Space Form Of Constant Curvature Research Articles

Outer billiards in the spaces of oriented geodesics of the three‐dimensional space forms

AbstractLet be the three‐dimensional space form of constant curvature , that is, Euclidean space , the sphere , or hyperbolic space . Let be a smooth, closed, strictly convex surface in . We define an outer billiard map on the four‐dimensional space of oriented complete geodesics of , for which the billiard table is the subset of consisting of all oriented geodesics not intersecting . We show that is a diffeomorphism when is quadratically convex. For , has a Kähler structure associated with the Killing form of . We prove that is a symplectomorphism with respect to its fundamental form and that can be obtained as an analogue to the construction of Tabachnikov of the outer billiard in defined in terms of the standard symplectic structure. We show that does not preserve the fundamental symplectic form on associated with the cross product on , for . We initiate the dynamical study of this outer billiard in the hyperbolic case by introducing and discussing a notion of holonomy for periodic points.

Real hypersurfaces of complex space forms satisfying Fischer–Marsden equation

Let M be a real hypersurface of a complex space form of constant curvature c. In this paper, we study the hypersurface M which admits a nontrivial solution to Fischer–Marsden equation, that is, the induce metric g of M satisfies $$ Hess _g(\nu )=(\varDelta _g \nu )g+\nu S_g$$ , where $$\nu $$ is a nontrivial function. We prove that there does not exist a complete Hopf real hypersurface in a non-flat complex space form satisfying Fischer–Marsden equation. Finally, we show that a complete real hypersurface with $$A\xi =\alpha \xi $$ , $$\alpha \ne 0$$ , of a complex Euclidean space $${\mathbb {C}}^n$$ satisfying Fischer–Marsden equation is locally congruent to a sphere or $${\mathbb {S}}^1 \times {\mathbb {R}}^{2n-2}$$ .

On the Principal Curvatures of Complete Minimal Hypersurfaces in Space Forms

In recent decades, there has been an increase in the number of publications related to the hypersurfaces of real space forms with two principal curvatures. The works focus mainly on the case when one of the two principal curvatures is simple. The purpose of this paper is to study a slightly more general class of complete minimal hypersurfaces in real space forms of constant curvature c, namely those with $$\mathrm{n}-1$$ principal curvatures having the same sign everywhere. From assumptions on the scalar curvature R and the Gauss–Kronecker curvature K we characterize Clifford tori if $$c > 0$$ and prove that K is identically zero if $$c \le 0$$ .

Characterization of minimal ascreen null hypersurfaces of (LCS)-space forms

In the present paper, we study nontotally geodesic minimal ascreen null hypersurface, M, of a Lorentzian concircular structure (LCS)-space form of constant curvature 0 or 1. We prove that; if the Ricci tensor of M is parallel with respect to a given leaf of its screen distribution, then M is isometric to a product of a null curve and spheres.

Quantitative volume space form rigidity under lower Ricci curvature bound I

Let $M$ be a compact $n$-manifold of $\mathrm{Ric}_M \geq (n - 1) H$ ($H$ is a constant). We are concerned with the following space form rigidity: $M$ is isometric to a space form of constant curvature $H$ under either of the following conditions: (i) There is $ \rho \gt 0$ such that for any $x \in M$, the open $ \rho $-ball at $x^{\ast}$ in the (local) Riemannian universal covering space, $ (U^{\ast}_{\rho} , x^{\ast}) \to (B_{\rho} (x) , x)$, has the maximal volume, i.e., the volume of a $\rho$-ball in the simply connected $n$-space form of curvature $H$. (ii) For $H = -1$, the volume entropy of $M$ is maximal, i.e., $n - 1$ ([LW1]). The main results of this paper are quantitative space form rigidity, i.e., statements that $M$ is diffeomorphic and close in the Gromov–Hausdorff topology to a space form of constant curvature $H$, if $M$ almost satisfies, under some additional condition, the above maximal volume condition. For $H = 1$, the quantitative spherical space form rigidity improves and generalizes the diffeomorphic sphere theorem in [CC2].

