Sectional Curvature Research Articles

Positivity and the Kodaira embedding theorem

Kodaira embedding theorem provides an effective characterization of projectivity of a Kahler manifold in terms the second cohomology. Recently X. Yang [21] proved that any compact Kahler manifold with positive holomorphic sectional curvature must be projective. This gives a metric criterion of the projectivity in terms of its curvature. In this note, we prove that any compact Kahler manifold with positive 2nd scalar curvature (which is the average of holomorphic sectional curvature over 2-dimensional subspaces of the tangent space) must be projective. In view of generic 2-tori being non-abelian, this new curvature characterization is sharp in certain sense.

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).

Numerical Simulation of Curvature and Confining Pressure Effect on Progressive Failure of Surrounding Rock in Arch Tunnel with Straight Wall

Considering the influence of tensile plastic strain of intact rock on tensile strength and elastic modulus. The influence of arch curvature and height of the straight wall on the progressive failure of surrounding rock under different confining pressures are simulated based on the Mohr-Coulomb constitutive model and the nonlinear CWFS criterion of friction strengthening and cohesion weakening. The results show that when λ 1 = 0.2, the difference between vertical load and horizontal load is large, the buckling failure of rock burst and slab cracking is obvious, and the sectional curvature effect is obvious. When λ 3 = 0.5, the buckling failure of rock burst and slab cracking is very limited, the sectional curvature effect is reduced, and the confining pressure effect of rock burst and slab cracking is obvious. The shear yield localization of the surrounding rock is sensitive to sectional curvature, and the sectional curvature effect of shear failure under different lateral pressure coefficients is obvious. The numerical calculation results are consistent with the failure law of surrounding rock obtained from field and laboratory tests, which can provide a reference for tunnel excavation and support design.

Curvatures of strongly convex Kähler-Finsler manifolds

In this article, we study curvatures on a strongly convex (weakly) Kähler-Finsler manifold. First, we prove that the holomorphic sectional curvature is just half of the flag curvature in a holomorphic plane section on a strongly convex weakly Kähler-Finsler manifold. Second, we compare curvatures associated to the Rund connection with curvatures associated to the Chern-Finsler connection or the complex Berward connection on a strongly convex Kähler-Finsler manifold. Finally, we discuss relationships between flag curvatures and holomorphic bisectional curvatures, and compare two kinds of S-curvatures on a strongly convex Kähler-Finsler manifold.

The isoperimetric problem on Riemannian manifolds via Gromov–Hausdorff asymptotic analysis

In this paper, we prove the existence of isoperimetric regions of any volume in Riemannian manifolds with Ricci bounded below assuming Gromov–Hausdorff asymptoticity to the suitable simply connected model of constant sectional curvature. The previous result is a consequence of a general structure theorem for perimeter-minimizing sequences of sets of fixed volume on noncollapsed Riemannian manifolds with a lower bound on the Ricci curvature. We show that, without assuming any further hypotheses on the asymptotic geometry, all the mass and the perimeter lost at infinity, if any, are recovered by at most countably many isoperimetric regions sitting in some (possibly nonsmooth) Gromov–Hausdorff limits at infinity. The Gromov–Hausdorff asymptotic analysis allows us to recover and extend different previous existence theorems. While studying the isoperimetric problem in the smooth setting, the nonsmooth geometry naturally emerges, and thus our treatment combines techniques from both the theories.

Riemann solitons on para-Sasakian geometry

The goal of the present article is to investigate almost Riemann soliton and gradient almost Riemann soliton on 3-dimensional para-Sasakian manifolds. At first, it is proved that if $(g, Z,\lambda)$ is an almost Riemann soliton on a para-Sasakian manifold $M^3$, then it reduces to a Riemann soliton and $M^3$ is of constant sectional curvature $-1$, provided the soliton vector $Z$ has constant divergence. Besides these, we prove that if $Z$ is pointwise collinear with the characteristic vector field $\xi$, then $Z$ is a constant multiple of $\xi$ and the manifold is of constant sectional curvature $-1$. Moreover, the almost Riemann soliton is expanding. Furthermore, it is established that if a para-Sasakian manifold $M^3$ admits gradient almost Riemann soliton, then $M^3$ is locally isometric to the hyperbolic space $H^{3}(-1)$. Finally, we construct an example to justify some results of our paper.

Non-quasiconvex subgroups of hyperbolic groups via Stallings-like techniques

We provide a new method of constructing non-quasiconvex subgroups of hyperbolic groups by utilizing techniques inspired by Stallings’ foldings. The hyperbolic groups constructed are in the natural class of right-angled Coxeter groups (RACGs for short) and can be chosen to be 2 2 -dimensional. More specifically, given a non-quasiconvex subgroup of a (possibly non-hyperbolic) RACG, our construction gives a corresponding non-quasiconvex subgroup of a hyperbolic RACG. We use this to construct explicit examples of non-quasiconvex subgroups of hyperbolic RACGs including subgroups whose generators are as short as possible (length two words), finitely generated free subgroups, non-finitely presentable subgroups, and subgroups of fundamental groups of square complexes of nonpositive sectional curvature.

