Finite Dimensional Riemannian Manifold Research Articles

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.

Poincaré inequality meets Brezis–Van Schaftingen–Yung formula on metric measure spaces

Let (X,ρ,μ) be a metric measure space of homogeneous type, and let Cc⁎(X) denote the set of all Lipschitz functions f on X such that limr→0+⁡supy∈B(⋅,r)⁡|f(⋅)−f(y)|/r converges uniformly to a compactly supported continuous function Lip f on X. Let p∈[1,∞) and let f∈Cc⁎(X) be such that the pair (f,Lipf) satisfies a certain Poincaré inequality. Assume that α∈R∖{0} if p∈(1,∞), and α∈(0,∞) if p=1. Under these assumptions, the authors prove the following result which extends a recent formula of H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung on finite dimensional Euclidean spaces:supλ∈(0,∞)⁡λp∬Dλ[U(x,y)]α−1dμ(x)dμ(y)∼∫X[Lipf(x)]pdμ(x), whereDλ:={(x,y)∈X×X:|f(x)−f(y)|>λρ(x,y)[U(x,y)]αp} for any λ∈(0,∞), V(x,y):=μ(B(x,ρ(x,y))) and U(x,y):=min⁡{V(x,y),V(y,x)} for any x,y∈X, and the positive constants of equivalence are independent of f. A similar result remains true if p∈(1,∞) and f lies in a wider class of functions in the Hajłasz–Sobolev space. As an application, the authors establish new fractional Sobolev and Gagliardo–Nirenberg inequalities on X. These results are applicable to many classical examples, such as Euclidean spaces, with weighted Lebesgue measure, and complete finite dimensional Riemannian manifolds, with non-negative Ricci curvature.

Probabilistic morphisms and Bayesian nonparametrics

In this paper we develop a functorial language of probabilistic morphisms and apply it to some basic problems in Bayesian nonparametrics. First we extend and unify the Kleisli category of probabilistic morphisms proposed by Lawvere and Giry with the category of statistical models proposed by Chentsov and Morse-Sacksteder. Then we introduce the notion of a Bayesian statistical model that formalizes the notion of a parameter space with a given prior distribution in Bayesian statistics. {We revisit the existence of a posterior distribution, using probabilistic morphisms}. In particular, we give an explicit formula for posterior distributions of the Bayesian statistical model, assuming that the underlying parameter space is a Souslin space and the sample space is a subset in a complete connected finite dimensional Riemannian manifold. Then we give a new proof of the existence of Dirichlet measures over any measurable space using a functorial property of the Dirichlet map constructed by Sethuraman.

Linear instability for periodic orbits of non-autonomous Lagrangian systems

Inspired by the classical Poincaré criterion about the instability of orientation preserving minimizing closed geodesics on surfaces, we investigate the relation intertwining the instability and the variational properties of periodic solutions of a non-autonomous Lagrangian on a finite dimensional Riemannian manifold. We establish a general criterion for a priori detecting the linear instability of a periodic orbit on a Riemannian manifold for a (maybe not Legendre convex) non-autonomous Lagrangian simply by looking at the parity of the spectral index, which is the right substitute of the Morse index in the framework of strongly indefinite variational problems and defined in terms of the spectral flow of a path of Fredholm quadratic forms on a Hilbert bundle.

Austere and arid properties for PF submanifolds in Hilbert spaces

Austere submanifolds and arid submanifolds constitute respectively two different classes of minimal submanifolds in finite dimensional Riemannian manifolds. In this paper we introduce the concepts of these submanifolds into a class of proper Fredholm (PF) submanifolds in Hilbert spaces, discuss their relation and show examples of infinite dimensional austere PF submanifolds and arid PF submanifolds in Hilbert spaces. We also mention a classification problem of minimal orbits in hyperpolar PF actions on Hilbert spaces.

Energy-preserving methods on Riemannian manifolds

The energy preserving discrete gradient methods are generalized to finite-dimensional Riemannian manifolds by definition of a discrete approximation to the Riemannian gradient, a retraction, and a coordinate center function. The resulting schemes are intrinsic and do not depend on a particular choice of coordinates, nor on embedding of the manifold in a Euclidean space. Generalizations of well-known discrete gradient methods, such as the average vector field method and the Itoh--Abe method are obtained. It is shown how methods of higher order can be constructed via a collocation-like approach. Local and global error bounds are derived in terms of the Riemannian distance function and the Levi-Civita connection. Some numerical results on spin system problems are presented.

Riemannian manifolds are KKM spaces

Let (M,g) be a complete, finite-dimensional Riemannian manifold. Based on the fact that any geodesic convex subset of M is a KKM space, we establish the KKM theory on such subsets originated from the Knaster-Kuratowski-Mazurkiewitz theorem in 1929.

The Poincaré variational principle in the Lagrange–Poincaré reduction of mechanical systems with symmetry

The local Lagrange–Poincaré equations (the reduced Euler–Lagrange equations) for the mechanical system describing the motion of a scalar particle on a finite-dimensional Riemannian manifold with a given free isometric smooth action of a compact semi-simple Lie group are obtained. The equations are written in terms of dependent coordinates which are used to represent the local dynamic given on the orbit space of the principal fiber bundle. The derivation of the equations is performed with the help of the variational principle developed by Poincaré for mechanical systems with symmetry.