The geometry of generalized Cheeger–Gromoll metrics on the total space of transitive Euclidean Lie algebroids

Natural metrics (Sasaki metric, Cheeger–Gromoll metric, Kaluza–Klein metrics etc. ) on the tangent bundle of a Riemannian manifold is a central topic in Riemannian geometry. Generalized Cheeger–Gromoll metrics is a family hp,q of natural metrics on the tangent bundle depending on two parameters with p∈R and q≥0. This family possesses interesting geometric properties. If p=q=0 we recover the Sasaki metric and when p=q=1 we recover the classical Cheeger–Gromoll metric. A transitive Euclidean Lie algebroid is a transitive Lie algebroid with a Euclidean product on its total space. In this paper, we show that the analogous of natural metrics 11Natural metrics on the tangent bundles come from a classification of natural transformation of Riemannian metrics on manifolds to metrics on tangent bundles. We use the some terminology in this paper to designate metrics obtained on the total space of a Euclidean Lie algebroid by imitating the expression of g-natural metrics on the tangent bundle.can be built in a natural way on the total space of transitive Euclidean Lie algebroids. Then we study the properties of generalized Cheeger–Gromoll metrics on this new context. We show a rigidity result of these metrics which generalizes some rigidity results known in the case of the tangent bundle. We show also that considering natural metrics on the total space of transitive Euclidean Lie algebroids opens new interesting horizons. For instance, Atiyah Lie algebroids constitute an important class of transitive Lie algebroids and we will show that natural metrics on the total space of Atiyah Euclidean Lie algebroids have interesting properties. In particular, if M is a Riemannian manifold of dimension n, then the Atiyah Lie algebroid associated to the O(n)-principal bundle of orthonormal frames over M possesses a family depending on a parameter k>0 of transitive Euclidean Lie algebroids structures say AO(M,k). When M is a space form of constant curvature c, we show that there exist two constants Cn<0 and K(n,c)>0 such that (AO(M,k),h1,1) is a Riemannian manifold with positive scalar curvature if and only if c>Cn and 0<k≤K(n,c).

The diffeomorphism type of manifolds with almost maximal volume

The smallest $r$ so that a metric $r$-ball covers a metric space $M$ is called the radius of $M$. The volume of a metric $r$-ball in the space form of constant curvature $k$ is an upper bound for the volume of any Riemannian manifold with sectional curvature $\geq k$ and radius $\leq r$. We show that when such a manifold has volume almost equal to this upper bound, it is diffeomorphic to a sphere or a real projective space.

A remark on compact hypersurfaces with constant mean curvature in space forms

In this note we characterize compact hypersurfaces of dimension n≥2 with constant mean curvature H immersed in space forms of constant curvature and satisfying an optimal integral pinching condition: they are either totally umbilical or, when n≥3 and H≠0, they are locally contained in a rotational hypersurface. In dimension two, the integral pinching condition reduces to a topological assumption and we recover the classical Hopf–Chern result.

On the curvature ellipse of minimal surfaces in $N^3(c)\times R$

We discuss the curvature ellipse of minimal surfaces in the product space $N^3(c)\times R$, where $N^3(c)$ is the $3$-dimensional simply connected space form of constant curvature $c$.

Einstein half lightlike submanifolds of a Lorentzian space form with a semi-symmetric metric connection

In this paper, we prove a classification theorem for Einstein half lightlike sbmanifolds M of a Lorentzian space form of constant curvature c equipped with a semi-symmetric metric connection subject such that the second fundamental forms of M and its screen distribution S(T M) are conformally related by some non-zero constant and the canonical normal vector field of M is conformal Killing one.

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).

