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We study the asymptotic behavior of convex Cauchy hypersurfaces on maximal globally hyperbolic spatially compact space–times of constant curvature. We generalise the result of Belraouti (Annales de l’institut Fourier 64(2):457–466, 2015) to the (2+1) de Sitter and anti de Sitter cases. We prove that in these cases the level sets of quasi-concave times converge in the Gromov equivariant topology, when time goes to 0, to a real tree. Moreover, this limit does not depend on the choice of the time function. We also consider the problem of asymptotic behavior in the flat \((n+1)\) dimensional case. We prove that the level sets of quasi-concave times converge in the Gromov equivariant topology, when time goes to 0, to a CAT(0) metric space. Moreover, this limit does not depend on the choice of the time function.

In this paper we introduce two new notions of sectional curvature for Riemannian manifolds with density. Under both notions of curvature we classify the constant curvature manifolds. We also prove generalizations of the theorems of Cartan-Hadamard, Synge, and Bonnet-Myers as well as a generalization of the (non-smooth) 1/4-pinched sphere theorem. The main idea is to modify the radial curvature equation and second variation formula and then apply the techniques of classical Riemannian geometry to these new equations., Comment: 19 pages, The expositiion of the paper has been shortened by a few pages and some of the arguments streamlined at the suggestion of the referee. Final version, to appear in Geometriae Dedicata

We obtain an explicit cellular decomposition of the quaternionic spherical space forms, manifolds of positive constant curvature that are factors of an odd sphere by a free orthogonal action of a generalized quaternionic group. The cellular structure gives and explicit description of the associated cellular chain complex of modules over the integral group ring of the fundamental group. As an application we compute the Whitehead torsion of these spaces for any representation of the fundamental group.

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).

Let Mn be an n-dimensional complete noncompact oriented submanifold with finite total curvature, i.e., \({\int_M(|A|^2-n|H|^2)^{\frac n2} < \infty}\), in an (n + p)-dimensional simply connected space form Nn+p(c) of constant curvature c, where |H| and |A|2 are the mean curvature and the squared length of the second fundamental form of M, respectively. We prove that if M satisfies one of the following: (i) n ≥ 3, c = 0 and \({\int_M|H|^n < \infty}\); (ii) n ≥ 5, c = −1 and \({|H| < 1-\frac{2}{\sqrt n}}\); (iii) n ≥ 3, c = 1 and |H| is bounded, then the dimension of the space of L2 harmonic 1-forms on M is finite. Moreover, in the case of (i) or (ii), M must have finitely many ends.

This work states some half-space type theorems in a warped product space of the form I ×ρM, where \({I \subseteq {\bf R}}\) is an open interval and M is either a compact n-manifold, or a complete simply connected surface with constant curvature c ≤ 0. Such theorems generalize the classical half-space theorem for minimal surfaces in R3, obtained by Hoffmann and Meeks (Invent Math 101:373–377, 1990), and recent results for surfaces contained in a slab of R ×ρM, obtained by Dajczer and Alias (Comment Math Helvetici 81:653–663, 2006).

In this article, we begin a systematic study of conformal properties of codimension-1 foliations. We first define and study local conformal invariants. A case of particular interest is that of harmonic foliations of the plane. Then we study existence of totally umbilical and “Dupin” foliations on compact 3-manifolds of constant curvature.

We prove that $${\mathcal{C}}^\infty$$ curves of constant curvature satisfy, in the sense of Gromov, the relative $${\mathcal{C}}^1$$ -dense h-principle in the space of immersed curves in Euclidean space R n ≥ 3. In particular, in the isotopy class of any given $${\mathcal{C}}^1$$ knot f there exists a $${\mathcal{C}}^\infty$$ knot f of constant curvature which is $${\mathcal{C}}^1$$ -close to f. More importantly, we show that if f is $${\mathcal{C}}^2$$ , then the curvature of f may be set equal to any constant c which is not smaller than the maximum curvature of f. We may also require that f be tangent to f along any finite set of prescribed points, and coincide with f over any compact set with an open neighborhood where f has constant curvature c. The proof involves some basic convexity theory, and a sharp estimate for the position of the average value of a parameterized curve within its convex hull.

