REDUCTIVE COMPACT HOMOGENEOUS CR MANIFOLDS

We define and investigate a class of compact homogeneous CR manifolds, that we call $ \mathfrak{n} $ -reductive. They are orbits of minimal dimension of a compact Lie group K ₀ in algebraic affine homogeneous spaces of its complexification K. For these manifolds we obtain canonical equivariant fibrations onto complex flag manifolds, generalizing the Hopf fibration $ {S^3}\to \mathbb{C}{{\mathbb{P}}^1} $ . These fibrations are not, in general, CR submersions, but satisfy the weaker condition of being CR-deployments; to obtain CR submersions we need to strengthen their CR structure by lifting the complex stucture of the base.     

Cycles of Bott–Samelson Type for Taut Representations

Bott and Samelson constructed cycles which are concrete representativesof a basis for the Z₂-homology of the orbits ofvariationally complete representations of compact Lie groups (theseinclude isotropy representations of symmetric spaces; in this case theorbits are the so-called generalized real flag manifolds). Then theyused these cycles to show that the orbits of those representations aretaut submanifolds. We adapt the construction of Bott and Samelson to theorbits of three representations which are not variationally complete. Inthis case, it also follows that the orbits are taut.     

Complete nondegenerate locally standard CR manifolds

Abstract. Let M be a complete nondegenerate locally standard CR manifold. We show that a necessary and sufficient condition for M to be compact is that the Lie algebra of its infinitesimal CR automorphisms is semisimple. In general we realize M as a Mostow fibration over a compact CR manifoldB whose universal covering is a Cartesian product of Hermitian symmetric spaces and compact nondegenerate standard CR manifolds. Received: 22 July 1998 / Published online: 8 May 2000     

On Orbits in Double Flag Varieties for Symmetric Pairs

Let G be a connected, simply connected semisimple algebraic group over the complex number field, and let K be the fixed point subgroup of an involutive automorphism of G so that (G, K) is a symmetric pair. We take parabolic subgroups P of G and Q of K, respectively, and consider the product of partial flag varieties G/P and K/Q with diagonal K-action, which we call a double flag variety for a symmetric pair. It is said to be of finite type if there are only finitely many K-orbits on it. In this paper, we give a parametrization of K-orbits on G/P × K/Q in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of P ? G or Q ? K is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of K-spherical flag varieties G/P and G-spherical homogeneous spaces G/Q.     

Torus Actions of Complexity 1 and Their Local Properties

We consider an effective action of a compact (n ? 1)-torus on a smooth 2n-manifold with isolated fixed points. We prove that under certain conditions the orbit space is a closed topological manifold. In particular, this holds for certain torus actions with disconnected stabilizers. There is a filtration of the orbit manifold by orbit dimensions. The subset of orbits of dimensions less than n ? 1 has a specific topology, which is axiomatized in the notion of a sponge. In many cases the original manifold can be recovered from its orbit manifold, the sponge, and the weights of tangent representations at fixed points. We elaborate on the introduced notions using specific examples: the Grassmann manifold G_4,2, the complete flag manifold F₃, and quasitoric manifolds with an induced action of a subtorus of complexity 1.     

Tranversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the restricted three-body problem

The restricted three-body problem is considered for values of the Jacobi constant C near the value C₂ associated to the Euler critical point L₂. A Lyapunov family of periodic orbits near L₂, the so-called family (c), is born for C = C₂ and exists for values of C less than C₂. These periodic orbits are hyperbolic. The corresponding invariant manifolds meet transversally along homoclinic orbits. In this paper the variation of the transversality is analyzed as a function of the Jacobi constant C and of the mass parameter μ. Asymptotical expressions of the invariant manifolds for C ? C₂ and μ ? 0 are found. Several numerical experiments provide accurate information for the manifolds and a good agreement is found with the asymptotical expressions. Symbolic dynamic techniques are used to show the existence of a large class of motions. In particular the existence of orbits passing in a random way (in a given sense) from the region near one primary to the region near the other is proved.     

