Manifolds Of Curvature Research Articles

A note on some remarkable differential equations on a Riemannian manifold

The Fischer-Marsden conjecture asserts that an n-dimensional compact manifold admitting a nontrivial solution of the so-called Fischer-Marsden differential equation is necessarily an Einstein space. If this were true, then a classical theorem of Obata would imply that the underlying manifold is either a standard sphere or a Ricci flat space. Although counterexamples to this conjecture have been found by Kobayashi and Lafontaine, it has recently been proved by Cernea and Guan that the Fischer-Marsden conjecture holds, provided that the space of nonconstant solutions of the Fischer-Marsden equation is of dimension at least n, the authors actually proving that in this case (M,g) is nothing but a standard sphere. The main aim of this article is to show that any compact Riemannian manifold of positive Ricci curvature that admits a nontrivial concircular vector field with the potential function satisfying the Fischer-Marsden equation must be isometric to a standard sphere and the converse is also valid. Moreover, we prove that the existence of a nontrivial solution to another remarkable differential equation on Riemannian manifolds, namely the stationary Schrödinger equation, it also leads to a characterization of the sphere, provided that some pinching conditions are satisfied.

НЕКОТОРЫЕ ВОПРОСЫ ГЕОМЕТРИИ ЛОКАЛЬНО КОНФОРМНО-КЕЛЕРОВЫХ МНОГООБРАЗИЙ

Torso-forming vector fields, as well as their special cases (concircular, special concircular and recurrent) are used in many areas of differential geometry, for example, in conformal, geodesic, almost geodesic, holomorphically projective and other mappings and transformations. The presence of torso-forming vector fields on the space under consideration makes the geometry of this space more meaningful. It is of interest to study the geometry of spaces that admit recurrent vector fields. In this paper, the authors consider locally conformally Kahler manifolds with a recurrent Lie vector. Such manifolds are called recurrent locally conformally Kahler manifolds. The Lie form and the Lie vector are calculated explicitly. It is proved that the Lie vector of a locally conformally Kahler manifold of constant curvature is a concircular field. A recurrence criterion for a conformally flat locally conformally Kahler manifold is obtained. Some properties of conformally flat locally conformally Kahler manifolds are proved. It is proved that a compact manifold of constant curvature does not admit its own recurrent locally conformal Kahler structure.

Some curvature characterizations on Kenmotsu metric spaces


 
 
 The purpose of this study is to characterize Kenmotsu manifolds that satisfy specific curvature conditions. We give the Kenmotsu manifold curvature tensors satisfying the conditions RW5 = 0, RW7 = 0, RW9 = 0 and φ-RW0* = 0. Also we consider a W0*-flat and a φ-W0*-flat Kenmotsu metric manifolds. As a result, M is an η-Einstein Kenmotsu metric manifold.
 
 

Minimal hypersurfaces in manifolds of Ricci curvature bounded below

Abstract In this paper, we study the angle estimate of distance functions from minimal hypersurfaces in manifolds of Ricci curvature bounded from below using Colding’s method in [T. H. Colding, Ricci curvature and volume convergence, Ann. of Math. (2) 145 1997, 3, 477–501]. With Cheeger–Colding theory, we obtain the Laplacian comparison for limits of distance functions from minimal hypersurfaces in the version of Ricci limit space. As an application, if a sequence of minimal hypersurfaces converges to a metric cone C ⁢ Y × ℝ n - k {CY\times\mathbb{R}^{n-k}} ( 2 ≤ k ≤ n {2\leq k\leq n} ) in a non-collapsing metric cone C ⁢ X × ℝ n - k {CX\times\mathbb{R}^{n-k}} obtained from ambient manifolds of almost nonnegative Ricci curvature, then we can prove a Frankel property for the cross section Y of CY. Namely, Y has only one connected component in X.

General Chen Inequalities for Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature

Chen’s first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature was obtained by B.-Y. Chen et al. Other particular cases of Chen inequalities in a statistical setting were given by different authors. The objective of the present article is to establish the general Chen inequalities for statistical submanifolds in Hessian manifolds of constant Hessian curvature.

Characterization of manifolds of constant curvature by ruled surfaces

We investigate ruled surfaces in 3d Riemannian manifolds, i.e., surfaces foliated by geodesics. In 3d space forms, we find the striction curve, distribution parameter, and the first and second fundamental forms, from which we obtain the Gaussian and mean curvatures. We also provide model-independent proof for the known fact that extrinsically flat surfaces in space forms are ruled. This proof allows us to identify the necessary and sufficient condition the curvature tensor must satisfy for an extrinsically flat surface in a generic 3d manifold to be ruled. Further, we show that if a 3d manifold has an extrinsically flat surface tangent to any 2d plane and if they are all ruled surfaces, then the manifold is a space form. As an application, we prove that there must exist extrinsically flat surfaces in the Riemannian product of the hyperbolic plane, or sphere, with the reals, and that do not make a constant angle with the real direction.

