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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 …
Tobias Diez's user avatar
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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 …
Tobias Diez's user avatar
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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 …
Tobias Diez's user avatar
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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 …
Tobias Diez's user avatar
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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 …
Tobias Diez's user avatar
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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 …
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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 …
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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), …
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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. …
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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 …
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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 …
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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 …
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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 …
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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 …
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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 …
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