Riemannian Manifold Of Dimension Research Articles

On the role of the surface geometry in convex billiards

Abstract This work presents a framework for billiards in convex domains on two dimensional Riemannian manifolds. These domains are contained in connected, simply connected open subsets which are totally normal. In this context, some properties that have long been known for billiards on the plane are established. We prove the twist property of the billiard maps and establish some conditions for the existence and non-existence of rotational invariant curves. Although we prove that Lazutkin’s and Hubacher’s theorems are valid for general surfaces, we also find that Mather’s theorem does not apply to surfaces of non-negative curvature.

Symbolic Dynamics for Nonuniformly Hyperbolic Maps with Singularities in High Dimension

We construct Markov partitions for non-invertible and/or singular nonuniformly hyperbolic systems defined on higher dimensional Riemannian manifolds. The generality of the setup covers classical examples not treated so far, such as geodesic flows in closed manifolds, multidimensional billiard maps, and Viana maps, and includes all the recent results of the literature. We also provide a wealth of applications.

Convexity of weakly regular surfaces of distributional nonnegative intrinsic curvature

We prove that the image of an isometric embedding into R3 of a two dimensional complete Riemannian manifold (Σ,g) without boundary is a convex surface, provided that, first, both the embedding and the metric g enjoy a C1,α regularity for some α>2/3, and second, the distributional Gaussian curvature of g is nonnegative and nonzero. The analysis must pass through some key observations regarding solutions to the very weak Monge-Ampère equation.

Systolic almost-rigidity modulo 2

No power law systolic freedom is possible for the product of mod 2 systoles of dimension 1 and codimension 1 . This means that any closed n -dimensional Riemannian manifold M of bounded local geometry obeys the following systolic inequality: the product of its mod 2 systoles of dimensions 1 and n-1 is bounded from above by c(n,\varepsilon) \,\mathrm{Vol}(M)^{1+\varepsilon} , if finite (if H_{1}(M; \mathbb{Z}/2) is non-trivial).

Manifolds whose Weyl spectral asymptotics have small but not tiny remainders

A compact, n -dimensional Riemannian manifold M has Weyl spectral asymptotics with remainder E_{M}(R) ; i.e., the spectral counting function satisfies \mathcal{N}(\Delta_{M},R)=C(M)R^{n}+ E_{M}(R) , with E_{M}(R)=o(R^{n}) . Generally, one actually has E_{M}(R)=O(R^{n-1}) , and one seeks geometrical conditions under which stronger estimates hold on the remainder and also conditions limiting how extra small the remainder can be. Here, we produce n -dimensional manifolds whose Weyl remainders are o(R^{n-1}) but not O(R^{n-1-\alpha}) for any \alpha>0 .

Characterization of a special type of Ricci–Bourguignon soliton on sequential warped product manifold

In this paper, we aim to characterize the sequential warped product [Formula: see text]-almost gradient conformal Ricci–Bourguignon soliton. We derive applications of some vector fields like conformal vector field, torse-forming vector field, torqued vector field on [Formula: see text]-almost conformal Ricci–Bourguignon soliton. The inheritance properties of the Einstein-like sequential warped product [Formula: see text]-almost gradient conformal Ricci–Bourguignon solitons of class types [Formula: see text] are investigated in this paper. Later we show that if [Formula: see text] is a sequential warped product of a complete connected [Formula: see text]-dimensional Riemannian manifold [Formula: see text] and one-dimensional Riemannian manifolds [Formula: see text] and [Formula: see text] admitting [Formula: see text]-almost conformal Ricci–Bourguignon soliton, then [Formula: see text] becomes a [Formula: see text]-dimensional sphere of radius [Formula: see text]. We have discovered that for a [Formula: see text]-almost gradient conformal Ricci–Bourguignon soliton sequential warped product, the warping functions are constants under certain conditions.

