Search Results

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 …

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 …

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 …

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), …

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. …

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 …

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 …

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 …

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 $\mu^{-1}(c)_{(H)} / G_c$ …

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 …

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 …

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 …

As an simple example with infinitely many different stabilizer subgroups you may take countably many disks. Since each disk may be rotated independently with a different speed, we obtain an action of …

An almost complex structure gives you a reduction of the frame bundle $LM$ from $Sp(2n)$ to $U(n)$. Hence the space $\mathcal{J}$ of all almost complex structures compatible with the symplectic form i …

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 …