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In the abstract index notation of Penrose, indicies serve as placeholders to indicate the type of a tensor field. For example, $X^i$ denotes a vector field. What is the commonly accepted notation for …

The 2-form $F$ has to be closed. Then you choose an open covering $U_i$ on which $F$ has a primitive $\theta_i$, i.e. $d \theta_i = F$. Now you try to glue together the trivial bundles $U_i \times U(1 …

Lets assume that $U$ is a compact submanifold of $\mathbb R^k$, so that we do not need to worry about boundary conditions and things happening at infinity. Then $C^\infty(U, V)$ is a smooth Fréchet ma …

If the holonomies of two connections coincide for all piecewise-smooth loops, then these connections are gauge equivalent. The basic idea is to consider the path bundle $PM$ over $M$, which is a princ …

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 …

Define the hyperbolic metric $g$ on $\mathbb{H}^2$ by $$ g = \frac{1}{2y^2} (d x^2 + d y^2). $$ The Iwasawa decomposition yields an isomorpism $SL(2)/SO(2)=\mathbb{H}^2$ which is given by $$ \pmatrix …

Given a Riemannian manifold $(M, g)$ of dimension $n$, the Hodge star operator $\star: \Omega^k(M) \to \Omega^{n-k}(M)$ is defined. What is the (formal) adjoint of $\star$ under the integration pairin …

Yes, they coincide (at least if I understand you correctly). Let $\check{\gamma}$ be a path in $M$. If we fix a lift $\gamma$ in P, then the horizontal lift $\gamma_\omega$ of $\check{\gamma}$ is nec …

I'm trying to understand the definition of a differential form on $M$ in the context of Fréchet spaces or, more generally, locally convex spaces. The standard procedure defines a k-form as a map $\alp …

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 …

There is a construction that works for connections on arbitrary principal bundles (and not just for the frame bundle): Let $P$ be a principal $G$-bundle over a $n$-dimensional manifold $M$. The space …

Note sure if you can call it "calculating" but there is a rather straightforward construction. Let us fix some generators $A_i, B_i$ of $\pi_1(\Sigma)$ (i.e. a canonical system of curves such that sli …

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 …