pseudo-Riemannian Space Forms Research Articles

On minimality and scalar curvature of CMC triharmonic hypersurfaces in pseudo-Riemannian space forms

On minimality and scalar curvature of CMC triharmonic hypersurfaces in pseudo-Riemannian space forms

A classification of triharmonic hypersurfaces in a pseudo-Riemannian space form

The purpose of this article is to investigate triharmonic hypersurfaces Mtn in a pseudo-Riemannian space form Nt1n+1(c). Assuming that the shape operator is diagonalizable, we show that every CMC triharmonic hypersurface either has constant scalar curvature or it is minimal if cε>0 and every CMC triharmonic hypersurface is minimal if cε<0, where ε is the inner product of the unit normal vector of Mtn. Furthermore, we prove that any CMC triharmonic hypersurface with diagonalizable shape operator in a pseudo-Euclidean space Et1n+1 is actually minimal provided that zero is a principal curvature of multiplicity at most one.

Triharmonic hypersurfaces with constant mean curvature in pseudo-Riemannian space forms

In this paper, triharmonic hypersurfaces with constant mean curvature in pseudo-Riemannian space forms are studied. Under the assumption that the shape operator is diagonalizable, we first classify completely the nonminimal hypersurfaces with at most two distinct principal curvatures and give some examples of non-biharmonic triharmonic hypersurfaces. Then, we prove that the hypersurfaces with at most four distinct principal curvatures have constant scalar curvature. As a consequence, we obtain that such triharmonic hypersurfaces in pseudo-Euclidean spaces are minimal, which gives an affirmative partial solution to the generalized Chen's conjecture in [21].

Inequalities on Isotropic Submanifolds in Pseudo-Riemannian Space Forms

Spacelike and timelike isotropic submanifolds of pseudo-Riemannian spaces have interesting properties, with important applications in Mathematics and Physics. The article presents inequalities for isotropic spacelike and timelike submanifolds of pseudo-Riemannian space forms and isotropic Lorentzian submanifolds are also considered.

Gap type results for spacelike submanifolds with parallel mean curvature vector

We deal with $n$-dimensional spacelike submanifolds immersed with parallel mean curvature vector (which is supposed to be either spacelike or timelike) in a pseudo-Riemannian space form $\mathbb L_q^{n+p}(c)$ of index $1\leq q\leq p$ and constant sectional curvature $c\in \{-1,0,1\}$. Under suitable constraints on the traceless second fundamental form, we adapt the technique developed by Yang and Li (Math. Notes 100 (2016) 298–308) to prove that such a spacelike submanifold must be totally umbilical. For this, we apply a maximum principle for complete noncompact Riemannian manifolds having polynomial volume growth, recently established by Alías, Caminha and Nascimento (Ann. Mat. Pura Appl. 200 (2021) 1637–1650).

On η-biharmonic hypersurfaces in pseudo-Riemannian space forms

Abstract In this paper, η-biharmonic hypersurfaces with constant scalar curvature in 5-dimensional pseudo-Riemannian space forms are studied. We prove that such hypersurfaces with diagonalizable shape operator have constant mean curvature, which gives an affirmative partial answer to the conjecture in [Arvanitoyeorgos, A.—Kaimakamis, F. G.: Hypersurfaces of type $\begin{array}{} \displaystyle M^3_2 \end{array}$ in $\begin{array}{} \displaystyle \mathbb{E}^4_2 \end{array}$ with proper mean curvature vector, J. Geom. Phys. 63 (2013), 99–106]. As a result, we give several partial classification results.

Crofton formulas in pseudo-Riemannian space forms

Crofton formulas on simply connected Riemannian space forms allow the volumes, or more generally the Lipschitz–Killing curvature integrals of a submanifold with corners, to be computed by integrating the Euler characteristic of its intersection with all geodesic submanifolds. We develop a framework of Crofton formulas with distributions replacing measures, which has in its core Alesker's Radon transform on valuations. We then apply this framework, and our recent Hadwiger-type classification, to compute explicit Crofton formulas for all isometry-invariant valuations on all pseudospheres, pseudo-Euclidean and pseudohyperbolic spaces. We find that, in essence, a single measure which depends analytically on the metric, gives rise to all those Crofton formulas through its distributional boundary values at parts of the boundary corresponding to the different indefinite signatures. In particular, the Crofton formulas we obtain are formally independent of signature.

