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Inspired by this question about conjugation of reql analytic maps to a holomorphic function and with a group action view point we ask the following question. The complex Lie group $H=\math …

Let $G$ be a compact topological group with normalized Haar measure $\mu$. An element $g\in G$ is an ergodic element if the mapping $L_g:G \to G $ with $x\mapsto gx$ is an ergodic map. The se …

Let $G$ be a Lie group whose Lie algebra is $\mathfrak{g}$ with exponential map $\exp:\mathfrak{g}\to G$. For what kind of Lie group $G$ the standard process of definition of rotation number …

Let $M$ be a Riemannian manifold. We denote by $\mathfrak{g}$ the space of all smooth function $f:TM\to \mathbb{R}$ with fibre wise polynomial growth. Is it a Lie algebra wrt the Poisson bracket …

1)Let $G$ be a countable discrete group. Can $G$ be embbeded in a locally connected Lie group? 2)let $G$ be a countable discrete group with a prescribed generating set and corresponding word metr …

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Sp{Sp}$Let $\omega=\sum dx_i\wedge dy_i$ be the standard symplectic structure of $\mathbb{R}^{2n}=\mathbb{R}^{n}\times \mathbb{R}^n$. We consider the …

Let $G$ be a compact Lie group. Is each conjugacy class a closed subset of $G$? Define the conjugacy equivalent relation $g\sim h$ if $g$ is conjugate to $h$.Is $G/\sim$ a Haussdoef space with …

Let $M$ be a Riemannian manifold or a Lie group whose corresponding exp map (in corresponding context) is denoted by "exp" which is a map $\exp:TM\to M$ We search for the set $\mathcal{H …

Edit: According to essential comment of YCore I revise the question. Let $A$ be a finite dimensional graded algebra which is a unital, super commutative and associative algebra. Is there a Lie group …

Edit: According to the comment of Ycor we remove the phrase "Naturally arises from a left invariant metric' Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. We fix an orientation for $G$. We d …

I call Poincaré $n$-half-space group the semidirect product of $\mathbb{R}^{n-1}$ and $\mathbb{R}^+$, where the action is by homotheties; equivalently as the group of translations and positive homothe …

Does there exist a real analytic vector field on $S^3$ all of whose orbits are dense? The second paragraph of page 285 Of this paper says that there is a vector field whose almost all orbits are dense …

We equip $\mathrm{GL}(n,\mathbb{R})$ and $\mathrm{O}(n)$ with their left-invariant metrics, whose restrictions to the corresponding neutral elements is the standard inner product $\mathrm{Trace}(AB^{ …

Let $G$ be a topological group and $X$ be a topological space. Let $\alpha$, $\beta:G\times X\to X$ be two group actions. We say that these two actions are homotopic actions if there is a continuous p …

Let $G$ be a Lie group with a left invariant metric. Assume that $N$ is a Lie subgroup of $G$. For a given $g\in G$, are $N$ and $g^{-1} N g$ necessarily isometric Riemannian manifold when they inher …