Global Schrödinger map flows to Kähler manifolds with small data in critical Sobolev spaces: Energy critical case

In this paper and the companion work \[J. Funct. Anal. 281 (2021)], we prove that the Schrödinger map flows from $\mathbb{R}^d$ with $d\ge 2$ to compact Kähler manifolds with small initial data in critical Sobolev spaces are global. The main difficulty compared with the constant sectional curvature case is that the gauged equation now is not self-contained due to the curvature part. Our main idea is to use a novel bootstrap-iteration scheme to reduce the gauged equation to an approximate constant curvature system in finite times of iteration. This paper together with the companion work solves the open problem raised by Tataru.

Graphical mean curvature flow with bounded bi-Ricci curvature

We consider the graphical mean curvature flow of strictly area decreasing maps $f:M\to N$, where $M$ is a compact Riemannian manifold of dimension $m>1$ and $N$ a complete Riemannian surface of bounded geometry. We prove long-time existence of the flow and that the strictly area decreasing property is preserved, when the bi-Ricci curvature $BRic_M$ of $M$ is bounded from below by the sectional curvature $\sigma_N$ of $N$. In addition, we obtain smooth convergence to a minimal map if $Ric_M\ge\sup\{0,{\sup}_N\sigma_N\}$. These results significantly improve known results on the graphical mean curvature flow in codimension $2$.

Ricci flow does not preserve positive sectional curvature in dimension four

We find examples of cohomogeneity one metrics on S^4 and mathbb {C}P^2 with positive sectional curvature that lose this property when evolved via Ricci flow. These metrics are arbitrarily small perturbations of Grove–Ziller metrics with flat planes that become instantly negatively curved under Ricci flow.

New lower bounds on the number of intersections of monotone Lagrangian submanifolds

Considering a closed monotone Lagrangian submanifold L , L, we give, under some hypotheses, a lower bound on the intersection number of L L with its image by a generic Hamiltonian isotopy. First for monotone Lagrangian submanifolds L L which are K ( π , 1 ) \mathbf {K}(\pi ,1) and, in particular, for monotone Lagrangian submanifolds with negative sectional curvature this bound is 1+ β 1 ( L ) . \beta _{1}(L). In more general cases the lower bound is weaker. We generalise some results previously obtained by L. Buhovsky in [J. Topol. Anal. 2 (2010), pp. 57–75] and P. Biran and O. Cornea in [Geom. Topol. 13 (2009), pp. 2881–2989].

One-dimensional exponential families with constant Hessian sectional curvature

One-dimensional exponential families with constant Hessian sectional curvature

Improved Technique for Autonomous Vehicle Motion Planning Based on Integral Constraints and Sequential Optimization

The study is dedicated to elaborating and analyzing a technique for autonomous vehicle (AV) motion planning based on sequential trajectory and kinematics optimization. The proposed approach combines the finite element method (FEM) basics and nonlinear optimization with nonlinear constraints. There were five main innovative aspects introduced in the study. First, a 7-degree polynomial was used to improve the continuity of piecewise functions representing the motion curves, providing 4 degrees of freedom (DOF) in a node. This approach allows using the irregular grid for roadway segments, increasing spans where the curvature changes slightly, and reducing steps in the vicinity of the significant inflections of motion boundaries. Therefore, the segment length depends on such factors as static and moving obstacles, average road section curvature, camera sight distance, and road conditions (adhesion). Second, since the method implies splitting the optimization stages, a strategy for bypassing the moving obstacles out of direct time dependency was developed. Thus, the permissible area for maneuvering was determined using criteria of safety distance between vehicles and physical limitation of tire–road adhesion. Third, the nodal inequality constraints were replaced by the nonlinear integral equality constraints. In contrast to the generally distributed approach of restricting the planning parameters in nodes, the technique of integral equality constraints ensures the disposition of motion parameters’ curves strictly within the preset boundaries, which is especially important for quite long segments. In this way, the reliability and stability of predicted parameters are improved. Fourth, the seamless continuity of both the sought parameters and their derivatives is ensured in transitional nodes between the planning phases and adjacent global coordinate systems. Finally, the problem of optimization rapidity to match real-time operation requirements was addressed. For this, the quadrature integration approach was implemented to represent and keep all the parameters in numerical form. The study considered cost functions, limitations stipulated by the vehicle kinematics and dynamics, as well as initial and transient conditions between the planning stages. Simulation examples of the predicted trajectories and curves of kinematic parameters are demonstrated. The advantages and limitations of the proposed approach are highlighted.