The Gauss–Kronecker curvature of minimal hypersurfaces in four-dimensional space forms

Let \({{\mathbb{Q}^4}(c)}\) be a four-dimensional space form of constant curvature c. In this paper we show that the infimum of the absolute value of the Gauss–Kronecker curvature of a complete minimal hypersurface in \({\mathbb{Q}^4(c), c\leq 0}\), whose Ricci curvature is bounded from below, is equal to zero. Further, we study the connected minimal hypersurfaces M 3 of a space form \({{\mathbb{Q}^4}(c)}\) with constant Gauss–Kronecker curvature K. For the case c ≤ 0, we prove, by a local argument, that if K is constant, then K must be equal to zero. We also present a classification of complete minimal hypersurfaces of \({\mathbb{Q}^4(c)}\) with K constant.

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].

Affine differential geometry of the unit normal vector fields of hypersurfaces in the real space forms

In this paper, for a hypersurface in the real space form of constant curvature, we prove that the unit normal vector field is an affine imbedding into a certain sphere bundle with canonical metric. Moreover, we study the relations between a hypersurface and its unit normal vector field as an affine imbedding. In particular, several hypersurfaces are characterized by affine geometric conditions which are independent of the choice of the transversal bundle.

The relative volume growth of minimal submanifolds

The volume growth of certain well-defined subsets of minimal submanifolds in riemannian spaces are compared with the volume growth of balls and spheres in space forms of constant curvature.

The Schläfli formula in Einstein manifolds with boundary

We give a smooth analogue of the classical Schläfli formula, relating the variation of the volume bounded by a hypersurface moving in a general Einstein manifold and the integral of the variation of the mean curvature. We extend it to variations of the metric in a Riemannian Einstein manifold with boundary, and apply it to Einstein cone-manifolds, to isometric deformations of Euclidean hypersurfaces, and to the rigidity of Ricci-flat manifolds with umbilic boundaries. Résumé. On donne un analogue régulier de la formule classique de Schläfli, reliant la variation du volume borné par une hypersurface se déplaçant dans une variété d’Einstein à l’intégrale de la variation de la courbure moyenne. Puis nous l’étendons aux variations de la métrique à l’intérieur d’une variété d’Einstein riemannienne à bord. On l’applique aux cone-variétés d’Einstein, aux déformations isométriques d’hypersurfaces de E n E^n , et à la rigidité des variétés Ricci-plates à bord ombilique.

ON THE INFINITESIMAL ISOMETRIC DEFORMATIONS OF SUBMANIFOLDS

In this paper, we consider the infinitesimal I-and II-isometry deformations of submanifolds immersed in a space form N of constant curvature. We obtain some results which are new even in the case of N being the Euclidean space. At the same time, we generalize some classical results in E3 to the submanifolds immersed in a space form of constant curvature.

Minimal surfaces with the Ricci condition in 4-dimensional space forms

Let X N ( c ) {X^N}(c) denote the N-dimensional simply connected space form of constant curvature c. We consider a problem to classify those minimal surfaces in X N ( c ) {X^N}(c) which are locally isometric to minimal surfaces in X 3 ( c ) {X^3}(c) . In this paper we solve this problem in the case where N = 4 N = 4 , and give a result also in higher codimensional cases.

Exceptional minimal surfaces with the Ricci condition

Let XN(c) denote the yV-dimensional simply connected space form of constant curvature c, and let M be a minimal surface in XN(c) with Gaussian curvature K { 0. In [1] Johnson studied a class of minimal surfaces in XN(c), called exceptional minimal surfaces. In thispaper, we discuss exceptional minimal surfaces in XN(c) which satisfiythe Ricci condition with respect to c. Our results are as follows:

Isotropic Immersions

Recently, Ferus ([5], [6]) classified connected Riemannian manifolds with parallel second fundamental form in a real space form of constant curvature . In this paper we may restrict our attention to isotropic submanifolds with parallel second fundamental form in the Euclidean sphere Sm(k) of constant curvature k. Due to Ferus, we find that an isotropic submanifold with parallel second fundamental form in Sm is locally congruent to one of compact symmetric spaces of rank one and the immersion is locally equivalent to the second or the first standard immersion according as M is a sphere or not. In Section 2, we characterize the first standard immersion of a complex projective space into a sphere in terms of isotropic immersions.