Riemannian Space Form Research Articles

Curves in Three Dimensional Riemannian Space Forms

In three dimensional Riemannian space forms, introducing a natural moving frame, we define the associate curve of a curve and study the properties and relations of a curve and its associate curve. We state necessary and sufficient condition that a Frenet curve is a Bertrand curve in three dimensional Riemannian space forms, especially in a Riemannian 3-dimensional sphere and in a 3-dimensional hyperbolic space, resp. At the same time we give an explicit expression of the partner curve of a Bertrand curve.

LINEAR WEINGARTEN HYPERSURFACES IN RIEMANNIAN SPACE FORMS

Abstract. In this note, we generalize the weak maximum principle in[4] to the case of complete linear Weingarten hypersurface in Riemannianspace form M n+1 (c) (c = 1,0,−1), and apply it to estimate the normof the total umbilicity tensor. Furthermore, we will study the linearWeingarten hypersurface in S n+1 (1) with the aid of this weak maximumprinciple and extend the rigidity results in Li, Suh, Wei [13] and Shu [15]to the case of complete hypersurface. 1. IntroductionIt is well known that many rigidity results have been obtained for hyper-surfaces in spheres ([1, 6, 10, 17]) and in space form ([2, 3, 12]) with constantscalar curvature or with constant mean curvature. As a natural generaliza-tion of hypersurface with constant scalar curvature or with constant meancurvature, linear Weingarten hypersurface has been studied in many places([5, 8, 9, 11, 13, 15, 16]). Recall that a hypersurface in a Riemannian spaceform is said to be linear Weingarten if its normalized scalar curvature R andmean curvature H satisfy R = aH +b for some constants a,b ∈ R. In [13], Li,Suh and Wei proved the first rigidity result for linear Weingarten hypersurfacein S

Explicit classification of biconservative surfaces in Lorentz 3-space forms

Biconservative submanifolds are defined as conservative stress–energy tensor with respect to the bienergy, which is a natural generalization of biharmonic submanifolds. R. Caddeo et al. classified biconservative surfaces in 3-dimensional Riemannian space forms. In this article, we give a complete explicit classification of biconservative surfaces in de Sitter 3-spaces and anti-de Sitter 3-spaces.

Hypersurfaces with a parallel higher fundamental form

By a “higher fundamental form” of a hypersurface is understood a form obtained by inserting a power of the shape operator into the first fundamental form. The hypersurfaces in a Riemannian space form or in a three-dimensional Bianchi–Cartan–Vranceanu space for which the covariant derivative of a higher fundamental form vanishes are classified.

On applications of Simons' type formula and reduction of codimension for complete submanifolds in space forms

We provide a survey of applications of Simons' type for- mula to submanifolds with constant mean curvature or with parallel mean curvature vector in Riemannian space forms. Also, we show a result of reduction of codimension for complete submanifolds such that the normalized mean curvature vector is parallel and the squared norm of the second fundamental form satises certain inequality. At the end, we give some open questions to submanifolds in general products of Riemannian space forms.

Harmonic Almost Complex Structures with Respect to General Natural Metrics

We continue here the study initiated in [9] on the harmonicity of certain geometric objects on the total space TM of the tangent bundle of a Riemannian space form (M(c), g). Precisely, in this paper we find all the general natural metrics on TM, with respect to which the canonical almost complex structure J on TM is harmonic. We also study the harmonicity of this tensor field with respect to the natural diagonal metrics. In particular, we obtain that J is harmonic with respect to the Sasaki metric on TM if and only if the base manifold is flat.

On the stability of hypersurfaces in space forms

We define the notion of (r,s)-stability concerning closed hypersurfaces with higher order mean curvatures linearly related in a Riemannian space form. By supposing that such a hypersurface Mn is contained either in an open hemisphere of the Euclidean sphere or in the hyperbolic space, we are able to show that Mn is (r,s)-stable if, and only if, Mn is a geodesic sphere. Moreover, we obtain a suitable characterization of the (r,s)-stability through the analysis of the first eigenvalue of the Jacobi operator associated to the corresponding variational problem.