In this short note we prove that the length spectrum of a compact constant curvature complex is discrete. After recalling the relevant definitions and reducing to a relatively simple situation, the result follows easily from a foundational result about real semi-algebraic sets and the Morse–Sard theorem. We conclude with a conjecture which remains open, a few remarks and an easy application.

This note describes an observation connecting Riemannian manifolds of constant sectional curvature with a particular class of Lie superalgebras. Specifically, it is shown that the structural equations of a space M with constant sectional curvature, of one variety or another, nearly coincide with some identities satisfied by tensors which can be used to construct some specific families of Lie superalgebras. In particular, one obtains either osp(n,2), spl(n,2), or osp(4,2n) if the Riemannian manifold has constant curvature, constant holomorphic curvature or constant quaternion-holomorphic curvature, respectively.

We consider a space X of constant curvature (Euclidean, elliptic, and hyperbolic space, and the sphere). The set of isometries of X transforming a surface such that it touches a second surface is called the configuration space of surface–surface contact. We investigate when this subset of Isom(X) is an immersed submanifold (which depends on the principal curvatures of the surfaces involved), and when it is an embedded submanifold (which depends on the ‘size’ and ‘shape’ of the given surfaces).

Let M be a compact orientable submanifold immersed in a Riemannian manifold of constant curvature with flat normal bundle. This paper gives intrinsic conditions for M to be totally umbilical or a local product of several totally umbilical submanifolds. It is proved especially that a compact hypersurface in the Euclidean space with constant scalar curvature and nonnegative Ricci curvature is a sphere.

For a closed curve in a CAT(K) space with given circumradius and upper bound on curvature, a basic lower bound on the length is established. The inequality is sharp, assumed only when the curve is the boundary of an isometric copy of a racetrack (the convex hull of two congruent circles) from a plane of constant curvature K. Previously such a theorem was proved for Euclidean plane curves by G.D.Chakerian, H.H. Johnson, and A. Vogt, and for curves in higher dimensional Euclidean spaces by A.D. Milka. A similar theorem is proved for nonclosed curves, with a notion of breadth replacing circumradius. Thus we illustrate how singular methods can extend classical Euclidean theorems to a large class of new spaces (including Riemannian manifolds of curvature bounded above) and also give significant strengthenings even in Euclidean space.

The complete homogeneous pseudo-Riemannian manifolds of constant non-zero curvature were classified up to isometry in 1961 [1]. In the same year, a structure theory [2] was developed for complete flat homogeneous pseudo-Riemannian manifolds. Here that structure theory is sharpened to a classification. This completes the classification of complete homogeneous pseudo-Riemannian manifolds of arbitrary constant curvature.

Every Riemann surfaceM g of genusg ⩾ 2 carries a canonical metric of constant curvature −1. We show

In this paper, we classify combinatorially different fundamental domains for any given planar discontinuous group and we give an algorithm for the complete enumeration of uniform tilings of any complete, simply connected, two-dimensional Riemannian manifold of constant curvature.

General methods for finding tile-k-transitive tilings of the three-dimensional Euclidean space with polyhedral bodies are discussed. Analogous methods for enumerating k-isohedral tilings of a two-dimensional plane of constant curvature have been obtained previously.

In hyperbolic, Euclidean and spherical n-space, we determine, for each positive number l, the largest interval of the form λ n (l)≤l ij≤l which guarantees the existence of an n-simplex p 1 p 2 ... p n+1 with edge-lengths p ipj=l ij. (In spherical geometry of curvature 1 the interval is empty unless l≤2 arcsin $$\sqrt {\left( {{{n + 1} \mathord{\left/ {\vphantom {{n + 1} {2n.}}} \right. \kern-\nulldelimiterspace} {2n.}}} \right)}$$ ) The assertion that these intervals are as large as possible is justified because each of them allows certain degenerate simplexes. We determine explicitly all of these critical configurations.