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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 …

What is an example of a $n$ dimensional manifold $M$ which is not a lie group or $S^{7}$ but satisfies the following property?: There is an $n$ dimensional sub vector space $V\su …

Let $G$ be a compact Lie group with invariant measure $\mu$. An odd function is a continuous function, $\phi:G\to \mathbb{C}$, such that $\int_{G} \phi d\mu=0$. An odd map is a continuous map, $f:G\to …

Assume that $G$ is a Lie group and at the same time it admits a symplectic structure. Does $G$ necessarily admit a symplectic structure such that the right multiplication preserves the symplectic …

I apologize in advance if my question is elementary. Before I present my question I mention my motivation: Motivation: A Lie group is a manifold. At the same time it is a Riemannian manifold equippe …

Is there a natural number $n$, a compact Lie group $G$ of dimension less than $n$ and a continuous map $f:S^n \to G$ with $f(-x)=f(x)^{-1}$, such that $f$ is not a null homotopic map? This quest …

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 …

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^{ …

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 …

Let $G$ be the set of all homeomorphisms $f$ of $\mathbb{R}^2$ such that both $f$ and $f^{-1}$ are polynomial maps. We equip $G$ with the compact open topology and the obvious group …

What is an example of a compact Lie group $G$ which is not isomorphic to a product of two nontrivial lie groups and satisfies the following property: There are two non zero vector fields $X, Y \in …

To what extent, the following types of Lie algebras are classified : Those Lie algebras $L$ such that every Lie Group $G$ with $Li(G)\sim L$, is necessarily compact.

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 …

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 …

Assume that $G$ is a Lie group with Lie algebra $\mathfrak{g}$. We fix an invariant Riemannian metric on $G$ and fix its corresponding $LC$ connection. Consider the natural right action of …