Differentiable Manifold Research Articles

Some results on Kenmotsu statistical manifolds

In this paper, we first investigate the Kenmotsu statistical structures built on a Kenmotsu space form and determine some special Kenmotsu statistical structures under two curvature conditions. Secondly, we show that if the holomorphic sectional curvature of the hypersurface orthogonal to the structure vector in a Kenmotsu statistical manifold is constant, then the $\phi-$sectional curvature of the ambient Kenmotsu statistical manifold must be constant $-1$, and the constant holomorphic sectional curvature of the hypersurface is $0$. In addition, some non-trivial examples are given to illustrate the results of this paper.

The Sasaki-Kenmotsu manifolds

In the present paper, we introduce a new class of structures on an even dimensional differentiable Riemannian manifold which combines, well known in literature, the Sasakian and Kenmotsu structures simultaneously. The structure will be called a Sasaki–Kenmotsu structure by us. Firstly, we discuss the normality of the Sasaki–Kenmotsu structure and give some basic properties. Secondly, we present some important results concerning with the curvatures of the Sasaki–Kenmotsu manifold. Finally, we show the existence of the Sasaki–Kenmotsu structure by giving some concrete examples.

Fractional Order Riemann Curvature Tensor in Differential Geometry

This study discussed some interesting aspects and features of fractional curvature in the differential manifold. In particular, Riemannian fractional curvature tensor, Livi-Civita fractional connection and Bianchi fractional identity are presented.

Sharp Hardy Inequalities via Riemannian Submanifolds

This paper is devoted to Hardy inequalities concerning distance functions from submanifolds of arbitrary codimensions in the Riemannian setting. On a Riemannian manifold with non-negative curvature, we establish several sharp weighted Hardy inequalities in the cases when the submanifold is compact as well as non-compact. In particular, these inequalities remain valid even if the ambient manifold is compact, in which case we find an optimal space of smooth functions to study Hardy inequalities. Further examples are also provided. Our results complement in several aspects those obtained recently in the Euclidean and Riemannian settings.

Free involutions on odd dimensional Wall manifolds and cohomology of orbit spaces

The main purpose of this paper is to compute all possible mod 2 cohomology algebra of orbit spaces of free involutions on odd dimensional Wall manifolds of type Q(m,n), which are the closed differentiable manifolds constructed as mapping torus of specific involutions on Dold manifolds P(m,n), for m even and n odd.

Environmental contours as Voronoi cells

Environmental contours are widely used as basis for design of structures exposed to environmental loads. The basic idea of the method is to decouple the environmental description from the structural response. This is done by establishing an envelope of joint extreme values representing critical environmental conditions, such that any structure tolerating loads on this envelope will have a failure probability smaller than a prescribed value. Specifically, given an n-dimensional random variable $$\mathbf {X}$$ and a target probability of failure $$p_{e}$$ , an environmental contour is the boundary of a set $$\mathcal {B} \subset \mathbb {R}^{n}$$ with the following property: For any failure set $$\mathcal {F} \subset \mathbb {R}^{n}$$ , if $$\mathcal {F}$$ does not intersect the interior of $$\mathcal {B}$$ , then the probability of failure, $$P(\mathbf {X} \in \mathcal {F})$$ , is bounded above by $$p_{e}$$ . We work under the assumption that failure sets are convex, which is relevant for many real-world applications. In this paper, we show that such environmental contours may be regarded as boundaries of Voronoi cells. This geometric interpretation leads to new theoretical insights and suggests a simple novel construction algorithm that guarantees the desired probabilistic properties. The method is illustrated with examples in two and three dimensions, but the results extend to environmental contours in arbitrary dimensions. Inspired by the Voronoi-Delaunay duality in the numerical discrete scenario, we are also able to derive an analytical representation where the environmental contour is considered as a differentiable manifold, and a criterion for its existence is established.

The Measurement and Analysis of Shapes

A de Rham p-current can be viewed as a map (the current map) between the set of embeddings of a closed p-dimensional manifold into an ambient n-manifold and the set of linear functionals on differential p-forms. We demonstrate that, for suitably chosen Sobolev topologies on both the space of embeddings and the space of p-forms, the current map is continuously differentiable, with an image that consists of bounded linear functionals on p-forms. Using the Riesz representation theorem, we prove that each p-current can be represented by a unique co-exact differential form that has a particular interpretation depending on p. Embeddings of a manifold can be thought of as shapes with a prescribed topology. Our analysis of the current map provides us with representations of shapes that can be used for the measurement and statistical analysis of collections of shapes. We consider two special cases of our general analysis and prove that: (1) if p=n-1 then closed, embedded, co-dimension one surfaces are naturally represented by probability distributions on the ambient manifold and (2) if p=1 then closed, embedded, one-dimensional curves are naturally represented by fluid flows on the ambient manifold. In each case, we outline some statistical applications using an {dot{H}}^{1} and L^{2} metric, respectively.

