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Results tagged with dg.differential-geometry
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user 17047
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
11
votes
Hodge decomposition of smooth n-forms: is it an isomorphism of topological vector spaces?
Yes, the Hodge decomposition is a topological decomposition with respect to the $C^\infty$-topology. One can argue, for example, that the Laplace-Beltrami $\Delta$ operator is elliptic and hence can b …
5
votes
Accepted
Quotient by freely acting group on Banach manifold
In finite dimensions, properness of the action is all you need for a slice theorem (and thus for the manifold structure of the quotient). However, in the Banach realm, properness is not enough. For ex …
4
votes
0
answers
110
views
Examples of non-equivariant momentum maps
What are examples of non-equivariant momentum maps?
Off the top of my hat, I know about the following examples:
the action of translations of a symplectic vector space (yielding the Heisenberg group …
5
votes
Understanding the slice theorem
Apart from some technical details, a slice is a (local) submanifold that it transversal to the orbit. For example, in the natural $SO(2)$-action on $R^2$ by rotations, a line segment in the radial dir …
6
votes
Compressible Ebin-Marsden?
The compressible case uses semidirect products of groups (group of diffeomorphisms times functions).
To my knowledge, the first paper that discusses this in detail is
Marsden, Ratiu, Weinstein: Semidi …
7
votes
Accepted
Equivariant implicit function theorem
The equivariant version of the implicit function theorem is the following.
Let $f: \mathbb{R}^p \times \mathbb{R}^n \to \mathbb{R}^m$ be a smooth function (possibly only defined on open neighborhoods …
7
votes
Accepted
General wedge-product for vector bundle valued forms
The most general definition I know is the following. Every fiberwise bilinear form $\eta: V_1 \times V_2 \to W$ of vector bundles $V_1, V_2, W$ over $M$ gives rise to the wedge product of vector-bundl …
1
vote
Stabilizer groups of Yang-Mills connections
For a Yang-Mills connection $A$, one indeed has a decomposition
$$H_A\bigl(Ad P \otimes \mathbb{C}\bigr) = \bigl(gau(P)_A\bigr)_{\mathbb{C}} \oplus \bigoplus_{\lambda > 0} H_A\bigl(Ad_\lambda P\bigr), …
5
votes
1
answer
380
views
Stabilizer groups of Yang-Mills connections
Let $G$ be a compact Lie group with complexification $G^c$, and consider a principal $G^c$-bundle $P^c \to M$ together with a reduction $P \subseteq P^c$ to $G$. Assume that $M$ is a Riemann surface.
…
1
vote
On the orbit of a Fréchet Lie group action
I'm not aware of a precise characterization of when the orbits are initial submanifolds in infinite dimensions. In fact, the manifold structure on the orbits is a hard problem even for $G$-actions on …
9
votes
2
answers
402
views
Differential refinement of homology
Differential cohomology is a refinement of ordinary cohomology by differential data. It's construction comes down to the observation that $H^2(M, \mathbb{Z})$ is isomorphic to the space of isomorphism …
9
votes
0
answers
97
views
Non-linear version of the Chevalley–Eilenberg complex
Let $\mathfrak{g}$ be a finite-dimensional Lie algebra. In small degrees, the differentials of the Chevalley–Eilenberg complex $C^\bullet(\mathfrak{g}, \mathfrak{g})$ with values in the adjoint repres …
4
votes
Accepted
A manifold or Riemannian structure on the space of all conjugacy classes of a compact Lie group
After you choose a maximal torus $T$, the space of conjugacy classes is identified with with the quotient $T / W(T)$ of $T$ by the Weyl group, see https://en.wikipedia.org/wiki/Maximal_torus#Weyl_grou …
7
votes
Functional approach vs jet approach to Lagrangian field theory
This is meant as a long comment to the very good answer by Pedro Ribeiro.
There is a nice analog of the variational bicomplex in the functional framework. Namely, the space of differential forms on $M …
1
vote
Accepted
Reference for working with the implicit function theorem
Shameless plug: In my thesis, I introduced the following notion of an (abstract) normal form. It consists of a tuple $(X, Y, \hat{f}, f_s)$ where:
$X$ and $Y$ finite-dimensional vector spaces with c …