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Yusuke Sakane

Publications and source records attributed to Yusuke Sakane.

18 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↗
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 $\SU(4+n)$ ($n \geq 2$).

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arXiv2020

Homogeneous Einstein metrics on non-K\"ahler C-spaces

We study homogeneous Einstein metrics on indecomposable non-K\"ahlerian C-spaces, i.e. even-dimensional torus bundles $M=G/H$ with $\mathsf{rank} G>\mathsf{rank} H$ over flag manifolds $F=G/K$ of a compact simple Lie group $G$. Based on the theory of painted Dynkin diagrams we present the classification of such spaces. Next we focus on the family \[ M_{\ell, m, n}:=\mathsf{SU}(\ell+m+n)/\mathsf{SU}(\ell)\times\mathsf{SU}(m)\times\mathsf{SU}(n)\,,\quad \ell, m, n\in\mathbb{Z}_{+} \] and examine several of its geometric properties. We show that invariant metrics on $M_{\ell, m, n}$ are not diagonal and beyond certain exceptions their parametrization depends on six real parameters. By using such an invariant Riemannian metric, we compute the diagonal and the non-diagonal part of the Ricci tensor and present explicitly the algebraic system of the homogeneous Einstein equation. For general positive integers $\ell, m, n$, by applying mapping degree theory we provide the existence of at least one $\mathsf{SU}(\ell+m+n)$-invariant Einstein metric on $M_{\ell, m, n}$. For $\ell=m$ we show the existence of two $\mathsf{SU}(2m+n)$ invariant Einstein metrics on $M_{m, m, n}$, and for $\ell=m=n$ we obtain four $\mathsf{SU}(3n)$-invariant Einstein metrics on $M_{n, n, n}$. We also examine the isometry problem for these metrics, while for a plethora of cases induced by fixed $\ell, m, n$, we provide the numerical form of all non-isometric invariant Einstein metrics.

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.

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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↗
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.

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arXiv2015

Non-naturally reductive Einstein metrics on exceptional Lie groups

Given an exceptional compact simple Lie group $G$ we describe new left-invariant Einstein metrics which are not naturally reductive. In particular, we consider fibrations of $G$ over flag manifolds with a certain kind of isotropy representation and we construct the Einstein equation with respect to the induced left-invariant metrics. Then we apply a technique based on Gr\"obner bases and classify the real solutions of the associated algebraic systems. For the Lie group ${\rm G}_2$ we obtain the first known example of a left-invariant Einstein metric, which is not naturally reductive. Moreover, for the Lie groups ${\rm E}_7$ and ${\rm E}_8$, we conclude that there exist non-isometric non-naturally reductive Einstein metrics, which are ${\rm Ad}(K)$-invariant by different Lie subgroups $K$.

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↗
arXiv2012

Homogeneous Einstein metrics on generalized flag manifolds with five isotropy summands

We construct the homogeneous Einstein equation for generalized flag manifolds $G/K$ of a compact simple Lie group $G$ whose isotropy representation decomposes into five inequivalent irreducible $\Ad(K)$-submodules. To this end we apply a new technique which is based on a fibration of a flag manifold over another flag manifold and the theory of Riemannian submersions. We classify all generalized flag manifolds with five isotropy summands, and we use Gr\"obner bases to study the corresponding polynomial systems for the Einstein equation. For the generalized flag manifolds E_6/(SU(4) x SU(2) x U (1) x U (1)) and E_7/(U(1) x U(6)) we find explicitely all invariant Einstein metrics up to isometry. For the generalized flag manifolds SO(2\ell +1)/(U(1) x U (p) x SO(2(\ell -p-1)+1)) and SO(2\ell)/(U(1) x U (p) x SO(2(\ell -p-1))) we prove existence of at least two non K\"ahler-Einstein metrics. For small values of $\ell$ and $p$ we give the precise number of invariant Einstein metrics.

