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Let $U \in U(n)$ be a generic unitary matrix. Since the unitary group $U(n)$ is compact and connected, I know that the exponential map is surjective, i.e. that every $U \in U(n)$ has the form $U = e^{...

$\DeclareMathOperator\ad{ad}\DeclareMathOperator\Ad{Ad}\DeclareMathOperator\Exp{Exp}\DeclareMathOperator\SL{SL}\DeclareMathOperator\sl{\mathfrak{sl}}$Let $G:=\SL(2, C) \ltimes_{\Ad} \SL(2,C)$, where $\...

$\newcommand\norm[1]{\lVert#1\rVert}$Given any point $p$ of a smooth Riemannian manifold $M$ there exists $r\in (0,\infty]$ such that the Riemannian exponential is a diffeomorphism in the geodesic ...

Let $A$ and $B$ be two anti-Hermitian operators on a finite-dimensional Hilbert space. BCH formula gives an explicit expression for $e^A e^B$ as $e^C=e^A e^B$, for $C$ in the Lie algebra generated by $...

For simplicity, let us work on an example. Regard $GL_r(\mathbb C)$ as a Lie group with associated Lie algebra $M_r(\mathbb C)$, then there exists a canonical so called exponential map: $$\exp: M_r(\...