Special directions on contact metric three-manifolds

Blair [5] has introduced special directions on a contact metric 3-manifolds with negative sectional curvature for plane sections containing the characteristic vector field and, when is Anosov, compared such directions with the Anosov directions. In this paper we introduce the notion of Anosov-like special directions on a contact metric 3-manifold. Such directions exist, on contact metric manifolds with negative -Ricci curvature, if and only if the torsion is -parallel, namely (1.1) is satisfied. If a contact metric 3-manifold M admits Anosov-like special directions, and is -parallel, where is the Berger-Ebin operator, then is Anosov and the universal covering of M is the Lie group (2,R). We note that the notion of Anosov-like special directions is related to that of conformally Anosow flow introduced in [9] and [14] (see [6]).Supported by funds of the M.U.R.S.T. and of the University of Lecce. 1991.     

On the Ricci and Einstein equations on the pseudo-euclidean and hyperbolic spaces

We consider tensors T=fg on the pseudo-euclidean space Rn and on the hyperbolic space Hn, where n?3, g is the standard metric and f is a differentiable function. For such tensors, we consider, in both spaces, the problems of existence of a Riemannian metric , conformal to g, such that , and the existence of such a metric which satisfies , where is the scalar curvature of . We find the restrictions on the Ricci candidate for solvability and we construct the solutions when they exist. We show that these metrics are unique up to homothety, we characterize those globally defined and we determine the singularities for those which are not globally defined. None of the non-homothetic metrics , defined on Rn or Hn, are complete. As a consequence of these results, we get positive solutions for the equation , where g is the pseudo-euclidean metric.     

Two classes of conformally flat contact metric 3-manifolds

We give a classification of 3—dimensional conformally flat contact metric manifolds satisfying: =0(=L g) orR(Y, Z)=k[(Z)Y–(Y)Z]+[(Z)hY]–(Y)hZ] wherek and are functions. It is proved that they are flat (the non-Sasakian case) or of constant curvature 1 (the Sasakian case).     

Totally complex submanifolds inH P m (1)

Let (resp.K) be the second fundamental form (resp. the sectional curvature) of a compact submanifoldM of the quaternion projective spaceH P ^m (1). We determine all compact totally complex submanifolds of complex dimensionn inH P ^m (1) satisfying either ||² n orK 1/8.Supported by the JSPS postdoctoral fellowship.     

Light-like CR hypersurfaces of indefinite Kähler manifolds

We study a new class of real hypersurfaces called Light-like CR hypersurfaces, of indefinite Kahler manifolds, and claim several new results of geometrical/physical significance. In particular, we show that our study has a direct relation with the physically important asymptotically flat spacetimes; which further lead to the Twistor theory of Penrose and the Heaven theory of Newman. As the induced connection, on the degenerate hypersurface, may not be a metric connection, we overcome this difficulty by using differential geometric technique and deduce the embedding conditions called Gauss-Codazzi equations. Finally, we find the integrability conditions for all the possible distributions and specialize the embedding conditions when the ambient space is a complex space form. We add to the list of totally umbilical nondegenerate hypersurfaces [16] the totally umbilical light-like cone, in the degenerate case, and prove the nonexistence of totally umbilical light-like CR hypersurfaces in ¯M(c) withc 0 (see Yano and Kon [22] and Tashiro and Tachibana [20] for the nondegenerate case).     

A splitting theorem for compact complex homogeneous spaces with a symplectic structure

In this note we prove a splitting theorem for compact complex homogeneous spaces with a cohomology 2 class [] such that the top power [ ⁿ ]0.Dedicated to Professor W. C. Hsiang on the occasion of his 60th birthdayPartially supported by NSF Grant DMS-9401755.     

Complete surfaces of constant curvature in H<Superscript>2</Superscript> × R and S<Superscript>2</Superscript> × R

We study isometric immersions of surfaces of constant curvature into the homogeneous spaces and . In particular, we prove that there exists a unique isometric immersion from the standard 2-sphere of constant curvature c > 0 into and a unique one into when c > 1, up to isometries of the ambient space. Moreover, we show that the hyperbolic plane of constant curvature c < −1 cannot be isometrically immersed into or . J.A. Aledo was partially supported by Ministerio de Education y Ciencia Grant No. MTM2004-02746 and Junta de Comunidades de Castilla-La Mancha, grant no. PAI-05-034. J.M. Espinar and J.A. Gálvez were partially supported by Ministerio de Education y Ciencia grant no. MTM2004-02746 and Junta de Andalucía Grant No. FQM325.     

