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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 …

Let $\pi: E \rightarrow M$ be a fiber bundle over the manifold M and denote by $\Gamma(E)$ the space of smooth sections of $E$. For compact $M$ it is well known (Hamilton 1982, Part II Corollary 1.3.9 …

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 …

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 …

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 …

Normally, in the context of pseudo-differential operators, a symbol on a vector bundle $E$ is defined as a smooth function on $E$ which in each trivializing chart fulfills the usual symbol estimates …

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. …

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 …

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 …

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 …

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 …

Let $P \to M$ be a principal $G$-bundle, assume as much regularity as you want (compact $G$ or compact base manifold, ect). Via parallel transport, a connection $A$ on $P$ gives rise to the holonomy m …

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), …

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 …

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 …