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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 …

Edit: I revise the question based on the comment conversations Let $\mathcal{F}$ be the set of all equivalence classes of finite groups under the "Isomorphism" equivalence relation. We define …

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 …

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 …

Edit: According to the answers to the linked MSE question and the comment of Holonomia, I understand that the answer to the second question is that " Every tangent bundles is a complex …

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 …

Edit 1: According to the comment of Andreas Cap I revise the integral formula in the question. Edit 2: I understand from the following post that some part of the previos version of my question has ne …

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Sp{Sp}$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 …

Edit: 28 January 2023 I just realized that this metric is frequently used in this paper https://hal.science/hal-01382281/document Let $$M=\{(x_1,x_2,\ldots,x_n)\in \mathbb{R}^n\mid x_i>0,\;i=1,2,\ldo …

Is there an orientation preserving smooth diffeomorphism $f$ on a compact manifold $M$ such that for every $n\in \mathbb{N}$, there is a smooth diffeomorphism $g:M \to M$, as $n$th root of …

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 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 …

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 …

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^{ …

A symplectic principal bundle is a principal bundle $(X,B, G)$ with projection map $q:X\to B$ such that $X$ and $B$ are symplectic manifolds and the right action of $G$ preserves the symplectic struct …