A Note on P—Sasakian Manifold

s51.IntroductionLetMbeadifferentiablemanifo1dofdimensionn,ifMadmitsa(l,l)-tensorfieldgl,aPoSitivedefiniteRiemannianmetricg,avectorfieldeandal-formVwhichsatisfythefollow-ingconditions.thensuchamanifoldMiscalledaPara-SasakianmanifoldorbrieflyaP-Sasakianmanifoldby7'.AdatiandK.Matsumoto[ljwhichareconsideredasspecialcaseofanalmostparacontactmanifoldintroducedbyI.Sato[2].WhereVdenotestheoperatorofcovariantdifferentiationwithrespecttothemetrictensorg'X(M)denotesthesetofdifferentiablevectorfiel…     

Equivariant Chern Classes of Orientable Toric Origami Manifolds∗

A toric origami manifold, introduced by Cannas da Silva, Guillemin and Pires, is a generalization of a toric symplectic manifold. For a toric symplectic manifold, its equivariant Chern classes can be described in terms of the corresponding Delzant polytope and the stabilization of its tangent bundle splits as a direct sum of complex line bundles. But in general a toric origami manifold is not simply connected, so the algebraic topology of a toric origami manifold is more difficult than a toric symplectic manifold. In this paper they give an explicit formula of the equivariant Chern classes of an oriented toric origami manifold in terms of the corresponding origami template. Furthermore, they prove the stabilization of the tangent bundle of an oriented toric origami manifold also splits as a direct sum of complex line bundles.     

Some constraints on Frobenius manifolds with a tt*-structure

The article gives a necessary and sufficient condition for a Frobenius manifold to be a CDV-structure. We show that there exists a positive definite CDV-structure on any semi-simple Frobenius manifold. We also compare three natural connections on a CDV-structure and conclude that the underlying Hermitian manifold of a non-trivial CDV-structure is not a K?hler manifold. Finally, we compute the harmonic potential of a harmonic Frobenius manifold.     

Liouville-type theorems for CC-harmonic maps from Riemannian manifolds to pseudo-Hermitian manifolds

In this paper, we introduce a horizontal energy functional for maps from a Riemannian manifold to a pseudo-Hermitian manifold. The critical maps of this functional will be called CC-harmonic maps. Under suitable curvature conditions on the domain manifold, some Liouville-type theorems are established for CC-harmonic maps from a complete Riemannian manifold to a pseudo-Hermitian manifold by assuming either growth conditions of the horizontal energy or an asymptotic condition at the infinity for the maps.     

On a Submersion Between Reinhart Lightlike Manifolds and Semi-Riemannian Manifolds

In this paper, we introduce a new submersion, namely, screen lightlike submersion between a lightlike manifold and a semi-Riemannian manifold. We give an example and obtain a characterization for lightlike manifold to be Reinhart under such submersion. Then, we investigate the geometry of a screen lightlike submersion when the total manifold is a Reinhart lightlike manifold. Received: March 6, 2007 Revised: October 4, 2007 and November 2, 2007 Accepted: December 6, 2007     

Cartan connection and its geodesics on a manifold of nondegenerate affinor fields

We consider the quotient manifold of the manifold of nondegenerate affinor fields on a compact manifold with respect to the action of the group of nowhere vanishing functions. On this manifold, we construct a Cartan connection and find its torsion tensor. We also find the geodesics of the Cartan connection.     

On manifolds of eigenfunctions and potentials generated by a family of periodic boundary-value problems

We consider a family of boundary-value problems with some potential as a parameter. We study the manifold of normalized eigenfunctions with even number of zeros in a period, and the manifold of potentials associated with double eigenvalues. In particular, we prove that the manifold of normalized eigenfunctions is a trivial fiber space over a unit circle and that the manifold of potentials with double eigenvalues is a homotopically trivial manifold trivially imbedded into the space of potentials.     

A manifold that does not contain a compact core

A core of a (noncompact) manifold is a submanifold with the property that the inclusion of the submanifold into the manifold is a homotopy equivalence. It is shown by example that a manifold may fail to contain a compact core even though the manifold has the homotopy type of a finite complex.     

A curvature condition for a twisted product¶to be a warped product

It is shown that a mixed Ricci-flat twisted product semi-Riemannian manifold can be expressed as a warped product semi-Riemannian manifold. As a consequence, any Einstein twisted product semi-Riemannian manifold is in fact, a warped product semi-Riemannian manifold. Received: 8 March 2001 / Revised version: 25 July 2001     

非线性发展方程的H(o)lder连续惯性流形

引入非线性发展方程的H\"older连续惯性流形的概念,为原来惯性流形概念的推广和修正.惯性流形是有限维不变的Lipschiz流形,是研究发展方程解的长时间性态的合适工具,其缺点是需要谱间隙条件.提出H\"older连续惯性流形也是有限维不变的,但光滑性减弱为H\"older连续,不需要谱间隙条件.该流形与指数吸引子交集具有指数吸引性,无穷维动力系统可在H\"older连续惯性流形上约化为有限维常微分方程组.     

RC-positivity and rigidity of harmonic maps into Riemannian manifolds Dedicated to Professor Lo Yang on the Occasion of His 80th Birthday

In this paper,we show that every harmonic map from a compact K?hler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant.In particular,there is no non-constant harmonic map from a compact Koahler manifold with positive holomorphic sectional curvature to a Riemannian manifold with non-positive complex sectional curvature.     

