Differentiable Manifold Research Articles

On jets, almost symmetric tensors, and traction hyper-stresses

The paper considers the formulation of higher-order continuum mechanics on differentiable manifolds devoid of any metric or parallelism structure. For generalized velocities modeled as sections of some vector bundle, a variational kth order hyper-stress is an object that acts on jets of generalized velocities to produce power densities. The traction hyper-stress is introduced as an object that induces hyper-traction fields on the boundaries of subbodies. Additional aspects of multilinear algebra relevant to the analysis of these objects are reviewed.

Topology of the intersections of ellipsoids in $$\mathbf{R}^n$$ R n

We present an overview of work, including unpublished results, on the characterization and description of the topological type of the differentiable manifolds that are diffeomorphic to a transverse intersection of ellipsoids in euclidean space, subject to certain conditions regarding the type and the number of them.

Model order reduction for dynamical systems: A geometric approach

The aim of this paper is to ask the question as whether it is possible, for a given dynamical system defined by a vector field over a finite dimensional inner product space, to construct a reduced-order model over a finite dimensional manifold. In order to give a positive answer to this question, we prove that if the manifold under consideration is an immersed submanifold of the vector space, considered as ambient manifold, then it is possible to construct explicitly a reduced-order vector field over this submanifold. In particular, we found that the reduced-order vector field satisfies the variational principle of Dirac–Frenkel and that we can formulate the Proper Orthogonal Decomposition under this framework. Finally, we propose a local-point estimator of the time-dependent error between the original vector field and the reduced-order one.

Extended Riemannian geometry II: local heterotic double field theory

We continue our exploration of local Double Field Theory (DFT) in terms of symplectic graded manifolds carrying compatible derivations and study the case of heterotic DFT. We start by developing in detail the differential graded manifold that captures heterotic Generalized Geometry which leads to new observations on the generalized metric and its twists. We then give a symplectic pre-NQ-manifold that captures the symmetries and the geometry of local heterotic DFT. We derive a weakened form of the section condition, which arises algebraically from consistency of the symmetry Lie 2-algebra and its action on extended tensors. We also give appropriate notions of twists — which are required for global formulations — and of the torsion and Riemann tensors. Finally, we show how the observed α′-corrections are interpreted naturally in our framework.

Singularities of mean convex level set flow in general ambient manifolds

We prove two new estimates for the level set flow of mean convex domains in Riemannian manifolds. Our estimates give control – exponential in time – for the infimum of the mean curvature, and the ratio between the norm of the second fundamental form and the mean curvature. In particular, the estimates remove a stumbling block that has been left after the work of White [16,17,20], and Haslhofer–Kleiner [9], and thus allow us to extend the structure theory for mean convex level set flow to general ambient manifolds of arbitrary dimension.

On the Linearization Covering Technique and its Application to Integrable Nonlinear Differential Systems

In this letter I analyze a covering jet manifold scheme, its relation to the invariant theory of the associated vector fields, and applications to the Lax-Sato-type integrability of nonlinear dispersionless differential systems. The related contact geometry linearization covering scheme is also discussed. The devised techniques are demonstrated for such nonlinear Lax-Sato integrable equations as Gibbons-Tsarev, ABC, Manakov-Santini and the differential Toda singular manifold equations.

Curvature Invariants for Statistical Submanifolds of Hessian Manifolds of Constant Hessian Curvature

We consider statistical submanifolds of Hessian manifolds of constant Hessian curvature. For such submanifolds we establish a Euler inequality and a Chen-Ricci inequality with respect to a sectional curvature of the ambient Hessian manifold.

Orbit Spaces Arising from Isometric Actions on Hyperbolic Spaces

Let be a differentiable action of a Lie group on a differentiable manifold and consider the orbit space with the quotient topology. Dimension of is called the cohomogeneity of the action of on . If is a differentiable manifold of cohomogeneity one under the action of a compact and connected Lie group, then the orbit space is homeomorphic to one of the spaces , , or . In this paper we suppose that the hyperbolic space is of cohomogeneity two under the action of , a connected and closed subgroup of Then we prove that its orbit space is homeomorphic to or Also we prove that either all orbits are diffeomorphic to or there are nonnegative integers such that some orbits are diffeomorphic to , and the other orbits are diffeomorphic to , where may be a sphere, a homogeneous hypersurface of sphere or a helix in some Euclidean space. ../files/site1/files/0Abstract6.pdf