What is a Lipschitzian Manifold?

We propose a definition of Lipschizian manifold that is more precise than the notion of Lipschitzian parameterization. It is modelled on the notion of differentiable manifold. We also give a notion of Lipschitzian submanifold and compare it with a notion devised by R.T. Rockafellar (Ann. I.H.P. Sect. C 2(3), 167–184, 1985). Among the examples we mention, the case of the graph of a maximally monotone operator and of the subjet of a convex function are the most notable.

Non-singular Morse–Smale flows on n-manifolds with attractor–repeller dynamics

In the present paper the exhaustive classification up to topological equivalence of non-singular Morse–Smale flows on n-manifolds M n with exactly two periodic orbits is presented. Denote by G 2(M n ) the set of such flows. Let a flow f t : M n → M n belongs to the set G 2(M n ). Hyperbolicity of periodic orbits of f t implies that among them one is an attracting and the other is a repelling orbit. Due to the Poincaré–Hopf theorem, the Euler characteristic of the ambient manifold M n is zero. Only the torus and the Klein bottle can be ambient manifolds for f t in case of n = 2. The authors established that there are exactly two classes of topological equivalence of flows in G 2(M 2) if M 2 is the torus and three classes if M 2 is the Klein bottle. For all odd-dimensional manifolds the Euler characteristic is zero. However, it is known that an orientable three-manifold admits a flow from G 2(M 3) if and only if M 3 is a lens space L p,q . In this paper it is proved that every set G 2(L p,q ) contains exactly two classes of topological equivalence of flows, except the case when L p,q is homeomorphic to the three-sphere or the projective space , where such a class is unique. Also, it is shown that the only non-orientable n-manifold (for n > 2), which admits flows from G 2(M n ) is the twisted I-bundle over the (n − 1)-sphere . Moreover, there are exactly two classes of topological equivalence of flows in . Among orientable n-manifolds only the product of the (n − 1)-sphere and the circle can be ambient manifolds for flows from G 2(M n ) and splits into two topological equivalence classes.

Finite displacement screw-based group analysis of 3PRS parallel mechanisms

When limbs of a parallel mechanism have been selected, the geometric relation among limbs determines the mechanism's motion. The limb arrangement which affects the mechanism's motion raises a fundamental and significant issue. If a mechanism only owns limbs with Lie group motions, it is convenient to predict how a limb arrangement determines the mechanism's motion. However, if a mechanism is mainly or purely constituted by limbs with motions of general differential manifolds, it is difficult to find the influence of a limb arrangement. On this regard, this paper provides an in-depth investigation by focusing on motion analysis of parallel mechanisms consisting of three PRS limbs with planar-spherical manifolds. By formulating and comparing the motion characteristics of these mechanisms which though could be modeled difficultly by a mathematical tool without using the finite displacement screw, a finite displacement screw-based group method is proposed in this paper to resolve the raised issue. The novel approach presents results in expressions of new types of manifold motions, providing a foundation for design and application of parallel mechanisms having the various three-planar-spherical manifolds’ intersection motions.

Some geometric inequalities for varifolds on Riemannian manifolds based on monotonicity identities

Using Rauch’s comparison theorem, we prove several monotonicity inequalities for Riemannian submanifolds. Our main result is a general Li–Yau inequality which is applicable in any Riemannian manifold whose sectional curvature is bounded above (possibly positive). We show that the monotonicity inequalities can also be used to obtain Simon-type diameter bounds, Sobolev inequalities and corresponding isoperimetric inequalities for Riemannian submanifolds with small volume. Moreover, we infer lower diameter bounds for closed minimal submanifolds as corollaries. All the statements are intrinsic in the sense that no embedding of the ambient Riemannian manifold into Euclidean space is needed. Apart from Rauch’s comparison theorem, the proofs mainly rely on the first variation formula and thus are valid for general varifolds.

Building three-dimensional differentiable manifolds numerically

A method is developed here for building differentiable three-dimensional manifolds on multicube structures. This method constructs a sequence of reference metrics that determine differentiable structures on the cubic regions that serve as non-overlapping coordinate charts on these manifolds. It uses solutions to the two- and three-dimensional biharmonic equations in a sequence of steps that increase the differentiability of the reference metrics across the interfaces between cubic regions. This method is algorithmic and has been implemented in a computer code that automatically generates these reference metrics. Examples of three-manifolds constructed in this way are presented here, including representatives from five of the eight Thurston geometrization classes, plus the well-known Hantzsche-Wendt, the Poincaré dodecahedral space, and the Seifert-Weber space.