On the rigidity of hypersurfaces into space forms

Our purpose is to study the rigidity of complete hypersurfaces immersed into a Riemannian space form. In this setting, first we use a classical characterization of the Euclidean sphere \(\mathbb S ^{n+1}\) due to Obata (J Math Soc Jpn 14:333–340, 1962) in order to prove that a closed orientable hypersurface \(\Sigma ^n\) immersed with null second-order mean curvature in \(\mathbb S ^{n+1}\) must be isometric to a totally geodesic sphere \(\mathbb S ^{n}\), provided that its Gauss mapping is contained in a closed hemisphere. Furthermore, as suitable applications of a maximum principle at the infinity for complete noncompact Riemannian manifolds due to Yau (Indiana Univ Math J 25:659–670, 1976), we establish new characterizations of totally geodesic hypersurfaces in the Euclidean and hyperbolic spaces. We also obtain a lower estimate of the index of minimum relative nullity concerning complete noncompact hypersurfaces immersed in such ambient spaces.

Curvature in sub-Riemannian geometry

We study curvature problems on a nearly Riemannian manifold, which is a sub-Riemannian manifold (M, HM, g, VM) whose adapted tensor field given by (2.2) vanishes identically. First, we prove the existence and uniqueness of what we call horizontal Riemannian connection, which is a torsion-free and metric linear connection ∇ on the horizontal distribution HM. Then we define the horizontal sectional curvature of M and study nearly Riemannian space forms, which are nearly Riemannian manifolds of constant horizontal sectional curvature. Several horizontal geometric objects are introduced and studied, for example, horizontal Ricci tensor, horizontal scalar curvature, horizontal Laplacian. Finally, as a prelude to applications to Kaluza–Klein theories we define the horizontal Einstein gravitational tensor field and show that its horizontal divergence vanishes identically on M. Some examples are presented to illustrate the theory developed in the paper.

Area preserving transformations in two-dimensional space forms and classical differential geometry

We generalize Scheffers’ method to construct area preserving transformations in the Euclidean plane to Riemannian and Lorentzian two-dimensional space forms in a unified way. We review and extend the classical applications of such transformations in classical differential geometry. We introduce two classes of surfaces in Lorentzian 3-space that admit holomorphic representation and are analogous to the classical Appell and Bonnet surfaces.

A maximum principle for hypersurfaces with constant scalar curvature and applications

In this article, we establish a weak maximum principle for complete hypersurfaces with constant scalar curvature into Riemannian space forms, and give some applications to estimate the norm of the traceless part of its second fundamental form.

Stability of compact constant mean curvature hypersurfaces in a wide class of Riemannian manifolds

Barbosa, do Carmo and Eschenburg proved in Barbosa and do Carmo (Math Z 185(3):339–353, 1984), Barbosa et al. (Math Z 197(1): 123–138, 1988) that the only stable compact hypersurfaces of constant mean curvature immersed in space forms are geodesic spheres. We give a more general result in a wide class of Riemannian manifold including space forms. The interest lies mostly on the simplicity of the proof.

ON THE CURVATURE GROUPS OF A RIEMANNIAN FOLIATION

We study basic curvature groups H • B (M, F ,� ) associated to a foliated Riemannian manifold (M, F ,g )[generalizingVaisman’s curvature groups of a Riemannian space form; cf. (The curvature groups of a space form, Ann. Sc. Norm. Super. Pisa 22 (1968), 331–341.)] and their relationship to the basic cohomology groups of (M, F ). We show that H • B (M, F ,� ) = 0 for any Einstein foliation F of non-zero transverse scalar curvature.