Equivariant cohomology of K-contact manifolds

We investigate the equivariant cohomology of the natural torus action on a K-contact manifold and its relation to the topology of the Reeb flow. Using the contact moment map, we show that the equivariant cohomology of this action is Cohen–Macaulay, the natural substitute of equivariant formality for torus actions without fixed points. As a consequence, generic components of the contact moment map are perfect Morse-Bott functions for the basic cohomology of the orbit foliation ${{\mathcal F}}$ of the Reeb flow. Assuming that the closed Reeb orbits are isolated, we show that the basic cohomology of ${{\mathcal F}}$ vanishes in odd degrees, and that its dimension equals the number of closed Reeb orbits. We characterize K-contact manifolds with minimal number of closed Reeb orbits as real cohomology spheres. We also prove a GKM-type theorem for K-contact manifolds which allows to calculate the equivariant cohomology algebra under the nonisolated GKM condition.     

Contact CR-Warped Product Submanifolds in Sasakian Space Forms

Recently, B.-Y. Chen studied warped products which are CR-submanifolds in Kaehler manifolds and established general sharp inequalities for CR-warped products in Kaehler manifolds. Afterwards, I. Hasegawa and the present author obtained a sharp inequality for the squared norm of the second fundamental form (an extrinsic invariant) in terms of the warping function for contact CR-warped products isometrically immersed in Sasakian manifolds. In this paper, we improve the above inequality for contact CR-warped products in Sasakian space forms. Some applications are derived. A classification of contact CR-warped products in spheres, which satisfy the equality case, identically, is given.Mathematics Subject Classifications (2000). 53C40, 53C25.     

A combinatorial rule for (co)minuscule Schubert calculus

We prove a root system uniform, concise combinatorial rule for Schubert calculus of minuscule and cominuscule flag manifolds G/P (the latter are also known as compact Hermitian symmetric spaces). We connect this geometry to the poset combinatorics of Proctor, thereby giving a generalization of Schützenberger's jeu de taquin formulation of the Littlewood-Richardson rule that computes the intersection numbers of Grassmannian Schubert varieties. Our proof introduces cominuscule recursions, a general technique to relate the numbers for different Lie types.     

General spherically symmetric Finsler metrics with constant Ricci and flag curvature

In this paper, we investigate the flag curvature of a special class of Finsler metrics called general spherically symmetric Finsler metrics, which are defined by a Euclidean metric and two related 1-forms. We find equations to characterize the class of metrics with constant Ricci curvature (tensor) and constant flag curvature. Moreover, we study general spherically symmetric Finsler metrics with the vanishing non-Riemannian quantity χ-curvature. In particular, we construct some new projectively flat Finsler metrics of constant flag curvature.     

Steepest Descent on Real Flag Manifolds

Real flag manifolds are the isotropy orbits of noncompact symmetricspaces G/K. Any such manifold M is acted on transitively bythe (noncompact) Lie group G, and it is embedded in euclideanspace as a taut submanifold. The aim of this paper is to showthat the gradient flow of any height function is a one-parametersubgroup of G, where the gradient is defined with respect toa suitable homogeneous metric s on M; this generalizes the Kählermetric on adjoint orbits (the so-called complex flag manifolds).2000 Mathematics Subject Classification 53C30, 53C35.     

Normal CR structures on compact 3-manifolds

We study normal CR compact manifolds in dimension 3. For a choice of a CR Reeb vector field, we associate a Sasakian metric on them, and we classify those metrics. As a consequence, the underlying manifolds are topologically finite quotients of or of a non-flat circle bundle over a Riemann surface of positive genus. In the latter case, we prove that their CR automorphisms group is a finite extension of , and we classify the normal CR structures on these manifolds. Received: 14 March 2000 / Published online: 17 May 2001     

A method to find generators of a semi-simple Lie group via the topology of its flag manifolds

In this paper we continue to develop the topological method to get semigroup generators of semi-simple Lie groups. Consider a subset \(\Gamma \subset G\) that contains a semi-simple subgroup \(G_{1}\) of G. If one can show that \( \Gamma \) does not leave invariant a contractible subset on any flag manifold of G, then \(\Gamma \) generates G if \(\mathrm {Ad}\left( \Gamma \right) \) generates a Zariski dense subgroup of the algebraic group \(\mathrm {Ad}\left( G\right) \). The proof is reduced to check that some specific closed orbits of \(G_{1}\) in the flag manifolds of G are not trivial in the sense of algebraic topology. Here, we consider three different cases of semi-simple Lie groups G and subgroups \(G_{1}\subset G\).     