Isoparametric foliations and the Pompeiu property

<abstract><p>A bounded domain $ \Omega $ in a Riemannian manifold $ M $ is said to have the Pompeiu property if the only continuous function which integrates to zero on $ \Omega $ and on all its congruent images is the zero function. In some respects, the Pompeiu property can be viewed as an overdetermined problem, given its relation with the Schiffer problem. It is well-known that every Euclidean ball fails to have the Pompeiu property while spherical balls have the property for almost all radii (Ungar's Freak theorem). In the present paper we discuss the Pompeiu property when $ M $ is compact and admits an isoparametric foliation. In particular, we identify precise conditions on the spectrum of the Laplacian on $ M $ under which the level domains of an isoparametric function fail to have the Pompeiu property. Specific calculations are carried out when the ambient manifold is the round sphere, and some consequences are derived. Moreover, a detailed discussion of Ungar's Freak theorem and its generalizations is also carried out.</p></abstract>

Some recent results in plastic structure on Riemannian manifold

The plastic ratio is a fascinating topic that continually generates new ideas. The purpose of this paper is to point out and find some applications of the plastic ratio in the differential manifold. Precisely, we say that an $(1,1)$-tensor field $P$ on a $m$-dimensional Riemannian manifold $(M, g)$ is a plastic structure if it satisfies the equation $ P^3 = P + I $, where $ I $ is the identity. We establish several properties of the plastic structure. Then we show that a plastic structure induces on every invariant submanifold a plastic structure, too.

Geometrical approach on set theoretical solutions of Yang-Baxter equation in Lie algebras

In this paper, we handle set-theoretical solutions of Yang-Baxter equation and Lyubashenko set theoretical solutions in Lie algebras. We present a new commutative binary operation on these structures, and we obtain new set theoretical solutions including this operation by using property of commutativity of it. Also, we show that some set theoretical solutions of Yang-Baxter equation corresponds to the Lyubashenko set theoretical solutions on these structures. Additionally, we give some relations to verify set theoretical solution of Yang-Baxter equation. Moreover, we put an interpretation for these solutions from the point of geometrical view in Euclidean space, Minkowski space and differentiable manifolds by using Lie algebras.

Geodesic fields for Pontryagin type C0-Finsler manifolds

Let M be a differentiable manifold, TxM be its tangent space at x ∈ M and TM = {(x, y);x ∈ M;y ∈ TxM} be its tangent bundle. A C0-Finsler structure is a continuous function F : TM → [0, ∞) such that F(x, ⋅) : TxM → [0, ∞) is an asymmetric norm. In this work we introduce the Pontryagin type C0-Finsler structures, which are structures that satisfy the minimum requirements of Pontryagin’s maximum principle for the problem of minimizing paths. We define the extended geodesic field ℰ on the slit cotangent bundle T*M\0 of (M, F), which is a generalization of the geodesic spray of Finsler geometry. We study the case where ℰ is a locally Lipschitz vector field. We show some examples where the geodesics are more naturally represented by ℰ than by a similar structure on TM. Finally we show that the maximum of independent Finsler structures is a Pontryagin type C0-Finsler structure where ℰ is a locally Lipschitz vector field.

Almost Bronze Structures on Differentiable Manifolds

This study introduces a novel structure that is not included in the metallic structure family. This new structure, which is called an almost bronze structure, has been defined using a (1,1) type tensor field φ which fulfills the requirement φ2 = mφ − Id on a differentiable manifold. We investigated the parallelism and integrability conditions of these almost bronze structures by use of an almost product structure corresponding to them. Also, we have defined an almost bronze Riemannian manifold.

Euler‐Lagrange Equation in Free Coordinates

In this paper, we introduce different equivalent formulations of variational principle. The language of differential forms and manifold has been utilized to deduce Euler–Lagrange equations in free coordinates. Thus, the expression is simple and global.

Global Strichartz estimates for the Dirac equation on symmetric spaces

Abstract In this paper, we study global-in-time, weighted Strichartz estimates for the Dirac equation on warped product spaces in dimension $n\geq 3$ . In particular, we prove estimates for the dynamics restricted to eigenspaces of the Dirac operator on the compact spin manifolds defining the ambient manifold under some explicit sufficient condition on the metric and estimates with loss of angular derivatives for general initial data in the setting of spherically symmetric and asymptotically flat manifolds.

Non-unimodular transversely homogeneous foliations

We give sufficient conditions for the tautness of a transversely homogenous foliation defined on a compact manifold, by computing its base-like cohomology. As an application, we prove that if the foliation is non-unimodular then either the ambient manifold, the closure of the leaves or the total space of an associated principal bundle fiber over $S^1$.