Curvature properties of compact spacelike hypersurvaces in de Sitter space

A sufficient condition for a compact spacelike hypersurface in de Sitter space to be spherical in terms of a lower bound for the square of its mean curvature is presented. Our result will be a consequence of the maximum principle for the Laplacian operator. We also derive some other applications and consequences of our main result. In particular, we establish another sufficient condition for a compact spacelike hypersurface in de Sitter space to be spherical in terms of a pinching condition for its scalar curvature, as well as in terms of the Ricci curvature and in terms of the higher order mean curvatures.

Luis J. Alias

Maximal surfaces in the Lorentz-Minkowski space and Calabi-Bernstein's theorem

A maximal surface in the Lorentz-Minkowski space is a spacelike surface with zero mean curvature everywhere. Calabi-Bernstein's theorem states that the only entire maximal graphs in the Lorentz-Minkwoski space are the spacelike planes. In this talk we will present a new approach to Calabi-Bernstein's theorem which is based on a local integral inequality for the Gaussian curvature of a maximal surface, involving the local geometry of the surface and its hyperbolic image. As an application of this, we will provide a new proof of Calabi-Bernstein's theorem.

Mitsunori Ara

The stability of F-harmonic maps

The notion of F-harmonic maps (which unifies p-harmonic and exponentially harmonic maps) is defined, and the stability of F-harmonic maps reported. In particular, we consider the following problem: Find compact Riemannian manifolds which are neither the domain nor target of any nonconstant stable F-harmonic map. We describe the properties of such a manifold. Lastly, we give a complete list of all compact irreducible symmetric spaces for which the identity map is unstable as an F-harmonic map.

Kadri Arslan

On contact metric manifolds

In this study we consider pseudosymmetric and pseudo Ricci symmetric manifolds which were introduced by M. C. Chaki. We show that, if M is a pseudosymmetric contact metric manifold of Chaki type with z belonging to (k,m)- nullity distribution then M is non-Sasakian or isometric to the product E^{n+1}x S^n(4), or has vanishing scalar curvature. We also show that, if M is a pseudo Ricci symmetric contact metric manifold of Chaki type with z belonging to (k,m)-nullity distribution then M has vanishing scalar curvature.

Livia Bejan

Pluriharmonic maps from the manifolds endowed with f-structures

The notion of pluriharmonicity is extended from the Hermitian manifolds to the manifolds endowed with a structure of K.Yano type. Some conditions under which a pluriharmonic map is f-holomorphic (or f-antiholomorphic) are given. Some special submanifolds are studied in this context.

Mohamed Belkhelfa

Sasakian manifolds and pseudosymmetry

It was proved by Deprez , Deszcz and Verstraelen that there are non semi symmetric pseudosymmetric Kahlerian manifolds, a similar theorem was obtained for Ricci pseudosymmetry by Olszak. However, the pseudosymmetry condition is non essential in Kahlerian manifolds. He introduced the following curvature condition of pseudosymmetry type on Kahlerian manifolds $(M,g,J)$: $$\bar{R}(X,Y)\cdot R= L(X\wedge_{g}Y+JX\wedge_{g}JY-2g(JX,JY)\cdot R,$$ where $L$ is a certain function on $M$; $g$, $J$,$R$ denote the Kahlerian metric, the comlex structure, the Riemannian curvature tensor and $\bar{R}(X,Y)=X\wedge_{g}Y+JX\wedge_{g}JY-2g(JX,JY).$ There are many examples of non semisymmetric Bochner flat manifolds satisfying the above condition. We investigate the pseudosymmetry of Sasakian manifolds.

Emma Carberry.

Harmonic tori: an algebraic perspective.

I will explain the method by which Hitchin reduced the study of harmonic tori in the 3-sphere to a problem in the theory of algebraic curves, and discuss approaches for further study.

Ryszard Deszcz

Examples of non-semisymmetric Ricci-semisymmetric hypersurfaces

We will present examples of non-semisymmetric Ricci-semisymmetric hypersurfaces of semi-Euclidean spaces.

Sorin Dragomir

Harmonic maps in CR geometry

Ugur Dursun

Minimal Lorentzian Hypersurfaces Foliated by Geodesics of 4-dimensional Lorentzian Space Forms.

Let M be a connected space-like surface in a 4-dimensional space form N. We define a Lorentzian hypersurface P which is the image of a subbundle of the normal bundle of M spanned by a time-like unit normal vector field in N under the normal exponential mapping of M in N. We show that P is minimal in N under some conditions on the components of the normal connection form. We also build up some examples.

Atsushi Fujioka

On some generalizations of constant mean curvature surfaces

We introduce some classes of surfaces in space forms as a natural generalization of constant mean curvature surfaces and consider basis properties for them.

