All Questions
Tagged with dg.differential-geometry differential-forms
58 questions
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Can every De Rham cohomology class be represented by a closed form $\alpha$ with $L_X \alpha=0$
Assume that $M$ is a manifold and $X$ is a vector field on $M$.
Is it true to say that every closed form is De Rham-cohomologue to a closed form $\alpha$ with $L_X \alpha =0$?
- 356
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90
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A curve in the bundle of two forms
Let $(M,g)$ be a closed Riemannian manifold. Fix a point $m\in M$ and a $2$-form $\omega$ at $m.$ Take a curve $\gamma$ in $M$ such that $\gamma(0)=m.$ Now we can get a $2$-form along $\gamma$ by ...
- 954
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The wedge product of two positive forms is positive
I have previously posted this question on MSE, but still didn't solve it.
Definition. A real $(p, p)$-form $\psi$ on a complex manifold $M^{n}$ is said to be (semi-) positive, if for any $x \in M$, ...
- 195
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Projection to trivial reduced cohomology class in $L^2(\mathbb{R})$
Given that I have had no success on the mathematics stackexchange (see here), I've decided to try my luck here.
I am attempting to solve the following exercise (original formulation here), which to my ...
- 111
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328
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Codifferential of wedge of two 1-forms
Let $\omega,\eta$ be two 1-forms on a manifold $M$. I'm interested in an expression for
$$
\delta(\omega\wedge\eta)
$$
where $\delta$ is the co-differential operator $\Lambda^2(M)\to\Lambda^1(M)$. ...
- 213
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604
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Exterior derivative on principal bundle [closed]
In Nakahara's Geometry, Topology and Physics on page 375, he constructs a Lie-algebra-valued one-form $\omega$ on a principal bundle $P$ by "lifting" a Lie-algebra-valued one-form $\mathcal A_i$ on an ...
- 103
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Behaviour of the Cartan Maurer form
Let there be a Lie-group $G$ and its Lie-algebra $g$. Then the Cartan Maurer form is an 1-form $\omega: T_gG \rightarrow T_eG$ for which holds:
$$ (L^\ast_g)\omega = \omega$$
In Shlomo Sternberg's ...
- 103
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Application of $k$-forms to differential equations
I was presented with a problem that, in my view, is somewhat difficult, and it relates $k$-forms to systems of differential equations.
Consider the ellipsoid, given by
$$f(u, v)=(a\sin(u) \cos(v), b \...
- 251