A sharp height estimate for compact spacelike hypersurfaces with constant r-mean curvature in the Lorentz-Minkowski space and application

In this paper we obtain a sharp height estimate concerning compact spacelike hypersurfaces Σn immersed in the (n+1)-dimensional Lorentz-Minkowski space Ln+1 with some nonzero constant r-mean curvature, and whose boundary is contained into a spacelike hyperplane of Ln+1. Furthermore, we apply our estimate to describe the nature of the end of a complete spacelike hypersurface of Ln+1.     

A characterization of isotropic immersions by extrinsic shapes of smooth curves

In this paper we show that an isometric immersion is isotropic in the sense of O'Neill if and only if it preserves logarithmic derivatives of first geodesic curvatures of some curves.     

The classification of 3-dimensional Lorentzian affine hypersurfaces with parallel cubic form

We study Lorentzian affine hypersurfaces in Rn+1 with parallel cubic form with respect to the Levi-Civita connection of the affine metric. As main result, a complete classification of such non-degenerate affine hypersurfaces in R⁴ is given.     

Conformal hypersurfaces with the same third fundamental form

We classify all hypersurfaces in a Euclidean space which allow conformal deformations, other than the ones obtained through conformal diffeomorphisms of the Euclidean space, preserving the third fundamental form.     

A rigidity theorem for complete CMC hypersurfaces in Lorentz manifolds

In this paper we use the standard formula for the Laplacian of the squared norm of the second fundamental form and the asymptotic maximum principle of H. Omori and S.T. Yau to classify complete CMC spacelike hypersurfaces of a Lorentz ambient space of nonnegative constant sectional curvature, under appropriate bounds on the scalar curvature.     

On submanifolds whose shape operator is unipotent

The object of this article is to characterize submanifolds of the Euclidean space whose shape operator A^ξ satisfies the equation (A^ξ)² = k||ξ ||²Id, where k > 0 is constant.     

A note on the stability of the Wulff shape

We give a new proof of Palmer’s result [6] that theWulff shapes are the only closed, oriented, stable hypersurfaces with constant anisotropic mean curvature. Our approach is based on the construction of a suitable testfunction in the anisotropic index form, thus generalizing the original proof of Barbosa, do Carmo [1]. Received: 3 August 2005     

On the rigidity of constant mean curvature complete vertical graphs in warped products

We investigate constant mean curvature complete vertical graphs in a warped product, which is supposed to satisfy an appropriated convergence condition. In this setting, under suitable restrictions on the values of the mean curvature and the norm of the gradient of the height function, we obtain rigidity theorems concerning to such graphs. Furthermore, applications to the hyperbolic and Euclidean spaces are given.     

Generalized dirac operators on nonsmooth manifolds and Maxwell's equations

We develop a function theory associated with Dirac type operators on Lipschitz subdomains of Riemannian manifolds. The main emphasis is on Hardy spaces and boundary value problems, and our aim is to identify the geometric and analytic assumptions guaranteeing the validity of basic results from complex function theory in this general setting. For example, we study Plemelj-Calderón-Seeley-Bojarski type splittings of Cauchy boundary data into traces of ‘inner’ and ‘outer’ monogenics and show that this problem has finite index. We also consider Szegö projections and the corresponding L^p-decompositions. Our approach relies on an extension of the classical Calderón-Zygmund theory of singular integral operators which allow one to consider Cauchy type operators with variable kernels on Lipschitz graphs. In the second part, where we explore connections with Maxwell's equations, the main novelty is the treatment of the corresponding electro-magnetic boundary value problem by recasting it as a ‘half’ Dirichlet problem for a suitable Dirac operator.     

Stability of spacelike hypersurfaces in foliated spacetimes

Given a generalized Robertson-Walker spacetime whose warping function verifies a certain convexity condition, we classify strongly stable spacelike hypersurfaces with constant mean curvature. More precisely, we will show that given a closed, strongly stable spacelike hypersurface of with constant mean curvature H, if the warping function ? satisfying ?^″?max{H?^′,0} along M, then Mn is either maximal or a spacelike slice Mt₀={t₀}×F, for some t₀∈I.     