A five-dimensional Riemannian manifold with an irreducible SO(3)-structure as a model of abstract statistical manifold

In the present paper, we consider a five-dimensional Riemannian manifold with an irreducible SO(3)-structure as an example of an abstract statistical manifold. We prove that if a five-dimensional Riemannian manifold with an irreducible SO(3)-structure is a statistical manifold of constant curvature, then the metric of the Riemannian manifold is an Einstein metric. In addition, we show that a five-dimensional Euclidean sphere with an irreducible SO(3)-structure cannot be a conjugate symmetric statistical manifold. Finally, we show some results for a five-dimensional Riemannian manifold with a nearly integrable SO(3)-structure. For example, we prove that the structure tensor of a nearly integrable SO(3)-structure on a five-dimensional Riemannian manifold is a harmonic symmetric tensor and it defines the first integral of third order of the equations of geodesics. Moreover, we consider some topological properties of five-dimensional compact and conformally flat Riemannian manifolds with irreducible SO(3)-structure.     

On doubly warped product Finsler manifolds

In this paper, we introduce horizontal and vertical warped product Finsler manifolds. We prove that every C-reducible or proper Berwaldian doubly warped product Finsler manifold is Riemannian. Then, we find the relation between Riemannian curvatures of doubly warped product Finsler manifold and its components, and consider the cases that this manifold is flat or has scalar flag curvature. We define the doubly warped Sasaki-Matsumoto metric for warped product manifolds and find a condition under which the horizontal and vertical tangent bundles are totally geodesic. We obtain some conditions under which a foliated manifold reduces to a Reinhart manifold. Finally, we study an almost complex structure on the tangent bundle of a doubly warped product Finsler manifold.     

Nonexistence of minimal Lagrangian spheres in hyperKähler manifolds

We prove that for one cannot immerse as a minimal Lagrangian manifold into a hyperK?hler manifold. More generally we show that any minimal Lagrangian immersion of an orientable closed manifold into a hyperK?hler manifold must have nonvanishing second Betti number and that if , is a K?hler manifold and more precisely a K?hler submanifold in w.r.t. one of the complex structures on . In addition we derive a result for the other Betti numbers. Received February 10, 1999 / Accepted April 23, 1999     

关于H. Wu问题

著名几何学家Ｈ．Ｗｕ在「４」中提出这样的问题：若一完备非紧的黎曼流形仅有两个符号相反的Ｂｕｓｅｍａｎｎ函数,则该流形的结构如何？本文在流形具非负Ｒｉｃｃｉ曲率或截曲率具下界的情况下部分地解决了这一问题。同时还讨论了流形在具非负曲率条件下有关的一些性质。     

Compact complex manifolds bimeromorphic to locally conformally Kähler manifolds

We study compact complex manifolds bimeromorphic to locally conformally Kähler (LCK) manifolds. This is an analogy of studying a compact complex manifold bimeromorphic to a Kähler manifold. We give a negative answer for a question of Ornea, Verbitsky, Vuletescu by showing that there exists no LCK current on blow ups along a submanifold (dim \(\ge 1\)) of Vaisman manifolds. We show that a compact complex manifold with LCK currents satisfying a certain condition can be modified to an LCK manifold. Based on this fact, we define a compact complex manifold with a modification from an LCK manifold as a locally conformally class C (LC class C) manifold. We give examples of LC class C manifolds that are not LCK manifolds. Finally, we show that all LC class C manifolds are locally conformally balanced manifolds.     

Lightlike Hypersurfaces of a Golden Semi-Riemannian Manifold

We introduce lightlike hypersurfaces of a golden semi-Riemannian manifold. We investigate several properties of lightlike hypersurfaces of a golden semi-Riemannian manifold. We prove that there is no radical anti-invariant lightlike hypersurface of a golden semi-Riemannian manifold. In particular, we obtain some results for screen semi-invariant lightlike hypersurfaces of a golden semi-Riemannian manifold. Moreover, we study screen conformal screen semi-invariant lightlike hypersurfaces.     

Schwarz Lemma and Hartogs Phenomenon in Complex Finsler Manifold

The authors prove the Schwarz lemma from a compact complex Finsler manifold to another complex Finsler manifold and any complete complex Finsler manifold with a non-positive holomorphic curvature obeying the Hartogs phenomenon.     

Regularity of harmonic maps into a static Lorentzian manifold

We study the regularity of harmonic maps from Riemannian manifold into a static Lorentzian manifold. We show that when the domain manifold is two-dimensional, any weakly harmonic map is smooth. We also show that when dimension n of the domain manifold is greater than two, there exists a weakly harmonic map for the Dirichlet problem which is smooth except for a closed set whose (n − 2)-dimensional Hausdorff measure is zero.     

On the class of generalized Landsberg manifolds

In 2000, Bejancu–Farran introduced the class of generalized Landsberg manifolds which contains the class of Landsberg manifolds. In this paper, we prove three global results for generalized Landsberg manifolds. First, we show that every compact generalized Landsberg manifold is a Landsberg manifold. Then we prove that every complete generalized Landsberg manifold with relatively isotropic Landsberg curvature reduces to a Landsberg manifold. Finally, we show that every generalized Landsberg manifold with vanishing Douglas curvature satisfies \(\mathbf{H}=0\).