Comparing the Morse index and the first Betti number of minimal hypersurfaces

By extending and generalizing previous work by Ros and Savo, we describe a method to show that in certain positively curved ambient manifolds the Morse index of every closed minimal hypersurface is bounded from below by a linear function of its first Betti number. The technique is flexible enough to prove that such a relation between the index and the topology of minimal hypersurfaces holds, for example, on all compact rank one symmetric spaces, on products of the circle with spheres of arbitrary dimension and on suitably pinched submanifolds of the Euclidean spaces. These results confirm a general conjecture due to Schoen and Marques–Neves for a wide class of ambient spaces.

Nonparametric Analysis of Non-Euclidean Data on Shapes and Images

The article presents some of the basic theory for nonparametric inference on non-Euclidean spaces using Frechet means that has been developed during the past two decades. Included are recent results on the asymptotic distribution theory of sample Frechet means on such spaces, especially differentiable and Riemannian manifolds. Apart from this main theme and its applications, a nonparametric Bayes theory on Riemannian manifolds is outlined for the purpose of density estimation and classification. A final section briefly discusses the problem of machine vision, or robotic recognition of images as Riemannian manifolds.

Calculation of the Behavior Utility of a Network System: Conception and Principle

Abstract The service and application of a network is a behavioral process that is oriented toward its operations and tasks, whose metrics and evaluation are still somewhat of a rough comparison. This paper describes scenes of network behavior as differential manifolds. Using the homeomorphic transformation of smooth differential manifolds, we provide a mathematical definition of network behavior and propose a mathematical description of the network behavior path and behavior utility. Based on the principle of differential geometry, this paper puts forward the function of network behavior and a calculation method to determine behavior utility, and establishes the calculation principle of network behavior utility. We also provide a calculation framework for assessment of the network’s attack-defense confrontation on the strength of behavior utility. Therefore, this paper establishes a mathematical foundation for the objective measurement and precise evaluation of network behavior.

On metric connections with torsion on the cotangent bundle with modified Riemannian extension

Let M be an n-dimensional differentiable manifold equipped with a torsion-free linear connection $$\nabla $$ and $$T^{*}M$$ its cotangent bundle. The present paper aims to study a metric connection $$\widetilde{ \nabla }$$ with nonvanishing torsion on $$T^{*}M$$ with modified Riemannian extension $${}\overline{g}_{\nabla ,c}$$ . First, we give a characterization of fibre-preserving projective vector fields on $$(T^{*}M,{}\overline{g} _{\nabla ,c})$$ with respect to the metric connection $$\widetilde{\nabla }$$ . Secondly, we study conditions for $$(T^{*}M,{}\overline{g}_{\nabla ,c})$$ to be semi-symmetric, Ricci semi-symmetric, $$\widetilde{Z}$$ semi-symmetric or locally conharmonically flat with respect to the metric connection $$ \widetilde{\nabla }$$ . Finally, we present some results concerning the Schouten–Van Kampen connection associated to the Levi-Civita connection $$ \overline{\nabla }$$ of the modified Riemannian extension $$\overline{g} _{\nabla ,c}$$ .

On the homotopy classification of proper Fredholm maps into a Hilbert space

Abstract We classify the homotopy classes of proper Fredholm maps from an infinite-dimensional Hilbert manifold into its model space in terms of a suitable version of framed cobordism. Our construction is an alternative approach to the classification introduced by Elworthy and Tromba in 1970 and does not make use of further structures on the ambient manifold, such as Fredholm structures. In the special case of index zero, we obtain a complete classification involving the Caccioppoli–Smale mod 2 degree and the absolute value of the oriented degree.

Upper bounds for geodesic periods over rank one locally symmetric spaces

Abstract We prove upper bounds for geodesic periods of automorphic forms over general rank one locally symmetric spaces. Such periods are integrals of automorphic forms restricted to special totally geodesic cycles of the ambient manifold and twisted with automorphic forms on the cycles. The upper bounds are in terms of the Laplace eigenvalues of the two automorphic forms, and they generalize previous results for real hyperbolic manifolds to the context of all rank one locally symmetric spaces.

