We study existence of invariant Einstein metrics on complex Stiefel manifolds $G/K = \SU(\ell+m+n)/\SU(n) $ and the special unitary groups $G = \SU(\ell+m+n)$. We decompose the Lie algebra $\frak g$ of $G$ and the tangent space $\frak p$ of $G/K$, by using the generalized flag manifolds $G/H = \SU(\ell+m+n)/\s(\U(\ell)\times\U(m)\times\U(n))$.
We parametrize scalar products on the 2-dimensional center of the Lie algebra of $H$, and we consider $G$-invariant and left invariant metrics determined by $\Ad(\s(\U(\ell)\times\U(m)\times\U(n))$-invariant scalar products on $\frak g$ and $\frak p$ respectively. Then we compute their Ricci tensor for such metrics.
We prove existence of $\Ad(\s(\U(1)\times\U(2)\times\U(2))$-invariant Einstein metrics on $V_3\bb{C}^{5}=\SU(5)/\SU(2)$, $\Ad(\s(\U(2)\times\U(2)\times\U(2))$-invariant Einstein metrics on $V_4\bb{C}^{6}=\SU(6)/\SU(2)$, and $\Ad(\s(\U(m)\times\U(m)\times\U(n))$-invariant Einstein metrics on $V_{2m}\bb{C}^{2m+n}=\SU(2m+n)/\SU(n)$. We also prove existence of $\Ad(\s(\U(1)\times\U(2)\times\U(2))$-invariant Einstein metrics on the compact Lie group $\SU(5)$, which are not naturally reductive. The Lie group $\SU(5)$ is the special unitary group of smallest rank known for the moment, admitting non naturally reductive Einstein metrics. Finally, we show that the compact Lie group $\SU(4+n)$ admits two non naturally reductive $\Ad(\s(\U(2)\times\U(2)\times\U(n)))$-invariant Einstein metrics for $ 2 \leq n \leq 25$, and four non naturally reductive Einstein metrics for $n\ge 26$. This extends previous results of K.~ Mori about non naturally reductive Einstein metrics on $\SU(4+n)$ ($n \geq 2$).
A generalized Stiefel manifold is the manifold of orthonormal frames in a vector space with a non-degenerated bilinear or hermitian form. In this article, the Isometry group of the generalized Stiefel manifolds are computed at least up to connected components in an explicit form. This is done by considering a natural non-associative algebra $\mathfrak{m}$ associated to the affine structure of the Stiefel manifold, computing explicitly the automorphism group $Aut(\mathfrak{m})$ and inducing this computation to the Isometry group.
We discuss conditions for the integrability of an almost complex structure defined on the total space of an induced Hopf S^3-bundle over a Sasakian manifold . As an application, we obtain an uncountable family of inequivalent complex structures on the Stiefel manifolds of orthonormal 2-frames in C^{n+1}, non compatible with its standard hypercomplex structure. Similar families of complex structures are constructed on the Stiefel manifold of oriented orthonormal 4-frames in R^{n+1}, as well as on some special Stiefel manifolds related to the groups G_2 and Spin(7).
We consider invariant Einstein metrics on the Stiefel manifold $V_q\bb{R} ^n$ of all orthonormal $q$-frames in $\bb{R}^n$. This manifold is diffeomorphic to the homogeneous space $\SO(n)/\SO(n-q)$ and its isotropy representation contains equivalent summands. %This causes difficulty in the description of all $\SO(n)$-invariant metrics. We prove, by assuming additional symmetries, that $V_4\bb{R}^n$ $(n\ge 6)$ admits at least four $\SO(n)$-invariant Einstein metrics, two of which are Jensen's metrics and the other two are new metrics. Moreover, we prove that $V_5\bb{R}^7$ admits at least six inv…
Let $V$ be a separable Hilbert space, possibly infinite dimensional. Let $\St(p,V)$ be the Stiefel manifold of orthonormal frames of $p$ vectors in $V$, and let $\Gr(p,V)$ be the Grassmann manifold of $p$ dimensional subspaces of $V$. We study the distance and the geodesics in these manifolds, by reducing the matter to the finite dimensional case. We then prove that any two points in those manifolds can be connected by a minimal geodesic, and characterize the cut locus.
We consider invariant Einstein metrics on the quaternionic Stiefel manifolds $V_p\mathbb{H} ^n$ of all orthonormal $p$-frames in $\mathbb{H}^n$. This manifold is diffeomorphic to the homogeneous space $\mathrm{Sp}(n) / \mathrm{Sp}(n-p)$ and its isotropy representation contains equivalent summands. We obtain new Einstein metrics on $V_p\mathbb{H}^n \cong \mathrm{Sp}(n)/\mathrm{Sp}(n-p)$, where $n = k_1 + k_2 + k_3$ and $p = n-k_3$. We view $V_p\mathbb{H}^n$ as a total space over the generalized Wallach space $\mathrm{Sp}(n) / (\mathrm{Sp}(k_1) \times \mathrm{Sp}(k_2) \times \mathrm{Sp}(k_3))$ a…
In the Stiefel manifold $X_{n,k}$, we replace Frankel linear height function by a quadratic one. We prove this is still a Morse-Bott function, whose structure of critical levels presents a dichotomy according to the sign of $n-2k$. The critical submanifolds are no longer Grassmannians but total spaces of fibrations of basis a product of two Grassmannians. We explicitly integrate the gradient flow.
