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Marina Statha

Publications and source records attributed to Marina Statha.

14 recordsLinked to original sources
arXiv2025

Non naturally reductive Einstein metrics on $\SU(N)$ via generalized flag manifolds

We obtain new invariant Einstein metrics on the compact Lie group $\SU(N)$ which are not naturally reductive. This is achieved by using the generalized flag manifold $G/K=\SU(k_1+\cdots +k_p)/\s(\U(k_1)\times\cdots\times\U(k_p))$ and by taking an appropriate choice of orthogonal basis of the center of Lie subalgebra $\frak k$ for $K$, which poses certain symmetry conditions to the $\Ad(K)$-invariant metrics of $\SU(N)$. We also study the isometry problem for the Einstein metrics found.

math.DG↗
arXiv2024

A review of compact geodesic orbit manifolds and the g.o. condition for $\SU(5)/\s(\U(2)\times \U(2))$

Geodesic orbit manifolds (or g.o. manifolds) are those Riemannian manifolds $(M,g)$ whose geodesics are integral curves of Killing vector fields. Equivalently, there exists a Lie group $G$ of isometries of $(M,g)$ such that any geodesic $γ$ has the simple form $γ(t)=e^{tX}\cdot p$, where $e$ denotes the exponential map on $G$. The classification of g.o. manifolds is a longstanding problem in Riemannian geometry. In this brief survey, we present some recent results and open questions on the subject focusing on the compact case. In addition we study the geodesic orbit condition for the space $\SU(5)/\s(\U(2)\times \U(2))$.

math.DG↗
arXiv2021

Geodesic orbit metrics in a class of homogeneous bundles over real and complex Stiefel manifolds

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 $(G/H,g)$, such that $G$ is one of the compact classical Lie groups $\SO(n)$, $U(n)$, and $H$ is a diagonally embedded product $H_1\times \cdots \times H_s$, where $H_j$ is of the same type as $G$. This class includes spheres, Stiefel manifolds, Grassmann manifolds and real flag manifolds. The present work is a contribution to the study of g.o. spaces $(G/H,g)$ with $H$ semisimple.

math.DG↗
arXiv2021

Equigeodesics on some classes of homogeneous spaces

We study homogeneous curves on some classes of reductive homogeneous spaces G=H which are geodesics with respect to any G-invariant metric on G=H. These curves are called equigeodesics. The spaces we consider are certain Stiefel manifolds VkRn, generalized Wallach spaces and spheres. We give a characterization for algebraic equigeodesics on V2Rn, V4R6, SO(6)= SO(3) ? SO(2), W6 = U(3)= U(1)3, W12 = Sp(3)= Sp(1)3, S2n+1 ?= U(n + 1)= U(n) and S4n+3 ?= Sp(n + 1)= Sp(n).

math.DG↗
arXiv2021

Geodesic orbit metrics in a class of homogeneous bundles over quaternionic Stiefel manifolds

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.

math.DG↗
arXiv2020

Ricci flow on certain homogeneous spaces

We study the behavior of the normalized Ricci flow of invariant Riemannian homogeneous metrics at infinity for generalized Wallach spaces, generalized flag manifolds with four isotropy summands and second Betti number equal to one, and the Stiefel manifolds $V_2\mathbb{R}^n$ and $V_{1+k_2}\mathbb{R}^n$, with $n = 1+k_2+k_3$. We use techniques from the theory of differential equations, in particular the Poincaré compactification.

math.DG↗
arXiv2020

Equigeodesics on generalized flag manifolds with $G_2$-type $t$-roots

We study homogeneous curves in generalized flag manifolds $G/K$ with $G_2$-type $t$-roots, which are geodesics with respect to each $G$-invariant metric on $G/K$. These curves are called equigeodesics. The tangent space of such flag manifolds splits into six isotropy summands, which are in one-to-one correspondence with $t$-roots. Also, these spaces are a generalization of the exceptional full flag manifold $G_2/T$. We give a characterization for structural equigeodesics for flag manifolds with $G_2$-type $t$-roots, and we give for each such flag manifold, a list of subspaces in which the vectors are structural equigeodesic vectors.

math.DG↗
arXiv2020

Invariant Einstein metrics on SU(N) and complex Stiefel manifolds

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 $\SU(4+n)$ ($n \geq 2$).

math.DG↗
arXiv2018

Homogeneous Einstein metrics on Stiefel manifolds associated to flag manifolds with two isotropy summands

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 $\mathrm{Ad}(\mathrm{U}(p) \times \mathrm{SO}(n-2p))$-invariant scalar products. Two of the metrics are Jensen's metrics and the other two are new Einstein metrics.

math.DG↗
arXiv2018

New homogeneous Einstein metrics on quaternionic Stiefel manifolds

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 $\mathrm{Sp}(n) / (\mathrm{Sp}(k_1) \times \mathrm{Sp}(k_2) \times \mathrm{Sp}(k_3))$ and over the generalized flag manifold $\mathrm{Sp}(n) / (\mathrm{U}(p) \times \mathrm{Sp}(n-p))$.

math.DG↗
arXiv2016

Invariant metrics on homogeneous spaces with equivalent isotropy summands

The space of $G$-invariant metrics on a homogeneous space $G/H$ is in one-to-one correspondence with the set of inner products on the tangent space $\fr{m}\cong T_{\it o}(G/H)$, which are invariant under the isotropy representation. When all the isotropy summands are inequivalent to each other, then the metric is called diagonal. We will describe a special class of $G$-invariant metrics %with additional symmetries. in the case where the isotropy representation of $G/H$ contains some equivalent isotropy summands. Even though this problem has been considered sporadically in the bibliography, in the present article we provide a more systematic and organized description of such metrics. This will enable us to simplify the problem of finding $G$-invariant Einstein metrics for homogeneous spaces. We also provide some applications.

math.DG↗
arXiv2015

New Einstein metrics on the Lie group $SO(n)$ which are not naturally reductive

We obtain new invariant Einstein metrics on the compact Lie groups $SO(n)$ ($n \geq 7$) which are not naturally reductive. This is achieved by imposing certain symmetry assumptions in the set of all left-invariant metrics on $SO(n)$ and by computing the Ricci tensor for such metrics. The Einstein metrics are obtained as solutions of systems polynomial equations, which we manipulate by symbolic computations using Gröbner bases.

math.DG↗
arXiv2013

New homogeneous Einstein metrics on Stiefel manifolds

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 invariant Einstein metrics, two of which are Jensen's metrics and the other four are new metrics.

math.DG↗