Diameter estimates for surfaces in conformally flat spaces

The aim of this paper is to give an upper bound for the intrinsic diameter of a surface with boundary immersed in a conformally flat three dimensional Riemannian manifold in terms of the integral of the mean curvature and of the length of its boundary. Of particular interest is the application of the inequality to minimal surfaces in the three-sphere and in the hyperbolic space. Here the result implies an a priori estimate for connected solutions of Plateau’s problem, as well as a necessary condition on the boundary data for the existence of such solutions. The proof follows a construction of Miura and uses a diameter bound for closed surfaces obtained by Topping and Wu–Zheng.

Dimension estimates and approximation in non-uniformly hyperbolic systems

Abstract Let $f: M\rightarrow M$ be a $C^{1+\alpha }$ diffeomorphism on an $m_0$ -dimensional compact smooth Riemannian manifold M and $\mu $ a hyperbolic ergodic f-invariant probability measure. This paper obtains an upper bound for the stable (unstable) pointwise dimension of $\mu $ , which is given by the unique solution of an equation involving the sub-additive measure-theoretic pressure. If $\mu $ is a Sinai–Ruelle–Bowen (SRB) measure, then the Kaplan–Yorke conjecture is true under some additional conditions and the Lyapunov dimension of $\mu $ can be approximated gradually by the Hausdorff dimension of a sequence of hyperbolic sets $\{\Lambda _n\}_{n\geq 1}$ . The limit behaviour of the Carathéodory singular dimension of $\Lambda _n$ on the unstable manifold with respect to the super-additive singular valued potential is also studied.

On Some Aspects of the Courant-Type Algebroids, the Related Coadjoint Orbits and Integrable Systems

Poisson structures related to affine Courant-type algebroids are analyzed, including those related with cotangent bundles on Lie-group manifolds. Special attention is paid to Courant-type algebroids and their related R structures generated by suitably defined tensor mappings. Lie–Poisson brackets that are invariant with respect to the coadjoint action of the loop diffeomorphism group are created, and the related Courant-type algebroids are described. The corresponding integrable Hamiltonian flows generated by Casimir functionals and generalizing so-called heavenly-type differential systems describing diverse geometric structures of conformal type in finite dimensional Riemannian manifolds are described.

Теоремы о дифференцируемых сферах для многообразий с ограниченными сверху кривизнами Риччи

In the present paper, we prove that if is an -dimensional compact Riemannian manifold and if where , and are the sectional and Ricci curvatures of respectively, then is diffeomorphic to a spherical space form where is a finite group of isometries acting freely. In particular, if is simply connected, then it is diffeo­mor­phic to the Euclidian sphere

The upper bound of the harmonic mean of the Steklov eigenvalues in curved spaces

AbstractConsidering an ‐dimensional Riemannian manifold whose sectional curvature is bounded above by and Ricci curvature is bounded below by , we obtain an upper bound of the harmonic mean of the first nonzero Steklov eigenvalues for domains contained in . This can be viewed as certain isoperimetric inequality and generalizes the result on comparing the first nonzero Steklov eigenvalues for such domains (Li–Wang–Wu, 2020, arXiv:2003.03093).

The anisotropic min‐max theory: Existence of anisotropic minimal and CMC surfaces

AbstractWe prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3–dimensional Riemannian manifolds. The constructed min‐max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard in dimension 3.

An inverse problem for the Riemannian minimal surface equation

In this paper we consider determining a minimal surface embedded in a Riemannian manifold Σ×R. We show that if Σ is a two dimensional Riemannian manifold with boundary, then the knowledge of the associated Dirichlet-to-Neumann map for the minimal surface equation determine Σ up to an isometry.

Myers-type theorems via m-Bakry–Émery Ricci curvatures of quartic decays

We establish several Myers-type theorems for [Formula: see text]-dimensional complete Riemannian manifolds assuming some quartic decays of the [Formula: see text]-Bakry–Émery and [Formula: see text]-modified Bakry–Émery Ricci curvatures with [Formula: see text] in terms of the distance function from a fixed point.

Macroscopic scalar curvature and codimension 2 width

We show that a complete [Formula: see text]-dimensional Riemannian manifold [Formula: see text] with finitely generated first homology has macroscopic dimension [Formula: see text] if it satisfies the following “macroscopic curvature” assumptions: every ball of radius [Formula: see text] in [Formula: see text] has volume at most [Formula: see text], and every loop in every ball of radius [Formula: see text] in [Formula: see text] is null-homologous in the concentric ball of radius [Formula: see text].

