Search Results

A gauge transformation can be viewed as a section of $P \times_G G$, where $G$ acts via conjugation. Hence the Lie algebra $\mathfrak{gau}$ of infinitesimal gauge transformation is the space of sectio …

Yes, the form $\Omega$ is closed and defines indeed a weak symplectic structure. This can be verified by a direct but a bit messy calculation. A cleaner way would be to generalize the ideas of Vizman: …

Let $ (M, \omega) $ be a symplectic manifold. The de Rham class of $\omega$ induces a homomorphism $[\omega]: H_2(M) \to \mathbb{R}$, whose image $\Gamma_{\omega} \subseteq \mathbb{R}$ is called the g …

There is indeed a deep relation between Lagrangian submanifolds, the Fourier transformation and microlocal analysis. This is extensively discussed in Bates and Weinstein: Lectures on the Geometry of Q …

Since the projection $\pi: T^* Q \to Q$ is equivariant, the stabilizer $G_p$ of a point $p \in T^*_q Q$ is indeed a subgroup of the stabilizer of the base point $G_q$. In fact, $G_p$ is also the stabi …

Let $M$ be compact and connected. For every closed $1$-form $\alpha$ on $M$ consider the Roger $2$-cocyle $\Psi_\alpha$ on the Lie algebra $\operatorname{ham}(M, \omega)$ defined by $$ \Psi_\alpha(X_f …

This is just a special case of Integration along fibers (http://en.wikipedia.org/wiki/Integration_along_fibers). However, notice that integrating of $f \omega^n$ along the natural fibration results in …

In the general case, the reduced space $\mu^{-1}(c) / G_c$ is what is called a stratified symplectic space. This means, that for every orbit type $(H)$ the orbit type subset $\mu^{-1}(c)_{(H)} / G_c$ …

There is no problem in defining the exterior differential $\delta$ on infinite-dimensional manifolds such as the function space. In particular, $\delta^2 = 0$ follows from a similar calculation as in …

I'm not aware of a momentum map interpretation of the Einsteins's equation, but you can bring Einstein's equations in a Hamiltonian form with momentum map constraints (this is due to Fischer & Marsden …

Here is an interpretation using symmetry reduction, but without explicitly using the Lenz-Runge vector (it's essentially an extended version of the example given in Cushman & Bates "Global aspects of …

The following textbooks contain a proof of the symplectic slice theorem: Juan-Pablo Ortega and Tudor S. Ratiu: Momentum Maps and Hamiltonian Reduction Gerd Rudolph and Matthias Schmidt: Differential …

Does there exists a symplectic formulation of statistical physics? I know that thermodynamics can be written in a symplectic language and of course classical mechanics is intrinsically formulated sym …

I'm looking for a reference for a special type of Lagrangian embedding in a prequantizable symplectic manifold. The setting is a symplectic manifold $(M, \omega)$, whose symplectic form is the curvat …

This is always the case (using "naturality" as Paul Bryan suggested in the comments). Let $f: M \to N$ be a smooth map between Riemannian manifolds $(M,g)$ and $(N, h)$. Let $g^\flat: TM \to T^*M$ de …