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arXiv2012

The classification of homogeneous Einstein metrics on flag manifolds with $b_2(M)=1$

Let $G$ be a simple compact connected Lie group. We study homogeneous Einstein metrics for a class of compact homogeneous spaces, namely generalized flag manifolds $G/H$ with second Betti number $b_{2}(G/H)=1$. There are 8 infinite families $G/H$ corresponding to a classical simple Lie group $G$ and 25 exceptional flag manifolds, which all have some common geometric features; for example they admit a unique invariant complex structure which gives rise to unique invariant K\"ahler--Einstein metric. The most typical examples are the compact isotropy irreducible Hermitian symmetric spaces for which the Killing form is the unique homogeneous Einstein metric (which is K\"ahler). For non-isotropy irreducible spaces the classification of homogeneous Einstein metrics has been completed for 24 of the 26 cases. In this paper we construct the Einstein equation for the two unexamined cases, namely the flag manifolds $\E_8/\U(1)\times \SU(4)\times \SU(5)$ and $\E_8/\U(1)\times \SU(2)\times \SU(3)\times \SU(5)$. In order to determine explicitly the Ricci tensors of an $\E_8$-invariant metric we use a method based on the Riemannian submersions. For both spaces we classify all homogeneous Einstein metrics and thus we conclude that any flag manifold $G/H$ with $b_{2}(M)=1$ admits a finite number of non-isometric non-K\"ahler invariant Einstein metrics. The precise number of these metrics is given in Table 1.

math.DG↗
arXiv2011

Gelfond-Bezier Curves

We show that the generalized Bernstein bases in Muntz spaces defined by Hirschman and Widder [7] and extended by Gelfond [6] can be obtained as limits of the Chebyshev-Bernstein bases in Muntz spaces with respect to an interval [a,1] as the real number, a, converges to zero. Such a realization allows for concepts of curve design such as de Casteljau algorithm, blossom, dimension elevation to be translated from the general theory of Chebyshev blossom in Muntz spaces to these generalized Bernstein bases that we termed here as Gelfond-Bernstein bases. The advantage of working with Gelfond-Bernstein bases lies in the simplicity of the obtained concepts and algorithms as compared to their Chebyshev-Bernstein bases counterparts.

math.NA↗
arXiv2011

A Muntz Type Theorem for a Family of Corner Cutting Schemes

By identifying a family of corner cutting schemes as a dimension elevation process of Gelfond-Bezier curves, we give a Muntz type condition for the convergence of the generated control polygons to the underlying curve. The surprising emergence of the Muntz condition in the problem raises the question of a possible connection between the density questions of nested Chebyshev spaces and the convergence of the corresponding dimension elevation algorithms.

math.NA↗
arXiv2011

Chebyshev Blossom in Muntz Spaces: Toward Shaping with Young Diagrams

The notion of blossom in extended Chebyshev spaces offers adequate generalizations and extra-utilities to the tools for free-form design schemes. Unfortunately, such advantages are often overshadowed by the complexity of the resulting algorithms. In this work, we show that for the case of Muntz spaces with integer exponents, the notion of Chebyshev blossom leads to elegant algorithms whose complexities are embedded in the combinatorics of Schur functions. We express the blossom and the pseudo-affinity property in Muntz spaces in term of Schur functions. We derive an explicit expression of the Chebyshev-Bernstein basis via an inductive argument on nested Muntz spaces. We also reveal a simple algorithm for the dimension elevation process. Free-form design schemes in Muntz spaces with Young diagrams as shape parameter will be discussed.

math.NA↗
arXiv2010

Homogeneous Einstein metrics on G_2/T

We construct the Einstein equation for an invariant Riemannian metric on the exceptional full flag manifold $M=G_2/T$. By computing a Gröbner basis for a system of polynomials of multi-variables we prove that this manifold admits exactly two non-Kähler invariant Einstein metrics. Thus $G_2/T$ turns out to be the first known example of an exceptional full flag manifold which admits at least one non-Kähler and not normal homogeneous Einstein metric.

math.DG↗
arXiv2009

Einstein metrics on compact Lie groups which are not naturally reductive

The study of left-invariant Einstein metrics on compact Lie groups which are naturally reductive was initiated by J. E. D'Atri and W. Ziller in 1979. In 1996 the second author obtained non-naturally reductive Einstein metrics on the Lie group SU(n) for $n \ge 6$, by using a method of Riemannian submersions. In the present work we prove existence of non-naturally reductive Einstein metrics on the compact simple Lie groups SO(n) ($n \geq 11$), $Sp(n)$ ($n \geq 3$), $E_6$, $E_7$, and $E_8$.

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