Sectional Curvature Research Articles

On the rigidity of cosmological space-times

In this paper, we analyse a family of geometrically well-behaved cosmological space-times (V^{n+1},g), which are foliated by intrinsically isotropic space-like hypersurfaces {M_t}_{tin mathbb {R}}, which are orthogonal to a family of co-moving observers defined by a global time-like vector field U. In particular, this implies such space-times satisfy several of the well-known criteria for isotropic cosmological space-times, although, in the family in question, the simultaneity spaces (M_t,g_t) associated with U can have as sectional curvature a sign-changing function k(t). Being this clearly impossible in the FLRW family of standard cosmological space-times, it motivates us to revisit the geometric rigidity consequences of different definitions of isotropy available in the literature. In this analysis, we divide such definition according to whether the isometries involved are taken to be (local) space-time (STI space-times) or (local) space isometries (SI space-times) of (M_t,g_t) for each t. This subtlety will be shown to be critical, proving that only when space-time isometries are considered one obtains the well-known rigidity properties associated with isotropic cosmological space-times. In particular, SI space-times will be shown to be a strictly larger class than the STI ones, allowing a family of basic cosmological curvature change models which are not even locally isometric to any FLRW space-time.

A note on area estimates for stable H-hypersurfaces and applications

We show some area estimates for stable CMC hypersurfaces immersed in Riemannian manifolds with scalar and sectional curvature bounded from below. In particular, we focus on immersions in three-dimensional Riemannian manifolds. As an application, we derive some upper estimates for the bottom spectrum of these hypersurfaces. This paper generalizes and builds on the techniques developed by Munteanu, Sung and Wang [19] for stable minimal surfaces.

Section performance analysis of vertical corrugated steel plate-reinforced concrete composite shear walls

A vertical corrugated steel plate embedded in steel-reinforced concrete composite shear wall demonstrates good seismic performance while adapting well to the section type and material of load-bearing members, thus demonstrating promising application prospects. This study analyses the stress distribution of a vertical corrugated steel plate-reinforced concrete composite shear wall (SPCWV) section and proposes a theoretical calculation method for the sectional yield, peak, and ultimate state curvatures controlled by the strain of the H-shaped steel section at the boundary element and confined concrete. Furthermore, the proposed method is extended to investigate the influence of structural settings, such as the geometric parameters of the vertical corrugated steel plates, H-shaped steel at the boundary element, and distributed rebars, and material properties, such as concrete and steel strength variations under different vertical loads on the section curvature and curvature ductility. The method is also used to determine the corrugated steel plate content ρcp and H-shaped steel content ρH to more effectively reflect the variations in the geometric parameters of the corrugated steel plate and H-shaped steel. Finally, simplified calculation methods for the yield, peak, and ultimate point curvatures of SPCWV are proposed, and their accuracy and applicability are verified.

On foliations of bounded mean curvature on closed three-dimensional Riemannian manifolds

The notion of systole of a foliation sys(ℱ) on an arbitrary foliated closed Riemannian manifold (M,ℱ) is introduced. A lower bound on sys(ℱ) of a bounded mean curvature foliation is given. As a corollary we prove that the number of Reeb components of a bounded mean curvature foliation on a closed oriented Riemannian 3-manifold M is bounded above by a constant depending on the volume, the radius of injectivity, and the maximum value of the sectional curvature of the manifold M.

Characterizations of weakly Ricci-symmetric spacetimes and f(ℛ)-gravity

In this paper, we characterize weakly Ricci-symmetric (shortly, [Formula: see text]) spacetimes and their solutions in [Formula: see text]-gravity. It is demonstrated that a [Formula: see text] spacetime represents a stiff matter fluid. In addition, we obtain that a conformally flat [Formula: see text] spacetime is a space of quasi-constant sectional curvature. Moreover, we establish that a Ricci symmetric [Formula: see text] spacetime represents a static spacetime. Finally, we investigate the effect of [Formula: see text] spacetime solutions in [Formula: see text]-gravity.

Norden Golden Manifolds with Constant Sectional Curvature and Their Submanifolds

This paper discusses the Norden golden manifold having a constant sectional curvature. First, it is shown that if a Norden golden manifold has a constant real sectional curvature, the manifold is flat. For this reason, the notions of holomorphic-like sectional curvature and holomorphic-like bisectional curvature on the Norden golden manifold are investigated, but it is seen that these notions do not work on the Norden golden manifold. This shows the need for a new concept of sectional curvature. In this direction, a new notion of sectional curvature (Norden golden sectional curvature) is proposed, an example is given, and if this new sectional curvature is constant, the curvature tensor field of the Norden golden manifold is expressed in terms of the metric tensor field. Since the geometry of the submanifolds of manifolds with constant sectional curvature has nice properties, the last section of this paper examines the semi-invariant submanifolds of the Norden golden space form.

