,320
1
vote
1
answer
80
views
Different (but equivalent) expression of a pullback
Consider the map $\varphi: M \to N$, $x^i$ a coordinate system on $M$ and $x'^i$ a coordinate system on $N$. $\alpha$ is a form.
I was given the "fact" that $$(\varphi^*\alpha)_i(p) =\frac{\...
- 845
0
votes
0
answers
55
views
On scattering theory of Riemann Surfaces
In this 2025 paper on Scattering theory Scattering theory on Riemann Surfaces, I have some technical questions when tying it together with what I have learned in my courses. Let $R$ be a compact ...
- 5,320
2
votes
1
answer
156
views
On notational conventions between Bott & Tu Vs. Lee for differential forms
Since I have been introduced to differential forms, I have seen (naively speaking) when you apply the exterior derivative, you "wedge" together one additional $d$ of the variable in question ...
- 5,320
1
vote
1
answer
55
views
Exact iff your integral over $n$ sphere is zero.
Show $\eta\in\Omega^n(\Bbb{S}^n)$ is exact iff $\int_{\Bbb{S}^n}\eta=0$.
Attempt:
($\Rightarrow)$ If $\eta$ is exact, there exists some $(n-1)-$form $\omega$ such that $\eta=d\omega$ then by Stokes',
$...
- 5,320
