Papers by Andreas Arvanitoyeorgos
Journal of Geometry and Physics, Jul 1, 2021
Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces (M = G/... more 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) × • • • × Sp(n s), g), with 0 < n 1 + • • • + n s ≤ 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.
Hypersurfaces of type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup></mml:math> in <mml:ma...
Journal of Geometry and Physics, 2013
ABSTRACT We show that if the mean curvature vector field of a hypersurface M23 in E24 satisfies t... more ABSTRACT We show that if the mean curvature vector field of a hypersurface M23 in E24 satisfies the equation ΔH→=αH→ (αα a constant) then M23 has constant mean curvature. This equation is a natural generalization of the biharmonic submanifold equation ΔH→=0→.
Journal of The Mathematical Society of Japan, Jul 1, 2007
Hypersurfaces satisfying <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mo>△</mml:mo><mml:mover accent="true"><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">→</mml:mo></mml:mrow></mml:mover><mml:mo linebreak="goodbreak" linebreakstyle="af...
Journal of Mathematical Analysis and Applications, Sep 1, 2023
arXiv (Cornell University), Jul 12, 2012
We construct the homogeneous Einstein equation for generalized flag manifolds G/K of a compact si... more 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 Gröbner bases to study the corresponding polynomial systems for the Einstein equation. For the generalized flag manifolds E 6 /(SU(4) × SU(2) × U(1) × U(1)) and E 7 /(U(1) × U(6)) we find explicitely all invariant Einstein metrics up to isometry. For the generalized flag manifolds SO(2ℓ + 1)/(U(1) × U(p) × SO(2(ℓ − p − 1) + 1)) and SO(2ℓ)/(U(1) × U(p) × SO(2(ℓ − p − 1))) we prove existence of at least two non Kähler-Einstein metrics. For small values of ℓ and p we give the precise number of invariant Einstein metrics.
arXiv (Cornell University), Jun 28, 2010
We find the precise number of non-Kähler SO(2n)-invariant Einstein metrics on the generalized fla... more We find the precise number of non-Kähler SO(2n)-invariant Einstein metrics on the generalized flag manifold M = SO(2n)/U (p) × U (n − p) with n ≥ 4 and 2 ≤ p ≤ n−2. We use an analysis on parametric systems of polynomial equations and we give some insight towards the study of such systems. We also examine the isometric problem for these Einstein metrics.
arXiv (Cornell University), Oct 21, 2010
We find the precise number of non-Kähler $Sp(n)$-invariant Einstein metrics on the generalized fl... more We find the precise number of non-Kähler $Sp(n)$-invariant Einstein metrics on the generalized flag manifold $M=Sp(n)/(U(p)\times U(n-p))$ with $n\geq 3$ and $1\leq p\leq n-1$. We use an analysis on parametric systems of polynomial equations and we give some insight towards the study of such systems.
arXiv (Cornell University), Mar 4, 2021
Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces (M = G/... more 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 H1 × • • • × Hs, where Hj 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.
Hypersurfaces satisfying △H→=λH→ in Es5
Journal of Mathematical Analysis and Applications
Non Naturally Reductive Einstein Metrics on the Symplectic Group via Quaternionic Flag Manifolds
WORLD SCIENTIFIC eBooks, Apr 19, 2022
Hypersurfaces of type M23 in E24 with proper mean curvature vector
Journal of Geometry and Physics, 2013
ABSTRACT We show that if the mean curvature vector field of a hypersurface M23 in E24 satisfies t... more ABSTRACT We show that if the mean curvature vector field of a hypersurface M23 in E24 satisfies the equation ΔH→=αH→ (αα a constant) then M23 has constant mean curvature. This equation is a natural generalization of the biharmonic submanifold equation ΔH→=0→.
arXiv (Cornell University), Mar 4, 2021
Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces (M = G/... more 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 H1 × • • • × Hs, where Hj 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.
Mediterranean Journal of Mathematics
In this paper we study biconservative hypersurfaces M in space forms $$\overline{M}^{n+1}(c)$$ M ... more In this paper we study biconservative hypersurfaces M in space forms $$\overline{M}^{n+1}(c)$$ M ¯ n + 1 ( c ) with four distinct principal curvatures whose second fundamental form has constant norm. We prove that every such hypersurface has constant mean curvature and constant scalar curvature.
Homogeneous Einstein Metrics on Complex Stiefel Manifolds and Special Unitary Groups
Contemporary Perspectives in Differential Geometry and its Related Fields, 2017
Einstein Metrics on the Symplectic Group Which Are Not Naturally Reductive
Current Developments in Differential Geometry and its Related Fields, 2015
The Student Mathematical Library, 2003
Let G be a complex reductive linear algebraic group and G0 G a real form. Suppose P is a paraboli... more Let G be a complex reductive linear algebraic group and G0 G a real form. Suppose P is a parabolic subgroup of G and assume that P has a Levi factor L such that G0 \ L = L0 is a real form of L. Using the minimal globalization Vmin of a finite length admissible representation for L0 , one can define a homogeneous analytic vector bundle on the G0 orbit S of P in the generalized flag manifold Y = G=P. Let A(P;Vmin) denote the corresponding sheaf of polarized sections. In this article we analyze the G0 representations obtained on the compactly supported sheaf cohomology groups H p c (S; A(P;Vmin)).
A Riemannian manifold (M,ρ) is called Einstein if the metric ρ satisfies the condition (ρ)=c·ρ fo... more A Riemannian manifold (M,ρ) is called Einstein if the metric ρ satisfies the condition (ρ)=c·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 p SO(n) (resp. Sp(n))-invariant Einstein metrics.
Let M=G/K be a generalized flag manifold, that is the adjoint orbit of a compact semisimple Lie g... more Let M=G/K be a generalized flag manifold, that is the adjoint orbit of a compact semisimple Lie group G. We use the variational approach to find invariant Einstein metrics for all flag manifolds with two isotropy summands. We also determine the nature of these Einstein metrics as critical points of the scalar curvature functional under fixed volume.
We consider the asymptotic behavior of the normalized Ricci flow on generalized Wallach spaces th... more We consider the asymptotic behavior of the normalized Ricci flow on generalized Wallach spaces that could be considered as special planar dynamical systems. All non symmetric generalized Wallach spaces can be naturally parametrized by three positive numbers a_1, a_2, a_3. Our interest is to determine the type of singularity of all singular points of the normalized Ricci flow on all such spaces. Our main result gives a qualitative answer for almost all points (a_1,a_2,a_3) in the cube (0,1/2]× (0,1/2]× (0,1/2].
We obtain new invariant Einstein metrics on the compact Lie groups SO(n) (n ≥ 7) which are not na... more We obtain new invariant Einstein metrics on the compact Lie groups SO(n) (n ≥ 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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Papers by Andreas Arvanitoyeorgos