The $$\bar \partial $$ -problem with support conditions on some weakly pseudoconvex domainswith support conditions on some weakly pseudoconvex domains

We consider a domain Ω with Lipschitz boundary, which is relatively compact in ann-dimensional Kähler manifold and satisfies some “logδ-pseudoconvexity” condition. We show that the\(\bar \partial \)-equation with exact support in ω admits a solution in bidegrees (p, q), 1≤q≤n?1. Moreover, the range of\(\bar \partial \) acting on smooth (p, n?1)-forms with support in\(\bar \Omega \) is closed. Applications are given to the solvability of the tangential Cauchy-Riemann equations for smooth forms and currents for all intermediate bidegrees on boundaries of weakly pseudoconvex domains in Stein manifolds and to the solvability of the tangential Cauchy-Riemann equations for currents on Levi flatCR manifolds of arbitrary codimension.     

Desingularizations of Schubert varieties in double Grassmannians

Let X = Gr(k, V) × Gr(l, V) be the direct product of two Grassmann varieties of k-and l-planes in a finite-dimensional vector space V, and let B ? GL(V) be the isotropy group of a complete flag in V. We consider B-orbits in X, which are an analog of Schubert cells in Grassmannians. We describe this set of orbits combinatorially and construct desingularizations for the closures of these orbits, similar to the Bott-Samelson desingularizations for Schubert varieties.     

Projective spherically symmetric Finsler metrics with constant flag curvature in R n

We investigate projective spherically symmetric Finsler metrics with constant flag curvature in R ⁿ and give the complete classification theorems. Furthermore, a new class of Finsler metrics with two parameters on n-dimensional disk is found to have constant negative flag curvature.     

Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds

 In this paper we study warped product CR-submanifolds in Kaehler manifolds and introduce the notion of CR-warped products. We prove several fundamental properties of CR-warped products in Kaehler manifolds and establish a general inequality for an arbitrary CR-warped product in an arbitrary Kaehler manifold. We then investigate CR-warped products in a general Kaehler manifold which satisfy the equality case of the inequality. Finally we classify CR-warped products in complex Euclidean space which satisfy the equality. (Received 24 August 2000; in revised form 19 February 2001)     

Almost contact metric manifolds whose Reeb vector field is a harmonic section

We investigate almost contact metric manifolds whose Reeb vector field is a harmonic unit vector field, equivalently a harmonic section. We first consider an arbitrary Riemannian manifold and characterize the harmonicity of a unit vector field ??, when ??? is symmetric, in terms of Ricci curvature. Then, we show that for the class of locally conformal almost cosymplectic manifolds whose Reeb vector field ?? is geodesic, ?? is a harmonic section if and only if it is an eigenvector of the Ricci operator. Moreover, we build a large class of locally conformal almost cosymplectic manifolds whose Reeb vector field is a harmonic section. Finally, we exhibit several classes of almost contact metric manifolds where the associated almost contact metric structures ?? are harmonic sections, in the sense of Vergara-Diaz and Wood?[25], and in some cases they are also harmonic maps.     

Equigeodesics on Generalized Flag Manifolds with b 2 (G / K) = 1

In this paper we provide a characterization of structural equigeodesics on generalized flag manifolds with second Betti number b ₂(G / K) = 1, and give examples of structural equigeodesics on generalized flag manifolds of the exceptional Lie groups F ₄, E ₆ and E ₇ with three isotropy summands.