Malgorzata Glogowska

On a class of semisymmetric manifolds

We will present curvature properties of semi-Riemannian manifolds (M,g), n>3, whose Weyl tensor is expressed by the square, in the sense of the product of Kulkarni-Nomizu, of the tensor S - (\kappa/(n-1))g.

Claudio Gorodski

The converse to a theorem by Bott and Samelson

We develop a technique based on methods of representation theory of compact Lie groups and the osculating spaces of orbits that enable us to study several geometrically interesting classes of manifolds, with some important applications including the converse to a theorem of Bott and Samelson and new results about taut manifolds, as well as simpler proofs of other classification results.

Martin Kilian

New Constant Mean Curvature Cylinders

Using the DPW method, I shall show how to solve period closing problems to obtain new examples of CMC cylinders of non-finite type.

Dorota Kowalczyk

Curvature properties of some spacetimes

We will present curvature properties of the Schwarzschild-type spacetimes.

Catherine McCune

Rational Minimal Surfaces

A special class of minimal surfaces of genus zero with finite total curvature, with Enneper type ends, and without umbilics will be described. The UP-iteration - a combination of Moebius transformations and Christoffel transformations - for the Gauss maps of these surfaces will be introduced. It will be used to produce infinitely many families of these surfaces, the proof relies on properties of the Schwarzian derivative.

Ian McIntosh

Algebraic harmonic tori

I will describe a way of looking at harmonic tori in some symmetric spaces which highlights the algebraic properties and raise a few questions concerning the existence of purely algebraic examples.

Hiroshi Matsuzoe

Semi-Weyl structures and conformal-projective transformations in affine differential geometry.

We introduce a semi-Weyl structure in order to construct a unified theory for Weyl structure and statistical structure. We also introduce conformal-projective transformations. We then give a necessary and sufficient condition for a semi-Weyl manifold to be a space of constant curvature.

Xiaohuan Mo

Harmonic maps from Finsler spaces

We introduce the energy, the Euler-Lagrange operator and the stress energy tensor for a smooth map $\phi$ from a Finisher manifold to a Romanian manifold. We show that $\phi$ is an extremal of the energy functional if and only if $\phi$ satisfies the corresponding Euler-Lagrange equation. We also characterize weak Landsberg manifolds in terms of harmonicity and horizontal conservativity. Using the representation of tensor fields in terms of geodesic coefficients, we construct new examples of harmonic maps from Berwald spaces which are neither Riemannian nor Minkowskian.

Stefano Montaldo

Biharmonic curves on a Surface

Zerrin Senturk

Generalized Einstein metric conditions

Einstein Manifolds form a natural subclass of various classes of Semi-Riemannian manifolds determined by a curvature condition imposed on their Ricci tensor. Certain pseudosymmetry type curvature conditions give rise to new examples of generalized Einstein metric conditions. An extension of the class of Einstein manifolds form quasi-Einstein manifolds. In this way we get other generalized Einstein metric conditions. After we construct examples of compact Ricci-pseudosymmetric manifolds, we present some results on Quasi-Einstein manifolds satisfying some curvature conditions of pseudosymmetry type. Next we introduce four new curvature conditions of pseudosymmetry type. We state that any Einstein manifold satisfies these conditions.

Hiroshi Tamaru

Symmetric spaces and two-step nilpotent Lie groups

We construct a class of two-step nilmanifolds from compact symmetric spaces. These nilmanifolds can be investigated in terms of the theory of symmetric spaces, similarly to the studies on H-type groups in terms of Clifford modules. We want to mention the following properties of our nilmanifolds: the construction, the isometry groups, singularity, weak symmetry, and so on.

Seiichi Udagawa

Harmonic maps of finite type into generalized flag manifolds and twistor fibrations

I will talk about the following : (1) An improvement of Burstall's result on harmonic maps of finite type into a sphere or a complex projective space and its generalization; (2) A classification of pluriharmonic maps from the n-dimensional complex torus into a complex projective space; (3) A new integrable system of harmonic maps from compact Riemann surfaces of higher genus into a k-symmetric space.

Marina Ville

Harmonic morphisms from 4-manifolds to surfaces.

If M and N are Riemannian manifolds, a harmonic morphism f:M->N is a map which pulls back local harmonic functions on N to local harmonic functions on M. If M is an Einstein 4-manifold and N is a Riemann surface, John Wood showed that f is holomorphic w.r.t. some integrable complex structure defined on M away from the singular points of f. In this paper we extend this complex structure to the whole manifold M. If M is compact, it admits a metric which is Kaehler w.r.t this complex structure. It follows that there are no non-constant harmonic morphisms from S^4 or CP^2 to a Riemann surface. The proof relies heavily on the real analyticity of the whole situation.