We study invariant Einstein metrics on the Stiefel manifold $V_k\mathbb{R}^n\cong \mathrm{SO}(n)/\mathrm{SO}(n-k)$ of all orthonormal $k$-frames in $\mathbb{R}^n$. The isotropy representation of this homogeneous space contains equivalent summands, so a complete description of $G$-invariant metrics is not easy. In this paper we view the manifold $V_{2p}\mathbb{R}^n$ as total space over a classical generalized flag manifolds with two isotropy summands and prove for $2\le p\le \frac25 n-1$ it admits at least four invariant Einstein metrics determined by $\mathrm{Ad}(\mathrm{U}(p) \times \mathrm{S…
Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces $(M=G/H,g)$ whose geodesics are orbits of one-parameter subgroups of $G$. The corresponding metric $g$ is called a geodesic orbit metric. We study the geodesic orbit spaces of the form $(\Sp(n)/\Sp(n_1)\times \cdots \times \Sp(n_s), g)$, with $0<n_1+\cdots +n_s\leq n$. Such spaces include spheres, quaternionic Stiefel manifolds, Grassmann manifolds and quaternionic flag manifolds. The present work is a contribution to the study of g.o. spaces $(G/H,g)$ with $H$ semisimple.
We give a new elementary proof of the parallelizability of closed orientable 3-manifolds. We use as the main tool the fact that any such manifold admits a Heegaard splitting.
In the paper we consider the Stiefel manifold $V_{n;k}$ as a principal $U(k)$- bundle over the Grassmann manifold and study the cut locus from the unit element. We gave the complete description of this cut locus on $V_{n;1}$ and presented the sufficient condition on the general case. At the end, we study the complement to the cut locus of $V_{2k;k}$.
This work aims to interpolate parametrized Reduced Order Model (ROM) basis constructed via the Proper Orthogonal Decomposition (POD) to derive a robust ROM of the system's dynamics for an unseen target parameter value. A novel non-intrusive Space-Time (ST) POD basis interpolation scheme is proposed, for which we define ROM spatial and temporal basis \emph{curves on compact Stiefel manifolds}. An interpolation is finally defined on a \emph{mixed part} encoded in a square matrix directly deduced using the space part, the singular values and the temporal part, to obtain an interpolated snapshot m…
A Riemannian manifold $(M,ρ)$ is called Einstein if the metric $ρ$ satisfies the condition $\Ric (ρ)=c\cdot ρ$ for some constant $c$. This paper is devoted to the investigation of $G$-invariant Einstein metrics with additional symmetries, on some homogeneous spaces $G/H$ of classical groups. As a consequence, we obtain new invariant Einstein metrics on some Stiefel manifolds $SO(n)/SO(l)$, and on the symplectic analogues $Sp(n)/Sp(l)$. Furthermore, we show that for any positive integer $p$ there exists a Stiefel manifold $SO(n)/SO(l)$ and a homogenous space $Sp(n)/Sp(l)$ which admit at least $…
We establish geometric properties of Stiefel and Grassmann manifolds which arise in relation to Slater type variational spaces in many-particle Hartree-Fock theory and beyond. In particular, we prove that they are analytic homogeneous spaces and submanifolds of the space of bounded operators on the single-particle Hilbert space. As a by-product we obtain that they are complete Finsler manifolds. These geometric properties underpin state-of-the-art results on existence of solutions to Hartree-Fock type equations.
Solving the so-called geodesic endpoint problem, i.e., finding a geodesic that connects two given points on a manifold, is at the basis of virtually all data processing operations, including averaging, clustering, interpolation and optimization. On the Stiefel manifold of orthonormal frames, this problem is computationally involved. A remedy is to use quasi-geodesics as a replacement for the Riemannian geodesics. Quasi-geodesics feature constant speed and covariant acceleration with constant (but possibly non-zero) norm. For a well-known type of quasi-geodesics, we derive a new representation …
The real symplectic Stiefel manifold is the manifold of symplectic bases of symplectic subspaces of a fixed dimension. It features in a large variety of applications in physics and engineering. In this work, we study this manifold with the goal of providing theory and matrix-based numerical tools fit for basic data processing. Geodesics are fundamental for data processing. However, these are so far unknown. Pursuing a Lie group approach, we close this gap and derive efficiently computable formulas for the geodesics both with respect to a natural pseudo-Riemannian metric and a novel Riemannian …