Exploring the effects of high curvature section distribution and density in termite tunnels on food transport efficiency: A simulation study

This study explores the food transport efficiency of termite using an individual-based model. Termites are believed to have evolved tunneling patterns that optimize food search and transport efficiency. However, few studies have investigated transport efficiency due to the difficulty of field observations. The model is characterized by four control variables: the number of simulated termites participating in transport (k1), the distribution of high curvature sections of the termite tunnel (k2), a quantity related to the density of the tunnel sections (k3), and the duration of traffic jams (k4). As k3 increases, the total length of the high curvature section decreases. Our simulation results show that the E(k1, k2) maps for k3 and k4 contain two modes: Mode A shows that E decreases with increasing k1 due to an increase in traffic jams, while Mode B shows E increasing with increasing k1 due to a decrease in the density of curved sections and an increase in jamming resolution time. The partial rank correlation coefficient analysis reveals that k1 and k2 have a negative effect on E, while k3 and k4 have a positive effect, with k1 having the greatest influence on E, followed by k3, k4, and k2. The ecological implications of the simulation results are briefly described, and the limitations of the model are discussed.

Diameter estimates for submanifolds in manifolds with nonnegative curvature

Given a closed connected manifold smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we estimate the intrinsic diameter of the submanifold in terms of its mean curvature field integral. On the other hand, for a compact convex surface with boundary smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we can estimate its intrinsic diameter in terms of its mean curvature field integral and the length of its boundary. These results are supplements of previous work of Topping, Wu-Zheng and Paeng.

Locally conformally flat affine hyperspheres with parallel Ricci tensor

Trying to extend the result of Vrancken et al. (1991) [20] which established the complete classification of locally strongly convex affine hyperspheres with constant sectional curvature, we consider the classification problem of n-dimensional locally strongly convex affine hyperspheres with locally conformally flat affine metrics. As the main result of this paper, we solved the problem under the additional condition that the affine metric has parallel Ricci tensor with respect to its Levi-Civita connection.

Parallel Codazzi tensors with submanifold applications

AbstractA decomposition theorem is established for a class of closed Riemannian submanifolds immersed in a space form of the constant sectional curvature. In particular, it is shown that if M has nonnegative sectional curvature and admits a Codazzi tensor with “parallel mean curvature”, then M is locally isometric to a direct product of irreducible factors determined by the spectrum of that tensor. This decomposition is global when M is simply connected, and generalizes what is known for immersed submanifolds with parallel mean curvature vector.

On the noncollapsedness of positively curved Type I ancient Ricci flows

AbstractIn this paper, we study complete Type I ancient Ricci flows with positive sectional curvature. Our main results are as follows: in the complete and noncompact case, all such ancient solutions must be noncollapsed on all scales; in the closed case, if the dimension is even, then all such ancient solutions must be noncollapsed on all scales. This furthermore gives a complete classification for three‐dimensional noncompact Type I ancient solutions without assuming the noncollapsing condition.

On $C$-totally real submanifolds of $\mathbb{S}^{2n+1}(1)$ with non-negative sectional curvature

On $C$-totally real submanifolds of $\mathbb{S}^{2n+1}(1)$ with non-negative sectional curvature

Anti-quasi-Sasakian manifolds

We introduce and study a special class of almost contact metric manifolds, which we call anti-quasi-Sasakian (aqS). Among the class of transversely Kähler almost contact metric manifolds (M,varphi , xi ,eta ,g), quasi-Sasakian and anti-quasi-Sasakian manifolds are characterized, respectively, by the varphi -invariance and the varphi -anti-invariance of the 2-form textrm{d}eta . A Boothby–Wang type theorem allows to obtain aqS structures on principal circle bundles over Kähler manifolds endowed with a closed (2, 0)-form. We characterize aqS manifolds with constant xi -sectional curvature equal to 1: they admit an Sp(n)times 1-reduction of the frame bundle such that the manifold is transversely hyperkähler, carrying a second aqS structure and a null Sasakian eta -Einstein structure. We show that aqS manifolds with constant sectional curvature are necessarily flat and cokähler. Finally, by using a metric connection with torsion, we provide a sufficient condition for an aqS manifold to be locally decomposable as the Riemannian product of a Kähler manifold and an aqS manifold with structure of maximal rank. Under the same hypothesis, (M, g) cannot be locally symmetric.

A cosine rule-based discrete sectional curvature for graphs

Abstract How does one generalize differential geometric constructs such as curvature of a manifold to the discrete world of graphs and other combinatorial structures? This problem carries significant importance for analysing models of discrete spacetime in quantum gravity; inferring network geometry in network science; and manifold learning in data science. The key contribution of this article is to introduce and validate a new estimator of discrete sectional curvature for random graphs with low metric-distortion. The latter are constructed via a specific graph sprinkling method on different manifolds with constant sectional curvature. We define a notion of metric distortion, which quantifies how well the graph metric approximates the metric of the underlying manifold. We show how graph sprinkling algorithms can be refined to produce hard annulus random geometric graphs with minimal metric distortion. We construct random geometric graphs for spheres, hyperbolic and Euclidean planes; upon which we validate our curvature estimator. Numerical analysis reveals that the error of the estimated curvature diminishes as the mean metric distortion goes to zero, thus demonstrating convergence of the estimate. We also perform comparisons to other existing discrete curvature measures. Finally, we demonstrate two practical applications: (i) estimation of the earth’s radius using geographical data; and (ii) sectional curvature distributions of self-similar fractals.

