Non-flat Space Form Research Articles

Two generalizations of an over-determined system on a surface

We introduce and study two generalizations of an over-determined system on a surface. The first generalization is an over-determined system of polynomial type, which is in the form of [Formula: see text], where each [Formula: see text] is a one-form; the second generalization is represented as [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text] are given by [Formula: see text] for a one-form [Formula: see text] and [Formula: see text] is given by [Formula: see text] for a nowhere zero two-form [Formula: see text]. Studying these systems, we will obtain generalizations and analogs of results we already have on over-determined systems on surfaces. In addition, we will obtain new kinds of results on the systems. Over-determined systems of the first generalization appear on admissible hypersurfaces in [Formula: see text] ([Formula: see text]). Results on the second generalization yield corollaries for over-determined systems on surfaces in three-dimensional non-flat space forms.

Normalized null hypersurfaces in nonflat Lorentzian space forms satisfying Lrx = Ux + b

In the present work, we classify normalized null hypersurfaces $x:(M,g,N)\to Q^{n+2}_1(c)$ immersed into one of the two real standard nonflat Lorentzian space-forms and satisfying the equation $L_r x=\mathcal U x+b$ for some field of screen constant matrices $\mathcal U$ and some field of screen constant vectors $b\in\mathbb{R}^{n+2}$, where $L_r$ is the linearized operator of the $(r+1)-$mean curvature of the normalized null hypersurface for $r=0,...,n$. We show that if the immersion $x$ is a solution of the equation $L_r x=\mathcal U x+b$ for $1\leq r\leq n$ and the normalization $N$ is quasi-conformal, then $M$ is either an $(r+1)-$maximal null hypersurface, or a totally umbilical (or geodesic) null hypersurface or an almost isoparametric normalized null hypersurface with at most two non-zero principal curvatures. We also show that a null hypersurface $M$, of a real standard semi-Riemannian nonflat space form $Q_t^{n+2}(c)$, admits a totally umbilical screen distribution (and then $M$ is totally umbilical or totally geodesic) if and only if $M$ is a section of $Q_t^{n+2}(c)$ by a hyperplane of $\mathbb{R}^{n+3}$. In particular a null hypersurface $M\to Q_t^{n+2}(c)$ is totally geodesic if and only if $M$ is a section of $Q_t^{n+2}(c)$ by a hyperplane of $\mathbb{R}^{n+3}$ passing through the origin.

A Characterization of Ruled Real Hypersurfaces in Non-Flat Complex Space Forms

The Levi-Civita connection and the k-th generalized Tanaka-Webster connection are defined on a real hypersurface M in a non-flat complex space form. For any nonnull constant k and any vector field X tangent to M the k-th Cho operator F X ( k ) is defined and is related to both connections. If X belongs to the maximal holomorphic distribution D on M, the corresponding operator does not depend on k and is denoted by F X and called Cho operator. In this paper, real hypersurfaces in non-flat space forms such that F X S = S F X , where S denotes the Ricci tensor of M and a further condition is satisfied, are classified.

Biharmonic hypersurfaces in 5-dimensional non-flat space forms

Abstract We prove that every biharmonic hypersurface having constant higher order mean curvature Hr for r > 2 in a space form M 5(c) is of constant mean curvature. In particular, every such biharmonic hypersurface in 𝕊5(1) has constant mean curvature. There exist no such compact proper biharmonic isoparametric hypersurfaces M in 𝕊5(1) with four distinct principal curvatures. Moreover, there exist no proper biharmonic hypersurfaces in hyperbolic space ℍ5 or in E 5 having constant higher order mean curvature Hr for r > 2.

Hopf hypersurfaces in spaces of oriented geodesics

A Hopf hypersurface in a (para-)Kaehler manifold is a real hypersurface for which one of the principal directions of the second fundamental form is the (para-)complex dual of the normal vector. We consider particular Hopf hypersurfaces in the space of oriented geodesics of a non-flat space form of dimension greater than 2. For spherical and hyperbolic space forms, the oriented geodesic space admits a canonical Kaehler-Einstein and para-Kaehler-Einstein structure, respectively, so that a natural notion of a Hopf hypersurface exists. The particular hypersurfaces considered are formed by the oriented geodesics that are tangent to a given convex hypersurface in the underlying space form. We prove that a tangent hypersurface is Hopf in the space of oriented geodesics with respect to this canonical (para-)Kaehler structure iff the underlying convex hypersurface is totally umbilic and non-flat. In the case of 3 dimensional space forms, however, there exists a second canonical complex structure which can also be used to define Hopf hypersurfaces. We prove that in this dimension, the tangent hypersurface of a convex hypersurface in the space form is always Hopf with respect to this second complex structure.

Deformations of 2 k-Einstein structures

It is shown that the space of infinitesimal deformations of 2 k-Einstein structures is finite dimensional on compact non-flat space forms. Moreover, spherical space forms are shown to be rigid in the sense that they are isolated in the corresponding moduli space.

Liouville and geodesic Ricci solitons

On a tangent bundle endowed with a pseudo-Riemannian metric of complete lift type two classes of Ricci solitons are obtained: a 1-parameter family of shrinking Liouville Ricci solitons if the base manifold is Ricci flat and a steady geodesic Ricci soliton if the base manifold is flat. A nonexistence result of geodesic Ricci solitons for the tangent bundle of a non-flat space form is also provided. To cite this article: M. Crasmareanu, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

An integral formula for compact hypersurfaces in space forms and its applications

AbstractIn this paper we establish an integral formula for compact hypersurfaces in non-flat space forms, and apply it to derive some interesting applications. In particular, we obtain a characterization of geodesic spheres in terms of a relationship between the scalar curvature of the hypersurface and the size of its Gauss map image. We also derive an inequality involving the average scalar curvature of the hypersurface and the radius of a geodesic ball in the ambient space containing the hypersurface, characterizing the geodesic spheres as those for which equality holds.

The third fundamental form metric for hypersurfaces in nonflat space forms

For hypersurfaces with regular Weingarten operator in nonflat space forms we study the relations between the intrinsic geometry of the third fundamental form metric and the extrinsic geometry of the hypersurface. We prove a theorema-egregium-type result for this metric and, in particular, give a local classification of hypersurfaces in case of an Einstein structure of this metric.