A Contribution of Liouville-Type Theorems to the Geometry in the Large of Hadamard Manifolds

A complete, simply connected Riemannian manifold of nonpositive sectional curvature is called a Hadamard manifold. In this article, we prove Liouville-type theorems for isometric and harmonic self-diffeomorphisms of Hadamard manifolds, as well as Liouville-type theorems for Killing–Yano, symmetric Killing and harmonic tensors on Hadamard manifolds.

On the $$\sigma _2$$-curvature and volume of compact manifolds

In this work, we are interested in studying deformations of the \(\sigma _2\)-curvature and the volume. For closed manifolds, we relate critical points of the total \(\sigma _2\)-curvature functional to the \(\sigma _2\)-Einstein metrics and, as a consequence of results of Gursky and Viaclovsky (Invent Math 145(2):251–278, 2001) and Hu and Li (Trans Am Math Soc 356(8):3005–3023, 2004), we obtain a sufficient and necessary condition for a critical metric to be Einstein. Moreover, we show a volume comparison result for Einstein manifolds with respect to \(\sigma _2\)-curvature which shows that the volume can be controlled by the \(\sigma _2\)-curvature under certain conditions. Next, for compact manifold with nonempty boundary, we study variational properties of the volume functional restricted to the space of metrics with constant \(\sigma _2\)-curvature and with fixed induced metric on the boundary. We characterize the critical points to this functional as the solutions of an equation and show that in space forms they are geodesic balls. Studying second-order properties of the volume functional, we show that there is a variation for which geodesic balls are indeed local minima in a natural direction.

Unbounded Pontryagin numbers on nonnegatively curved spin manifolds

We prove that any rational linear combination of Pontryagin numbers that does not factor through the universal elliptic genus is unbounded on connected closed spin manifolds of nonnegative sectional curvature.

Flow curvature manifold and energy of generalized Liénard systems

In his famous book entitled Theory of Oscillations, Nicolas Minorsky wrote: “each time the system absorbs energy the curvature of its trajectory decreases and vice versa”. By using the Flow Curvature Method, we establish that, in the ε-vicinity of the slow invariant manifold of generalized Liénard systems, the curvature of trajectory curve increases while the energy of such systems decreases. Hence, we prove Minorsky's statement for the generalized Liénard systems. These results are then illustrated with the classical Van der Pol and generalized Liénard singularly perturbed systems.

Casorati Inequalities for Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant ϕ-Sectional Curvature with Semi-Symmetric Metric Connection.

In this paper, we prove some inequalities between intrinsic and extrinsic curvature invariants, namely the normalized -Casorati curvatures and the scalar curvature of statistical submanifolds in Kenmotsu statistical manifolds of constant -sectional curvature that are endowed with semi-symmetric metric connection. Furthermore, we investigate the equality cases of these inequalities. We also describe an illustrative example.

Manifold Calculus in System Theory and Control—Second Order Structures and Systems

The present tutorial paper constitutes the second of a series of tutorials on manifold calculus with applications in system theory and control. The aim of the present tutorial, in particular, is to explain and illustrate some key concepts in manifold calculus such as covariant derivation and manifold curvature. Such key concepts are then applied to the formulation, to the control, and to the analysis of non-linear dynamical systems whose state-space are smooth (Riemannian) manifolds. The main flow of exposition is enriched by a number of examples whose aim is to clarify the notation used and the main theoretical findings through practical calculations.

Black holes with spindles at the horizon

We construct AdS4 × Σ and AdS2 × Σ × {Sigma}_{mathfrak{g}} solutions in F(4) gauged supergravity in six dimensions, where Σ is a two dimensional manifold of non-constant curvature with conical singularities at its two poles, called a spindle, and {Sigma}_{mathfrak{g}} is a constant curvature Riemann surface of genus mathfrak{g} . We find that the first solution realizes a “topologically topological twist”, while the second class of solutions gives rise to an “anti twist”. We compute the holographic free energy of the AdS4 × Σ solution and find that it matches the entropy computed by extremizing an entropy functional that is constructed by gluing gravitational blocks. For the AdS2 × Σ × {Sigma}_{mathfrak{g}} solution, we find that the Bekenstein-Hawking entropy is reproduced by extremizing an appropriately defined entropy functional, which leads us to conjecture that this solution is dual to a three dimensional SCFT on a spindle. A class of the AdS2 × Σ × {Sigma}_{mathfrak{g}} solutions can be embedded in four dimensional T3 gauged supergravity, which is a subtruncation of the six dimensional theory.