Betti numbers and the curvature operator of the second kind

AbstractWe show that compact, ‐dimensional Riemannian manifolds with ‐nonnegative curvature operators of the second kind are either rational homology spheres or flat. More generally, we obtain vanishing of the th Betti number provided that the curvature operator of the second kind is ‐positive. Our curvature conditions become weaker as increases. For , we have , and for , we exhibit a ‐positive algebraic curvature operator of the second kind with negative Ricci curvatures.

Peano property, two dimensional triangles, constant curvature

In the present note, we discuss the connection of the convex geometric and the analytic properties of the underlying manifold. As an application of the main results we show that in Riemannian manifolds of dimension at least three, local versions the Peano property, the drop completeness, or the “thinness” of triangles are equivalent with having constant curvature, and global versions characterize Euclidean and hyperbolic spaces among Cartan–Hadamard manifolds.

On complete hypersurfaces with negative Ricci curvature in Euclidean spaces

In this paper, we prove that if M^n , n\geq 3 , is a complete Riemannian manifold with negative Ricci curvature and f\colon M^n\to\mathbb{R}^{n+1} is an isometric immersion such that \mathbb{R}^{n+1}\backslash f(M) is an open set that contains balls of arbitrarily large radius, then \inf_M|A|=0 , where |A| is the norm of the second fundamental form of the immersion. In particular, an n -dimensional complete Riemannian manifold with negative Ricci curvature bounded away from zero cannot be properly isometrically immersed in a half-space of \mathbb{R}^{n+1} . This gives a partial answer to a question raised by Reilly and Yau.

Some notes on the tangent bundle with a Ricci quarter-symmetric metric connection

<abstract><p>Let $ (M, g) $ be an $ n $-dimensional (pseudo-)Riemannian manifold and $ TM $ be its tangent bundle $ TM $ equipped with the complete lift metric $ ^{C}g $. First, we define a Ricci quarter-symmetric metric connection $ \overline{\nabla } $ on the tangent bundle $ TM $ equipped with the complete lift metric $ ^{C}g $. Second, we compute all forms of the curvature tensors of $ \overline{\nabla } $ and study their properties. We also define the mean connection of $ \overline{\nabla } $. Ricci and gradient Ricci solitons are important topics studied extensively lately. Necessary and sufficient conditions for the tangent bundle $ TM $ to become a Ricci soliton and a gradient Ricci soliton concerning $ \overline{\nabla } $ are presented. Finally, we search conditions for the tangent bundle $ TM $ to be locally conformally flat with respect to $ \overline{\nabla } $.</p></abstract>

Hierarchical Geodesic Polynomial Model for Multilevel Analysis of Longitudinal Shape.

Longitudinal analysis is a core aspect of many medical applications for understanding the relationship between an anatomical subject's function and its trajectory of shape change over time. Whereas mixed-effects (or hierarchical) modeling is the statistical method of choice for analysis of longitudinal data, we here propose its extension as hierarchical geodesic polynomial model (HGPM) for multilevel analyses of longitudinal shape data. 3D shapes are transformed to a non-Euclidean shape space for regression analysis using geodesics on a high dimensional Riemannian manifold. At the subject-wise level, each individual trajectory of shape change is represented by a univariate geodesic polynomial model on timestamps. At the population level, multivariate polynomial expansion is applied to uni/multivariate geodesic polynomial models for both anchor points and tangent vectors. As such, the trajectory of an individual subject's shape changes over time can be modeled accurately with a reduced number of parameters, and population-level effects from multiple covariates on trajectories can be well captured. The implemented HGPM is validated on synthetic examples of points on a unit 3D sphere. Further tests on clinical 4D right ventricular data show that HGPM is capable of capturing observable effects on shapes attributed to changes in covariates, which are consistent with qualitative clinical evaluations. HGPM demonstrates its effectiveness in modeling shape changes at both subject-wise and population levels, which is promising for future studies of the relationship between shape changes over time and the level of dysfunction severity on anatomical objects associated with disease.