W^{1,2} Bott-Chern and Dolbeault Decompositions on Kähler Manifolds

Let (M,J,g,omega ) be a Kähler manifold. We prove a W^{1,2} weak Bott-Chern decomposition and a W^{1,2} weak Dolbeault decomposition of the space of W^{1,2} differential (p, q)-forms, following the L^2 weak Kodaira decomposition on Riemannian manifolds. Moreover, if the Kähler metric is complete and the sectional curvature is bounded, the W^{1,2} Bott-Chern decomposition is strictly related to the space of W^{1,2} Bott-Chern harmonic forms, i.e., W^{1,2} smooth differential forms which are in the kernel of an elliptic differential operator of order 4, called Bott-Chern Laplacian.

The twisted coarse Baum-Connes conjecture with coefficients in coarsely proper algebras

We introduce a notion of (approximately) coarsely proper algebras for coarse embeddings of metric spaces to formulate and prove a version of twisted coarse Baum-Connes conjecture with coefficients in coarsely proper algebras. This may be regarded as a coarse geometric version of the generalized Green-Julg Theorem in the Baum-Connes conjecture for countable discrete groups. It also provides a conceptual framework for the Dirac-dual-Dirac method to the coarse Novikov conjecture for coarse embeddings into several different spaces, including Hilbert spaces, simply connected complete Riemannian manifolds with non-positive sectional curvature, Banach spaces with property (H) and Hilbert-Hadamard spaces. Moreover, for a group extension 1→N→G→Q→1, we show that if N is coarsely embeddable into Hilbert space and Q is coarsely embeddable into an admissible Hilbert-Hadamard space, then the coarse Novikov conjecture holds for G.

Kenmotsu 3-manifolds and gradient solitons

The aim of this article is to characterize a Kenmotsu 3-manifold whose metric is either a gradient Yamabe soliton or gradient Einstein soliton. It is proven that in both cases this manifold is reduced to the manifold of constant sectional curvature. Finally, we verify the obtained results by an example.

Elliptic Weingarten hypersurfaces of Riemannian products

AbstractLet be either a simply connected space form or a rank‐one symmetric space of the noncompact type. We consider Weingarten hypersurfaces of , which are those whose principal curvatures and angle function satisfy a relation , being W a differentiable function which is symmetric with respect to . When on the positive cone of , a strictly convex Weingarten hypersurface determined by W is said to be elliptic. We show that, for a certain class of Weingarten functions W, there exist rotational strictly convex Weingarten hypersurfaces of which are either topological spheres or entire graphs over M. We establish a Jellett–Liebmann‐type theorem by showing that a compact, connected and elliptic Weingarten hypersurface of either or is a rotational embedded sphere. Other uniqueness results for complete elliptic Weingarten hypersurfaces of these ambient spaces are obtained. We also obtain existence results for constant scalar curvature hypersurfaces of and which are either rotational or invariant by translations (parabolic or hyperbolic). We apply our methods to give new proofs of the main results by Manfio and Tojeiro on the classification of constant sectional curvature hypersurfaces of and .

Poincaré type inequality for hypersurfaces and rigidity results

In this article, under mild constraints on the sectional curvature, we exploit a divergence formula for symmetric endomorphisms to deduce a general Poincaré type inequality. We apply such inequality to higher-order mean curvature of hypersurfaces of space forms and Einstein manifolds, to obtain several isoperimetric inequalities, as well as rigidity results for complete r-minimal hypersurfaces satisfying a suitable decay of the second fundamental form at infinity. Furthermore, using these techniques, we prove flatness and non-existence results for self-similar solutions to a large class of fully nonlinear curvature flows.

A Note on Nearly Sasakian Manifolds

A class of nearly Sasakian manifolds is considered in this paper. We discuss the geometric effects of some symmetries on such manifolds and show, under a certain condition, that the class of Ricci semi-symmetric nearly Sasakian manifolds is a subclass of Einstein manifolds. We prove that a Codazzi-type Ricci nearly Sasakian space form is either a Sasakian manifold with a constant ϕ-holomorphic sectional curvature H=1 or a 5-dimensional proper nearly Sasakian manifold with a constant ϕ-holomorphic sectional curvature H>1. We also prove that the spectrum of the operator H2 generated by the nearly Sasakian space form is a set of a simple eigenvalue of 0 and an eigenvalue of multiplicity 4, and we induce that the underlying space form carries a Sasaki–Einstein structure. We show that there exist integrable distributions with totally geodesic leaves on the same manifolds, and we prove that there are no proper nearly Sasakian space forms with constant sectional curvature.