Questions tagged [differential-forms]

Question 1

In a previous post, I used an unusual method to derive the Gradient Theorem (i.e. the FTC for Line Integrals), and I simply reduced to the case of $\mathcal{M}=\gamma\subset\mathbb{R}^n$ to take ...

Question 2

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 ...

Question 3

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 ...

Question 4

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 ...

Question 5

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 ...

Question 6

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 ...

Question 7

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 ...

Question 8

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 ...

Question 9

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{\...

Question 10

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 ...

Question 11

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 ...

Question 12

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', $...

Question 13

$\newcommand{\pd}{\partial}$ $\newcommand{\wdw}{\wedge \cdots \wedge}$ $\newcommand{\cdc}{, \cdots ,}$ I'm trying to compute the divergence in general coordinates for the case of pseudo Riemannian ...

Question 14

I was reading the Wikipedia page on symplectic manifolds, and I think there might be a notational inconsistency in the section on Lagrangian and other submanifolds. The page defines, for a subspace $V ...

Question 15

TL;DR Is it possible to act a real-valued 1-form on a vector-valued 1-form? If so, How? For context, in a previous post of mine, I proposed an admittedly unnecessary and lengthy approach to deriving/...

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Questions tagged [differential-forms]

Contracting Metric Tensor and $df$ [closed]

Where does this partition-of-unity argument not work in general?

Splitting of the Chiral de Rham differential for affine space

Why isn't $\mathbb{RP}^n$ always orientable?

Question if a lemma from Miranda is classical u-sub

Problem out of Miranda on Zero mean theorem

Miranda X.$2$.F.

IV.I.$1$ out of Miranda Algebraic curves & Riemann Surfaces.

Different (but equivalent) expression of a pullback

On scattering theory of Riemann Surfaces

On notational conventions between Bott & Tu Vs. Lee for differential forms

Exact iff your integral over $n$ sphere is zero.

Is there any sign mistake in computing the divergence in general coordinates?

Possible typo in Wikipedia’s definition of isotropic and coisotropic submanifolds on the ‘Symplectic manifold’ page

Real-Valued Differential Forms Acting on Vector-Valued Differential Forms

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