Geometric properties of rotation minimizing vector fields along curves in Riemannian manifolds

Rotation minimizing (RM) vector fields and frames were introduced by Bishop as an alternative to the Frenet frame. They are used in CAGD because they can be defined even when the curvature vanishes. Nevertheless, many other geometric properties have not been studied. In the present paper, RM vector fields along a curve immersed into a Riemannian manifold are studied when the ambient manifold is the Euclidean 3-space, the hyperbolic 3-space, and a Kahler manifold.

Fluid State of Dirac Quantum Particles

In our previous works, we suggest that quantum particles are composite physical objects endowed with the geometric and topological structures of their corresponding differentiable manifolds that would allow them to imitate and adapt to physical environments. In this work, we show that Dirac equation in fact describes quantum particles as composite structures that are in a fluid state in which the components of the wavefunction can be identified with the stream function and the velocity potential of a potential flow formulated in the theory of classical fluids. We also show that Dirac quantum particles can manifest as standing waves which are the result of the superposition of two fluid flows moving in opposite directions. However, for a steady motion a Dirac quantum particle does not exhibit a wave motion even though it has the potential to establish a wave within its physical structure, therefore, without an external disturbance a Dirac quantum particle may be considered as a classical particle defined in classical physics. And furthermore, from the fact that there are two identical fluid flows in opposite directions within their physical structures, the fluid state model of Dirac quantum particles can be used to explain why fermions are spin-half particles.

Optimization Methods on Riemannian Manifolds via Extremum Seeking Algorithms

This paper formulates the problem of Extremum Seeking for optimization of cost functions defined on Riemannian manifolds. We extend the conventional extremum seeking algorithms for optimization problems in Euclidean spaces to optimization of cost functions defined on smooth Riemannian manifolds. This problem falls within the category of online optimization methods. We introduce the notion of geodesic dithers which is a perturbation of the optimizing trajectory in the tangent bundle of the ambient state manifolds and obtain the extremum seeking closed loop as a perturbation of the averaged gradient system. The main results are obtained by applying closeness of solutions and averaging theory on Riemannian manifolds. The main results are further extended for optimization on Lie groups. Numerical examples on Riemannian manifolds (Lie groups) SO(3) and SE(3) are presented at the end of the paper.

Finite Difference Method for Modified Boussinesq Equation

In this paper a numerical of the solution of the Cauchy problem for the nonlinear modified Boussinesq equation (or IMBq equation) is studied. This equation with the boundary conditions models the propagation of waves in shallow water, taking into account capillary effects and the preservation of mass in the layer, filtration of water in the soil, as well as shock waves. In the case when the equation is nondegenerate, a global solution and a solution in the form of solitons are obtained. In the degenerate case, the existence of a unique local solution was proved by the methods of phase space and the theory of relatively limited ones developed by G. Sviridyuk and his students, as well as the theory of differentiable Banach manifolds. A numerical study of this problem by the modified Galerkin method has already been carried out earlier. However, the operation time of algorithms based on modified Galerkin method it rapidly increases with an increase amount of Galerkin sum. In this article, a numerical study is carried out by the finite difference method. The Cauchy -- Dirichlet problem for the IMBq equation is reduced to an implicit difference problem. A comparison is made of the speed of the modified Galerkin method and the finite difference method.

Upper bounds for geodesic periods over hyperbolic manifolds

We prove an upper bound for geodesic periods of Maass forms over hyperbolic manifolds. By definition, such periods are integrals of Maass forms restricted to a special geodesic cycle of the ambient manifold, against a Maass form on the cycle. Under certain restrictions, the bound will be uniform.

Topological classification of $Ω$-stable flows on surfaces by means of effectively distinguishable multigraphs

Structurally stable (rough) flows on surfaces have only finitely many singularities and finitely many closed orbits, all of which are hyperbolic, and they have no trajectories joining saddle points. The violation of the last property leads to \begin{document}$Ω$\end{document} -stable flows on surfaces, which are not structurally stable. However, in the present paper we prove that a topological classification of such flows is also reduced to a combinatorial problem. Our complete topological invariant is a multigraph, and we present a polynomial-time algorithm for the distinction of such graphs up to an isomorphism. We also present a graph criterion for orientability of the ambient manifold and a graph-associated formula for its Euler characteristic. Additionally, we give polynomial-time algorithms for checking the orientability and calculating the characteristic.