Ihor V. Mykytyuk | Searcharxiv

arXiv2016

Integrability of geodesic flows for metrics on suborbits of the adjoint orbits of compact groups

Let $G/K$ be an orbit of the adjoint representation of a compact connected Lie group $G$, $σ$ be an involutive automorphism of $G$ and $\tilde G$ be the Lie group of fixed points of $σ$. We find a sufficient condition for the complete integrability of the geodesic flow of the Riemannian metric on $\tilde G/(\tilde G\cap K)$, which is induced by the bi-invariant Riemannian metric on $\tilde G$. The integrals constructed here are real analytic functions, polynomial in momenta. It is checked that this sufficient condition holds when $G$ is the unitary group $U(n)$ and $σ$ is its automorphism defined by the complex conjugation.

math.DG↗

arXiv2016

Dirac brackets and reduction of invariant bi-Poisson structures

Let $X$ be a manifold with a bi-Poisson structure $\{η^t\}$ generated by a pair of $G$-invariant symplectic structures $ω_1$ and $ω_2$, where the Lie group $G$ acts properly on $X$. Let $H$ be some isotropy subgroup for this action representing the principle orbit type and $X^r_\mathfrak{h}$ be the submanifold of $X$ consisting of the points in $X$ with the stabilizer algebra equal to the Lie algebra $\mathfrak{h}$ of $H$ and with the stabilizer group conjugated to $H$ in $G$. We prove that the pair of symplectic structures $ω_1|_{X^r_\mathfrak{h}}$ and $ω_2|_{X^r_\mathfrak{h}}$ generates an $N(H^0)/H^0$-invariant bi-Poisson structure on $X^r_\mathfrak{h}$, where $N(H^0)$ is the normalizer in $G$ of the identity component $H^0$ of $H$. The action of $\widetilde G=N(H^0)/H^0$ on $X^r_\mathfrak{h}$ is locally free and proper and, moreover, the spaces $A^G$ of $G$-invariant functions on $X$ and $A^{\widetilde G}$ of $\widetilde G$-invariant functions on $X^r_\mathfrak{h}$ can be canonically identified and therefore the bi-Poisson structure $\{(η^t)'\}$ induced on $A^G\simeq A^{\widetilde G}$ can be treated as the reduction with respect to a {\em locally free} action of a Lie group which essentially simplifies the study of $\{(η^t)'\}$.

math.DG↗