Polyharmonic hypersurfaces into pseudo-Riemannian space forms

In this paper, we shall assume that the ambient manifold is a pseudo-Riemannian space form N^{m+1}_t(c) of dimension m+1 and index t (mge 2 and 1 le tle m). We shall study hypersurfaces {M}^{m}_{t'} which are polyharmonic of order r (briefly, r-harmonic), where rge 3 and either t'=t or t'=t-1. Let A denote the shape operator of {M}^{m}_{t'}. Under the assumptions that {M}^{m}_{t'} is CMC and {{rm{Tr}}},{A}^{2} is a constant, we shall obtain the general condition which determines that {M}^{m}_{t'} is r-harmonic. As a first application, we shall deduce the existence of several new families of proper r-harmonic hypersurfaces with diagonalizable shape operator, and we shall also obtain some results in the direction that our examples are the only possible ones provided that certain assumptions on the principal curvatures hold. Next, we focus on the study of isoparametric hypersurfaces whose shape operator is non-diagonalizable and also in this context we shall prove the existence of some new examples of proper r-harmonic hypersurfaces (r ge 3). Finally, we shall obtain the complete classification of proper r-harmonic isoparametric pseudo-Riemannian surfaces into a three-dimensional Lorentz space form.

On η-biharmonic hypersurfaces with constant scalar curvature in higher dimensional pseudo-Riemannian space forms

In this paper, η-biharmonic hypersurfaces Mrn with constant scalar curvature in a pseudo-Riemannian space form are studied. Under the assumption that Mrn has diagonalizable shape operator with at most six distinct principal curvatures, we prove that Mrn has constant mean curvature. As an application, we obtain a partial classification result of these hypersurfaces and show that (nc)-biharmonic hypersurfaces must be minimal.

Pythagorean Isoparametric Hypersurfaces in Riemannian and Lorentzian Space Forms

We introduce the notion of a Pythagorean hypersurface immersed into an n+1-dimensional pseudo-Riemannian space form of constant sectional curvature c∈−1,0,1. By using this definition, we prove in Riemannian setting that if an isoparametric hypersurface is Pythagorean, then it is totally umbilical with sectional curvature φ+c, where φ is the Golden Ratio. We also extend this result to Lorentzian ambient space, observing the existence of a non totally umbilical model.

Totally umbilical submanifolds in pseudo-Riemannian space forms

A totally umbilical submanifold in pseudo-Riemannian manifolds is a fundamental notion, which is characterized by the condition that the second fundamental form is proportional to the metric. It is also a generalization of the notion of a totally geodesic submanifold. In this paper, we classify congruent classes of full totally umbilical submanifolds in non-flat pseudo-Riemannian space forms, and consider their moduli spaces. As a consequence, we show that some moduli spaces of isometric immersions between space forms which one of the same constant curvature are non-Hausdorff.

Uniqueness of Curvature Measures in Pseudo-Riemannian Geometry

The recently introduced Lipschitz–Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz–Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.

Biharmonic Hypersurfaces in Pseudo-Riemannian Space Forms with at Most Two Distinct Principal Curvatures

In this paper, we show that biharmonic hypersurfaces with at most two distinct principal curvatures in pseudo-Riemannian space form Nsn+1c with constant sectional curvature c and index s have constant mean curvature. Furthermore, we find that such biharmonic hypersurfaces Mr2k−1 in even-dimensional pseudo-Euclidean space Es2k, Ms−12k−1 in even-dimensional de Sitter space Ss2kcc&gt;0, and Ms2k−1 in even-dimensional anti-de Sitter space ℍs2kcc&lt;0 are minimal.

Darboux Associated Curves of a Null Curve on Pseudo-Riemannian Space Forms

In this work, the Darboux associated curves of a null curve on pseudo-Riemannian space forms, i.e., de-Sitter space, hyperbolic space and a light-like cone in Minkowski 3-space are defined. The relationships of such partner curves are revealed including the relationship of their Frenet frames and the curvatures. Furthermore, the Darboux associated curves of k-type null helices are characterized and the conclusion that a null curve and its self-associated curve share the same Darboux associated curve is obtained.