Bounds for the first Dirichlet eigenvalue attained at an infinite family of Riemannian manifolds

LetM be a compact Riemannian manifold with smooth boundary M. We get bounds for the first eigenvalue of the Dirichlet eigenvalue problem onM in terms of bounds of the sectional curvature ofM and the normal curvatures of M. We discuss the equality, which is attained precisely on certain model spaces defined by J. H. Eschenburg. We also get analog results for Kähler manifolds. We show how the same technique gives comparison theorems for the quotient volume(P)/volume(M),M being a compact Riemannian or Kähler manifold andP being a compact real hypersurface ofM.Work partially supported by a DGICYT Grant No. PB94-0972 and by the E.C. Contract CHRX-CT92-0050 GADGET II.     

Harmonicity of unit vector fields with respect to Riemannian g-natural metrics

Let (M,g) be a compact Riemannian manifold and T₁M its unit tangent sphere bundle. Unit vector fields defining harmonic maps from (M,g) to , being the Sasaki metric on T₁M, have been extensively studied. The Sasaki metric, and other well known Riemannian metrics on T₁M, are particular examples of g-natural metrics. We equip T₁M with an arbitrary Riemannian g-natural metric , and investigate the harmonicity of a unit vector field V of M, thought as a map from (M,g) to . We then apply this study to characterize unit Killing vector fields and to investigate harmonicity properties of the Reeb vector field of a contact metric manifold.     

A remark on left invariant metrics on compact Lie groups

We show that a left invariant metric on a compact Lie group G with Lie algebra has some negative sectional curvature if it is obtained by enlarging a biinvariant metric on a subalgebra , unless the semi-simple part of is an ideal of This answers a question raised in [8]. Received: 7 May 2007     

New examples of weakly symmetric spaces

We provide some new examples of weakly symmetric spaces inside the class of complete, simply connected Riemannian manifolds equipped with a complete unit Killing vector field such that the reflections with respect to its flow lines are global isometries.Supported by the Consejería de Educación del Gobierno de Canarias     

Tits topology and fundamental groups of compact Riemannian manifolds

Let M be a compact Riemannian manifold without conjugate points. We generalize the Tits topology on the ideal boundary of the universal covering space of M. Then we show that if π₁(M) is amenable and is compact with respect to the Tits topology, then M is flat. This work was supported by Grant No.R01-2006-000-10047-0(2006) from the Basic Research Program of the Korea Science & Engineering Foundation.     

ParaHermitian and paraquaternionic manifolds

A set of canonical paraHermitian connections on an almost paraHermitian manifold is defined. ParaHermitian version of the Apostolov-Gauduchon generalization of the Goldberg-Sachs theorem in General Relativity is given. It is proved that the Nijenhuis tensor of a Nearly paraKähler manifolds is parallel with respect to the canonical connection. Salamon's twistor construction on quaternionic manifold is adapted to the paraquaternionic case. A hyper-paracomplex structure is constructed on Kodaira-Thurston (properly elliptic) surfaces as well as on the Inoe surfaces modeled on . A locally conformally flat hyper-paraKähler (hypersymplectic) structure with parallel Lee form on Kodaira-Thurston surfaces is obtained. Anti-self-dual non-Weyl flat neutral metric on Inoe surfaces modeled on is presented. An example of anti-self-dual neutral metric which is not locally conformally hyper-paraKähler is constructed.     

Circles and spheres in pseudo-Riemannian geometry

Summary A totally umbilical pseudo-Riemannian submanifold with the parallel mean curvature vector field is said to be an extrinsic sphere. A regular curve in a pseudo-Riemannian manifold is called a circle if it is an extrinsic sphere. LetM be ann-dimensional pseudo-Riemannian submanifold of index (0n) in a pseudo-Riemannian manifold with the metricg and the second fundamental formB. The following theorems are proved. For ₀ = +1 or –1, ₁ = +1, –1 or 0 (2–2 ₀+ ₁2n–2–2) and a positive constantk, every circlec inM withg(c, c) = ₀ andg( _c c, _c c) = ₁ k ² is a circle in iffM is an extrinsic sphere. For ₀ = +1 or –1 (–₀n–), every geodesicc inM withg(c, c) = ₀ is a circle in iffM is constant isotropic and B(x,x,x) = 0 for anyx T(M). In this theorem, assume, moreover, that 1n–1 and the first normal space is definite or zero at every point. Then we can prove thatM is an extrinsic sphere. When = 0 orn, this fact does not hold in general.     

On conformally flat contact metric manifolds

In the present paper we classify the conformally flat contact metric manifolds of dimension satisfying . We prove that these manifolds are Sasakian of constant curvature 1.     

Naturally reductive Riemannian homogeneous structures on some classes of generic submanifolds in complex space forms

We prove that the universal covering spaces of the generic submanifolds of C P _n and of C H _n are naturally reductive homogeneous spaces by determining explicitly tensor fields defining naturally reductive homogeneous structures on them.