A characterization for isometries and conformal mappings of pseudo-Riemannian manifolds

Generalizing theorems of Myers-Steenrod and of Hawking, we obtain characterizations for isometries and conformal mappings of pseudo-Riemannian spaces (M, g): Define a local distance function on convex normal neighbourhoods by (p, q) =g(exp _p ^–1 q, exp _p ^–1 q). Then every homeomorphismf locally preserving these functions is an isometry. If (M, g) has indefinite signature andf locally preserves distance zero, it is a conformal diffeomorphism.     

Higher order parallel surfaces in the Heisenberg group

We give a classification of k-parallel surfaces in the three-dimensional Heisenberg group. In particular, we prove that every k-parallel surface in the Heisenberg group is a vertical cylinder over a polynomial spiral of degree at most k−1.     

On the volume of the intersection of two geodesic balls

The first author and D. Kunszenti-Kovács (2010) [1] proved that if the volume of the intersection of three geodesic balls of a complete connected Riemannian manifold depends only on the center-center distances and the radii of the balls, then the manifold is one of the simply connected spaces of constant curvature. In this paper, we study the geometrical consequences of the analogous condition for pairs of geodesic balls. We show that in a complete, connected and simply connected Riemannian manifold, the volume of the intersection of two small geodesic balls depends only on the distance between the centers and the radii if and only if the space is harmonic. It is also shown that if in a Riemannian manifold the volume of the intersection of two small geodesic balls of equal radii depends only on the distance between the centers and the common value of the radii, then the space is Einstein, and if we assume in addition that the space is symmetric, then it must be Osserman and hence two-point homogeneous.     

On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds

It is well known that if the tangent bundle TM of a Riemannian manifold (M,g) is endowed with the Sasaki metric gs, then the flatness property on TM is inherited by the base manifold [Kowalski, J. Reine Angew. Math. 250 (1971) 124-129]. This motivates us to the general question if the flatness and also other simple geometrical properties remain “hereditary” if we replace gs by the most general Riemannian “g-natural metric” on TM (see [Kowalski and Sekizawa, Bull. Tokyo Gakugei Univ. (4) 40 (1988) 1-29; Abbassi and Sarih, Arch. Math. (Brno), submitted for publication]). In this direction, we prove that if (TM,G) is flat, or locally symmetric, or of constant sectional curvature, or of constant scalar curvature, or an Einstein manifold, respectively, then (M,g) possesses the same property, respectively. We also give explicit examples of g-natural metrics of arbitrary constant scalar curvature on TM.     

On spacelike hypersurfaces with constant scalar curvature in the de Sitter space

We classify spacelike hypersurfaces of the de Sitter space with constant scalar curvature and with two principal curvatures. Moreover, we prove that if Mn is a complete spacelike hypersurface with constant scalar curvature n(n−1)R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n−1, then R<(n−2)c/n. Additionally, we prove several rigidity theorems for such hypersurfaces.     

Complete hypersurfaces immersed in a semi-Riemannian warped product

The aim of this paper is to study the uniqueness of complete hypersurfaces immersed in a semi-Riemannian warped product whose warping function has convex logarithm and such that its fiber has constant sectional curvature. By using as main analytical tool a suitable maximum principle for complete noncompact Riemannian manifolds and supposing a natural comparison inequality between the r-th mean curvatures of the hypersurface and that ones of the slices of the region where the hypersurface is contained, we are able to prove that a such hypersurface must be, in fact, a slice.     

δ-invariants and their applications to centroaffine geometry

We introduce the notion of δ-invariant for curvature-like tensor fields and establish optimal general inequalities in case the curvature-like tensor field satisfies some algebraic Gauss equation. We then study the situation when the equality case of one of the inequalities is satisfied and prove a dimension and decomposition theorem. In the second part of the paper, we apply these results to definite centroaffine hypersurfaces in Rn+1. The inequality is specified into an inequality involving the affine δ-invariants and the Tchebychev vector field. We show that if a centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is a proper affine hypersphere. Furthermore, we prove that if a positive definite centroaffine hypersurface in , satisfies the equality case of one of the inequalities, it is foliated by ellipsoids. And if a negative definite centroaffine hypersurface satisfies the equality case of one of the inequalities, then it is foliated by two-sheeted hyperboloids. Some further applications of the inequalities are also provided in this article.     

Warped product structure of submanifolds with nonpositive extrinsic curvature in space forms

By means of a simple warped product construction we obtain examples of submanifolds with nonpositive extrinsic curvature and minimal index of relative nullity in any space form. We then use this to extend to arbitrary space forms four known splitting results for Euclidean submanifolds with nonpositive sectional curvature.