Surfaces in pseudo-Riemannian space forms with zero mean curvature vector

We characterize a space-like surface in a pseudo-Riemannian space form with zero mean curvature vector, in terms of complex quadratic differentials on the surface as sections of a holomorphic line bundle. In addition, combining them, we have a holomorphic quartic differential. If the ambient space is $S^4$, then this differential is just one given in [5]. If the space is $S^4_1$, then the differential coincides with a holomorphic quartic differential in [6] on a Willmore surface in $S^3$ corresponding to the original surface through the conformal Gauss map. We define the conformal Gauss maps of surfaces in $E^3$ and $H^3$, and space-like surfaces in $S^3_1$, $E^3_1$, $H^3_1$ and the cone of future-directed light-like vectors of $E^4_1$, and have results which are analogous to those for the conformal Gauss map of a surface in $S^3$.

On biconservative hypersurfaces in pseudo-Riemannian space forms and their Gauss map

We first present a survey about recent results on biconservative hypersurfaces in the Minkowski space E4 1, pseudo-Euclidean space E5 2 and Rieamnnian space-form H4. Then we obtain some geometrical properties of these hypersurface families concerning their mean curvature and Gauss map.

Biharmonic Submanifolds with Parallel Normalized Mean Curvature Vector Field in Pseudo-Riemannian Space Forms

In this paper, we investigate biharmonic submanifolds with parallel normalized mean curvature vector field in pseudo-Riemannian space forms and classify completely such pseudo-umbilical submanifolds. Also, we prove that such submanifolds have parallel mean curvature vector field under the assumption that they have diagonalizable shape operator with at most two distinct principal curvatures in the direction of the mean curvature vector field, and apply it to obtain a partial classification result.

Hypersurfaces Satisfying τ 2 (ϕ) = ητ(ϕ) in Pseudo-Riemannian Space Forms

In this paper, we derived the equations for the hypersurface \({M^{n}_{r}}\) of a pseudo-Riemannian space form \(N^{n+1}_{q}(c)\) to satisfy τ2(ϕ) = ητ(ϕ) (η a constant) with τ(ϕ) and τ2(ϕ) be the tension and bitension fields of \({M^{n}_{r}}\). As applications, we prove that a hypersurface \({M^{n}_{r}}\) satisfying τ2(ϕ) = ητ(ϕ) in \(N^{n+1}_{q}(c)\) has constant mean curvature, under the assumption that \({M^{n}_{r}}\) has diagonalizable shape operator with at most three distinct principal curvatures. Then, using this result, we classify partially such hypersurface. We also make a preliminary study of hypersurfaces satisfying τ2(ϕ) = fτ(ϕ) with f be function.

Biharmonic submanifolds of pseudo-Riemannian manifolds

In this paper, we derived biharmonic equations for pseudo-Riemannian submanifolds of pseudo-Riemannian manifolds which include the biharmonic equations for submanifolds of Riemannian manifolds as special cases. As applications, we proved that a pseudo-umbilical biharmonic pseudo-Riemannian submanifold of a pseudo-Riemannian manifold has constant mean curvature, we completed the classifications of biharmonic pseudo-Riemannian hypersurfaces with at most two distinct principal curvatures, which were used to give four construction methods to produce proper biharmonic pseudo-Riemannian submanifolds from minimal submanifolds. We also made some comparison study between biharmonic hypersurfaces of Riemannian space forms and the space-like biharmonic hypersurfaces of pseudo-Riemannian space forms.

On transforms of timelike isothermic surfaces in pseudo-Riemannian space forms

We provide the basic theory on the conformal geometry of timelike surfaces in pseudo-Riemannian space forms, which is to generalize the fundamental work of Burstall et al. for spacelike surfaces. Then we have a discussion on the transforms of timelike \({(\pm)}\)-isothermic surfaces (or real isothermic, complex isothermic surfaces), including \({c}\)-polar transforms, Darboux transforms and spectral transforms. The first main result is that c-polar transforms preserve timelike \({(\pm)}\)-isothermic surfaces, which are generalizations of the classical Christoffel transforms. The next main result is that a Darboux pair of timelike isothermic surfaces can also be characterized as a Lorentzian \({O(n-r+1,r+1)/O(n-r,r)\times O(1,1)}\)-type curved flat. Finally two permutability theorems of c-polar transforms are established.