Generalized Wintgen Inequality for Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature

The geometry of Hessian manifolds is a fruitful branch of physics, statistics, Kaehlerian and affine differential geometry. The study of inequalities for statistical submanifolds in Hessian manifolds of constant Hessian curvature was truly initiated in 2018 by Mihai, A. and Mihai, I. who dealt with Chen-Ricci and Euler inequalities. Later on, Siddiqui, A.N., Ahmad K. and Ozel C. came with the study of Casorati inequality for statistical submanifolds in the same ambient space by using algebraic technique. Also, Chen, B.-Y., Mihai, A. and Mihai, I. obtained a Chen first inequality for such submanifolds. In 2020, Mihai, A. and Mihai, I. studied the Chen inequality for δ(2,2)-invariant. In the development of this topic, we establish the generalized Wintgen inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature. Some examples are also discussed at the end.

Gallagherian PGT on Some Compact Riemannian Manifolds of Negative Curvature

Gallagherian PGT on Some Compact Riemannian Manifolds of Negative Curvature

Emergent time, cosmological constant and boundary dimension at infinity in combinatorial quantum gravity

Combinatorial quantum gravity is governed by a discrete Einstein-Hilbert action formulated on an ensemble of random graphs. There is strong evidence for a second-order quantum phase transition separating a random phase at strong coupling from an ordered, geometric phase at weak coupling. Here we derive the picture of space-time that emerges in the geometric phase, given such a continuous phase transition. In the geometric phase, ground-state graphs are discretizations of Riemannian, negative-curvature Cartan-Hadamard manifolds. On such manifolds, diffusion is ballistic. Asymptotically, diffusion time is soldered with a manifold coordinate and, consequently, the probability distribution is governed by the wave equation on the corresponding Lorentzian manifold of positive curvature, de Sitter space-time. With this asymptotic Lorentzian picture, the original negative curvature of the Riemannian manifold turns into a positive cosmological constant. The Lorentzian picture, however, is valid only asymptotically and cannot be extrapolated back in coordinate time. Before a certain epoch, coordinate time looses its meaning and the universe is a negative-curvature Riemannian “shuttlecock” with ballistic diffusion, thereby avoiding a big bang singularity. The emerging coordinate time leads to a de Sitter version of the holographic principle relating the bulk isometries with boundary conformal transformations. While the topological boundary dimension is (D − 1), the so-called “dimension at infinity” of negative curvature manifolds, i.e. the large-scale spectral dimension seen by diffusion processes with no spectral gap, those that can probe the geometry at infinity, is always three.

Local symmetry rank bound for positive intermediate Ricci curvatures

We use a local argument to prove if an $r$-dimensional torus acts isometrically and effectively on a connected $n$-dimensional manifold which has positive $k^\mathrm{th}$-intermediate Ricci curvature at some point, then $r \leq \lfloor \frac{n+k}{2} \rfloor$. This symmetry rank bound generalizes those established by Grove and Searle for positive sectional curvature and Wilking for quasipositive curvature. As a consequence, we show that the symmetry rank bound in the Maximal Symmetry Rank Conjecture for manifolds of non-negative sectional curvature holds for those which also have positive intermediate Ricci curvature at some point. In the process of proving our symmetry rank bound, we also obtain an optimal dimensional restriction on isometric immersions of manifolds with non-positive intermediate Ricci curvature into manifolds with positive intermediate Ricci curvature, generalizing a result by Otsuki.

A new characterization of Kenmotsu manifolds with respect to Q tensor

This paper introduces a new characterization of Kenmotsu manifolds and analyzes curvature conditions of Kenmotsu manifolds admitting the Q tensor whose trace is the well-known Z tensor. New types of Kenmotsu manifold are defined which are named ϕ−Q symmetric Kenmotsu manifold and ϕ−Q recurrent Kenmotsu manifold. Various properties of such a 2m+1-dimensional Kenmotsu manifold are studied. A theorem is given which states that a three-dimensional locally ϕ−Q recurrent and locally ϕ−Q symmetric Kenmotsu manifold admitting Q tensor is a manifold of constant curvature. Some examples of locally ϕ−Q symmetric Kenmotsu manifolds and Kenmotsu manifolds admitting Q tensor are provided.

Manifolds of positive Ricci curvature, quadratically asymptotically nonnegative curvature, and infinite Betti numbers

Manifolds of positive Ricci curvature, quadratically asymptotically nonnegative curvature, and infinite Betti numbers

Variational aspects of phase transitions with prescribed mean curvature

We study the spectrum of phase transitions with prescribed mean curvature in Riemannian manifolds. These phase transitions are solutions to an inhomogeneous semilinear elliptic PDE that give rise to diffuse objects (varifolds) that limit to hypersurfaces, possibly with singularities, whose mean curvature is determined by the "prescribed mean curvature" function and the limiting multiplicity. We establish upper bounds for the eigenvalues of the diffuse problem, as well as the more subtle lower bounds when the diffuse problem converges with multiplicity one. For the latter, we also establish asymptotics that are sharp to order $o(\varepsilon^2)$ and $C^{2,\alpha}$ estimates on multiplicity-one phase transition layers.