Equivariant deformations of meromorphic actions on compact complex manifolds

Meromorphicity is the most basic property for holomorphic -actions on compact complex manifolds. We prove that the meromorphicity of -actions on compact complex manifolds are not necessarily preserved by small deformations, if the complex dimension of complex manifolds is greater than two. In contrast, we also show that the meromorphicity of -actions on compact complex surface depends only on the topology (the first Betti number) of the surface. We construct such examples of dimension greater than two by studying an equivariant deformation of certain complex threefold, so called a twistor space. Received January 25, 2000 / Published online October 30, 2000     

On transverse Killing fields of metric foliations¶of manifolds with positive curvature

In this paper, we extend a result by H. Takagi on the non-existence of mutually commuting and linearly independent Killing vector fields on positively curved Riemannian manifolds. Further, a kind of “Compact Leaf Theorem” is proved for metric foliations of closed manifolds with positive sectional curvature. Received: 26 May 2000 / Revised version: 28 February 2001     

Almost hypercomplex pseudo-Hermitian manifolds and a 4-dimensional Lie group with such structure

Almost hypercomplex pseudo-Hermitian manifolds are considered. Isotropic hyper-K?hler manifolds are introduced. A 4-parametric family of 4-dimensional manifolds of this type is constructed on a Lie group. This family is characterized geometrically. The condition a 4-manifold to be isotropic hyper-K?hler is given.      

Geometric diffeomorphism finiteness in low dimensions and homotopy group finiteness

The main results of this note consist in the following two geometric finiteness theorems for diffeomorphism types and homotopy groups of closed simply connected manifolds: 1. For any given numbers C and D the class of closed smooth simply connected manifolds of dimension which admit Riemannian metrics with sectional curvature bounded in absolute value by $\vert K \vert\le C$ and diameter bounded from above by D contains at most finitely many diffeomorphism types. In each dimension there exist counterexamples to the preceding statement. 2. For any given numbers C and D and any dimension m there exist for each natural number up to isomorphism always at most finitely many groups which can occur as the k-th homotopy group of a closed smooth simply connected m-manifold which admits a metric with sectional curvature and diameter . Received: 21 August 1999 / Accepted: 20 April 2001 / Published online: 19 October 2001     

On the spectral flow in Lorentzian manifolds

In this paper we use functional analytical techniques to determine the differential equation satisfied by the eigenvalues of a smooth family of Fredholm operators, obtained from the index form along a Lorentzian geodesic. The formula is then applied to the study of the evolution of the index function, and, using a perturbation argument, we prove a version of the classical Morse index theorem for stationary Lorentzian manifolds. Received: January 31, 2000; in final form: March 13, 2002?Published online: February 20, 2003 The second author is partially sponsored by CNPq (Brazil), Grant 200615/01-7.     

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)     

Asymptotically hyperbolic metrics on the unit ball with horizons

In this paper, we construct a family of three-dimensional asymptotically hyperbolic manifolds with horizons and with scalar curvature equal to −6. The manifolds we construct can be arbitrarily close to anti-de Sitter-Schwarzschild manifolds at infinity. Hence, the mass of our manifolds can be very large or very small. The main arguments we use in this paper are gluing methods which are used by Miao in (Proc Am Math Soc 132(1):217–222, 2004).     

Four-dimensional Einstein-like manifolds and curvature homogeneity

The aim of this paper is to classify 4-dimensional Einstein-like manifolds whose Ricci tensor has constant eigenvalues (this being a special kind of curvature homogeneity condition). We give a full classification when the Ricci tensor is of Codazzi type; when the Ricci tensor is cyclic parallel, we classify all such manifolds when not all Ricci curvatures are distinct. In this second case we find a one-parameter family of Riemannian metrics on a Lie groupG as the only possible ones which are irreducible and non-symmetric.     

Representing products of manifolds by edge-coloured graphs: the boundary case

In this paper we generalize the construction - introduced by Gagliardi and Grasselli in the closed case - of a coloured-graph representing the product of two manifolds, starting by two coloured graphs representing the manifolds themselves, to the boundary case. In particular we study the genus of the graph product of low dimensional manifold ( resp. n-spheres ) with m-disks. Received September 28, 1998; in final form January 5, 2000 / Published online October 11, 2000     

Second variation and Legendrian stabilities of minimal Legendrian submanifolds in Sasakian manifolds

First, we derive a new second variation formula which holds for minimal Legendrian submanifolds in Sasakian manifolds. Using this, we prove that any minimal Legendrian submanifold in an η-Einstein Sasakian manifold with “nonpositive” η-Ricci constant is stable. Next we introduce the notion of the Legendrian stability of minimal Legendrian submanifolds in Sasakian manifolds. Using our second variation formula, we find a general criterion for the Legendrian stability of minimal Legendrian submanifolds in η-Einstein Sasakian manifolds with “positive” η-Ricci constant.     

The classification of 4-dimensional homogeneous D'Atri spaces revisited

In this short note we correct the (incomplete) classification theorem from [F. Podestà, A. Spiro, Four-dimensional Einstein-like manifolds and curvature homogeneity, Geom. Dedicata 54 (1995) 225-243], we improve a result from [P. Bueken, L. Vanhecke, Three- and four-dimensional Einstein-like manifolds and homogeneity, Geom. Dedicata 75 (1999) 123-136] and we announce the final solution of the classification problem for 4-dimensional homogeneous D'Atri spaces.     

Generalized quasi-Kaehlerian manifolds and axioms ofCR-submanifolds in generalized Hermitian geometry,I

The class of almost Hermitian manifolds is widely generalized. The different characterizations of the isotropy of such manifolds are studied.     

Generalized quasi-Kaehlerian manifolds and axioms ofCR-submanifolds in generalized Hermitian geometry,II

The geometry of generalized quasi-Kaehlerian and generalized nearly Kaehlerian manifolds is studied. Some classes of such manifolds under the different suppositions of their isotropy are completely classified.     

On Gromov's large Riemannian manifolds

Gromov introduced several notions of largeness of Riemannian manifolds and proved that they are all equivalent for manifolds of nonnegative sectional curvature. In this paper, we study the equivalence of these notions in the case of nonnegative Ricci curvature and in particular we give an affirmative answer to one of Gromov's open questions.     

Conformal holonomy of C-spaces, Ricci-flat, and Lorentzian manifolds

The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebra of a C-space is a Berger algebra. For Ricci-flat spaces we show how the conformal holonomy can be obtained by the holonomy of the ambient metric and get results for Riemannian manifolds and plane waves.     

2-step nilpotent Lie groups of higher rank

We construct a family of simply connected 2-step nilpotent Lie groups of higher rank such that every geodesic lies in a flat. These are as Riemannian manifolds irreducible and arise from real representations of compact Lie algebras. Moreover we show that groups of Heisenberg type do not even infinitesimally have higher rank. Received: 2 July 2001 / Revised version: 19 October 2001     

The geometry of a bi-Lagrangian manifold

This is a survey on bi-Lagrangian manifolds, which are symplectic manifolds endowed with two transversal Lagrangian foliations. We also study the non-integrable case (i.e., a symplectic manifold endowed with two transversal Lagrangian distributions). We show that many different geometric structures can be attached to these manifolds and we carefully analyze the associated connections. Moreover, we introduce the problem of the intersection of the two leaves, one of each foliation, through a point and show a lot of significative examples.