D’Atri spaces and the total scalar curvature of hemispheres, tubes and cylinders

Csikós and Horváth proved in J Geom Anal 28(4): 3458-3476, (2018) that if a connected Riemannian manifold of dimension at least 4 is harmonic, then the total scalar curvatures of tubes of small radius about an arbitrary regular curve depend only on the length of the curve and the radius of the tube, and conversely, if the latter condition holds for cylinders, i.e., for tubes about geodesic segments, then the manifold is harmonic. In the present paper, we show that in contrast to the higher dimensional case, a connected 3-dimensional Riemannian manifold has the above mentioned property of tubes if and only if the manifold is a D’Atri space, furthermore, if the space has bounded sectional curvature, then it is enough to require the total scalar curvature condition just for cylinders to imply that the space is D’Atri. This result gives a negative answer to a question posed by Gheysens and Vanhecke. To prove these statements, we give a characterization of D’Atri spaces in terms of the total scalar curvature of geodesic hemispheres in any dimension.

Collapsing to Alexandrov spaces with isolated mild singularities

Let Mj be a sequence of Riemannian manifolds with sectional curvature bound below collapsing to a compact Alexandrov space X of dimension k. Suppose that all but finitely many points of X are (k,δ)-strained and that the space of directions at each exceptional point contains k+1 directions making obtuse angles with each other. We prove that Mj admits a structure of locally trivial fibration over X for sufficiently large j. The same is true for collapsing sequences of Alexandrov spaces such that the infimum of the volume of the spaces of directions is sufficiently large relative to δ.

Linear isoperimetric inequalities for free boundary submanifolds in a geodesic ball

We derive linear isoperimetric inequalities for free boundary submanifolds in a geodesic ball of a Riemannian manifold in terms of the modified volume. It is known that the twice of the area of a free boundary minimal surface in a Euclidean unit ball is equal to the length of its boundary. This can be extended to space forms by using our linear isoperimetric inequalities for the modified volume. Moreover, we obtain a sharp lower bound for the modified volume of free boundary minimal surfaces in a geodesic ball of the upper hemisphere mathbb {S}_+^3. Finally, it is proved that the monotonicity property still holds for the modified volume of any submanifold in a complete simply connected Riemannian manifold with sectional curvature bounded above by a constant.

Einstein Hypersurfaces of Warped Product Spaces

We consider Einstein hypersurfaces of warped products \(I\times _\omega {\mathbb {Q}}_\epsilon ^n,\) where \(I\subset {\mathbb {R}}\) is an open interval and \({\mathbb {Q}}_\epsilon ^n\) is the simply connected space form of dimension \(n\ge 2\) and constant sectional curvature \(\epsilon \in \{-1,0,1\}\). We show that, for all \(c\in {\mathbb {R}}\) (resp. \(c>0\)), there exist rotational hypersurfaces of constant sectional curvature c in \(I\times _\omega {\mathbb {H}}^n\) and \(I\times _\omega {\mathbb {R}}^n\) (resp. \(I\times _\omega {\mathbb {S}}^n\)), provided that \(\omega \) is nonconstant. We also show that the gradient T of the height function of any Einstein hypersurface of \(I\times _\omega \mathbb {Q} _\epsilon ^n\) (if nonzero) is one of its principal directions. Then, we consider a particular type of Einstein hypersurface of \(I\times _\omega {\mathbb {Q}}_\epsilon ^n\) with non vanishing T—which we call ideal—and prove that, for \(n>3,\) such a hypersurface \(\Sigma \) has either precisely two or precisely three distinct principal curvatures everywhere. We show that, in the latter case, there exist such a \(\Sigma \) for certain warping functions \(\omega ,\) whereas in the former case \(\Sigma \) is necessarily of constant sectional curvature and rotational, regardless the warping function \(\omega .\) We also characterize ideal Einstein hypersurfaces of \(I\times _\omega {\mathbb {Q}}_\epsilon ^n\) with no vanishing angle function as local graphs on families of isoparametric hypersurfaces of \({\mathbb {Q}}_\epsilon ^n.\)

Index theory and deformations of open nonnegatively curved manifolds

We use an index-theoretic technique of Hitchin to show that the space of complete Riemannian metrics of nonnegative sectional curvature on certain open spin manifolds has nontrivial homotopy groups in infinitely many degrees. A new ingredient of independent interest is homotopy density of the subspace of metrics with cylindrical ends.

On Almost Nonpositive k-Ricci Curvature

Motivated by the recent work of Chu–Lee–Tam on the nefness of canonical line bundle for compact Kähler manifolds with nonpositive k-Ricci curvature, we consider a natural notion of almost nonpositive k-Ricci curvature, which is weaker than the existence of a Kähler metric with nonpositive k-Ricci curvature. When \(k=1\), this is just the almost nonpositive holomorphic sectional curvature introduced by Zhang. We firstly give a lower bound for the existence time of the twisted Kähler-Ricci flow when there exists a Kähler metric with k-Ricci curvature bounded from above by a positive constant. As an application, we prove that a compact Kähler manifold of almost nonpositive k-Ricci curvature must have nef canonical line bundle.

Bergman metric on the symmetrized bidisk and its consequences

On the symmetrized bidisk [Formula: see text] with the Bergman metric, the holomorphic sectional curvature is negatively pinched and the holomorphic bisectional curvature is not. The consequences of invariant metrics are provided.