The geometry of closed conformal vector fields on Riemannian spaces

In this paper we examine different aspects of the geometry of closed conformal vector fields on Riemannian manifolds. We begin by getting obstructions to the existence of closed conformal and nonparallel vector fields on complete manifolds with nonpositive Ricci curvature, thus generalizing a theorem of T.K. Pan. Then we explain why it is so difficult to find examples, other than trivial ones, of spaces having at least two closed, conformal and homothetic vector fields. We then focus on isometric immersions, firstly generalizing a theorem of J. Simons on cones with parallel mean curvature to spaces furnished with a closed, Ricci null conformal vector field; then we prove general Bernstein-type theorems for certain complete, not necessarily cmc, hypersurfaces of Riemannian manifolds furnished with closed conformal vector fields. In particular, we obtain a generalization of theorems J. Jellett and A. Barros and P. Sousa for complete cmc radial graphs over finitely punctured geodesic spheres of Riemannian space forms.

Classification Theorems for Space-Like Surfaces in 4-Dimensional Indefinite Space Forms with Index 2

Surfaces in 4D Riemannian space forms have been investigated extensively. In contrast, only few results are known for surfaces in 4D neutral indefinite space forms $R^4_2(c)$. Thus, in this paper we study space-like surfaces in $R^4_2(c)$ satisfying certain simple geometric properties. In particular, we classify space-like surfaces in $\mathbb E^4_2$ with constant mean and Gauss curvatures and null normal curvature. We also classify Wintgen ideal surfaces in $R^4_2(c)$ whose Gauss and normal curvatures satisfy $K^D = 2K$.

An estimate for the scalar curvature of constant mean curvature hypersurfaces in space forms

In this paper we derive a sharp estimate for the supremum of the scalar curvature (or, equivalently, the infimum of the squared norm of the second fundamental form) of a constant mean curvature hypersurface with two principal curvatures immersed into a Riemannian space form of constant curvature. Our results will be an application of the generalized Omori-Yau maximum principle, following the approach by Pigola et al. (Memoirs Am Math Soc 822, 2005).

Pseudo-Riemannian manifolds modelled on symmetric spaces

We construct irreducible pseudo-Riemannian manifolds (M, g) of arbitrary signature (p, q) with the same curvature tensor as a pseudo-Riemannian symmetric space which is a direct product of a two-dimensional Riemannian space form M2(c) and a pseudo-Euclidean space with the signature (p, q − 2), or (p − 2, q), respectively.

COMPLETE CLASSIFICATION OF LORENTZ SURFACES WITH PARALLEL MEAN CURVATURE VECTOR IN ARBITRARY PSEUDO-EUCLIDEAN SPACE

Surfaces with parallel mean curvature vector play important roles in the theory of harmonic maps, differential geometry as well as in physics. Surfaces with parallel mean curvature vector in Riemannian space forms were classified in the early 1970s by Chen and Yau. Recently, space-like surfaces with parallel mean curvature vector in arbitrary indefinite space forms were completely classified by Chen in two papers in 2009. In this paper, we completely classify Lorentz surfaces with parallel mean curvature vector in a pseudo-Euclidean space Ems with arbitrary dimension m and arbitrary index s. Our main result states that there are 23 families of Lorentz surfaces with parallel mean curvature vector in a pseudo-Euclidean m-space Ems . Conversely, every Lorentz surface with parallel mean curvature vector in Ems is obtained from the 23 families.

On the scalar curvature of constant mean curvature hypersurfaces in space forms

In this paper we study the behavior of the scalar curvature S of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of S. Our results will be an application of a weak Omori–Yau maximum principle due to Pigola, Rigoli, Setti (2005) [17].

Compact graphs over a sphere of constant second order mean curvature

The aim of this work is to show that a compact smooth star-shaped hypersurface Σ n in the Euclidean sphere S n+1 whose second function of curvature S 2 is a positive constant must be a geodesic sphere S n (ρ). This generalizes a result obtained by Jellett in 1853 for surfaces Σ 2 with constant mean curvature in the Euclidean space ℝ 3 as well as a recent result of the authors for this type of hypersurface in the Euclidean sphere S n+1 with constant mean curvature. In order to prove our theorem we shall present a formula for the operator L r (g) = div(P r ∇g) associated with a new support function g defined over a hypersurface M n in a Riemannian space form M n+1 c .