A Kernel Regression Procedure in the 3D Shape Space with an Application to Online Sales of Children’s Wear

This paper is focused on kernel regression when the response variable is the shape of a 3D object represented by a configuration matrix of landmarks. Regression methods on this shape space are not trivial because this space has a complex finite-dimensional Riemannian manifold structure (non-Euclidean). Papers about it are scarce in the literature, the majority of them are restricted to the case of a single explanatory variable, and many of them are based on the approximated tangent space. In this paper, there are several methodological innovations. The first one is the adaptation of the general method for kernel regression analysis in manifold-valued data to the three-dimensional case of Kendall’s shape space. The second one is its generalization to the multivariate case and the addressing of the curse-of-dimensionality problem. Finally, we propose bootstrap confidence intervals for prediction. A simulation study is carried out to check the goodness of the procedure, and a comparison with a current approach is performed. Then, it is applied to a 3D database obtained from an anthropometric survey of the Spanish child population with a potential application to online sales of children’s wear.

Dissipative Numerical Schemes on Riemannian Manifolds with Applications to Gradient Flows

This paper concerns an extension of discrete gradient methods to finite-dimensional Riemannian manifolds termed discrete Riemannian gradients, and their application to dissipative ordinary differen...

Viscosity solutions of second order partial differential equations with boundary conditions on Riemannian manifolds

The comparison, uniqueness and existence for viscosity solution to a class of fully nonlinear second order partial differential equations with boundary condition on a finite dimensional Riemannian manifold are studied. Some results regarding the regularity of the boundary are obtained. These results show that every compact manifold with C1,1 boundary satisfies the regularity condition.

Entropy, volume growth and SRB measures for Banach space mappings

We consider $$C^2$$ Frechet differentiable mappings of Banach spaces leaving invariant compactly supported Borel probability measures, and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces. SRB measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds, equivalently the sum of positive Lyapunov exponents of the map. In addition to numerous difficulties incurred by our infinite-dimensional setting, a crucial aspect to the proof is the technical point that the volume elements induced on unstable manifolds are regular enough to permit distortion control of iterated determinant functions. The results here generalize previously known results for diffeomorphisms of finite dimensional Riemannian manifolds, and are applicable to dynamical systems defined by large classes of dissipative parabolic PDEs.

Numerical treatment of a geometrically nonlinear planar Cosserat shell model

We present a new way to discretize a geometrically nonlinear elastic planar Cosserat shell. The kinematical model is similar to the general six-parameter resultant shell model with drilling rotations. The discretization uses geodesic finite elements (GFEs), which leads to an objective discrete model which naturally allows arbitrarily large rotations. GFEs of any approximation order can be constructed. The resulting algebraic problem is a minimization problem posed on a nonlinear finite-dimensional Riemannian manifold. We solve this problem using a Riemannian trust-region method, which is a generalization of Newton's method that converges globally without intermediate loading steps. We present the continuous model and the discretization, discuss the properties of the discrete model, and show several numerical examples, including wrinkling of thin elastic sheets in shear.

On Entropy of Graph Maps That Give Hereditarily Indecomposable Inverse Limits

We prove that if f:Grightarrow G is a map on a topological graph G such that the inverse limit varprojlim (G,f) is hereditarily indecomposable, and entropy of f is positive, then there exists an entropy set with infinite topological entropy. When G is the circle and the degree of f is positive then the entropy is always infinite and the rotation set of f is nondegenerate. This shows that the Anosov-Katok type constructions of the pseudo-circle as a minimal set in volume-preserving smooth dynamical systems, or in complex dynamics, obtained previously by Handel, Herman and Chéritat cannot be modeled on inverse limits. This also extends a previous result of Mouron who proved that if G=[0,1], then h(f)in {0,infty }, and combined with a result of Ito shows that certain dynamical systems on compact finite-dimensional Riemannian manifolds must either have zero entropy on their invariant sets or be non-differentiable.

The $$k$$ k -means algorithm for 3D shapes with an application to apparel design

Clustering of objects according to shapes is of key importance in many scientific fields. In this paper we focus on the case where the shape of an object is represented by a configuration matrix of landmarks. It is well known that this shape space has a finite-dimensional Riemannian manifold structure (non-Euclidean) which makes it difficult to work with. Papers about clustering on this space are scarce in the literature. The basic foundation of the $$k$$k-means algorithm is the fact that the sample mean is the value that minimizes the Euclidean distance from each point to the centroid of the cluster to which it belongs, so, our idea is integrating the Procrustes type distances and Procrustes mean into the $$k$$k-means algorithm to adapt it to the shape analysis context. As far as we know, there have been just two attempts in that way. In this paper we propose to adapt the classical $$k$$k-means Lloyd algorithm to the context of Shape Analysis, focusing on the three dimensional case. We present a study comparing its performance with the Hartigan-Wong $$k$$k-means algorithm, one that was previously adapted to the field of Statistical Shape Analysis. We demonstrate the better performance of the Lloyd version and, finally, we propose to add a trimmed procedure. We apply both to a 3D database obtained from an anthropometric survey of the Spanish female population conducted in this country in 2006. The algorithms presented in this paper are available in the Anthropometry R package, whose most current version is always available from the Comprehensive R Archive Network.

On the Calculus of Limiting Subjets on Riemannian Manifolds

In this paper fuzzy calculus rules for subjets of order two on finite dimensional Riemannian manifolds are obtained. Then a second order singular subjet derived from a sequence of efficient subsets of symmetric matrices is introduced. Employing fuzzy calculus rules for subjets of order two and various qualification assumptions based on a second order singular subjet, calculus rules for limiting subjets on a finite dimensional Riemannian manifold are obtianed.

Convergence of variational regularization methods for imaging on Riemannian manifolds

We consider abstract operator equations Fu = y, where F is a compact linear operator between Hilbert spaces U and V, which are function spaces on closed, finite-dimensional Riemannian manifolds, respectively. This setting is of interest in numerous applications such as computer vision and non-destructive evaluation. In this work, we study the approximation of the solution of the ill-posed operator equation with Tikhonov-type regularization methods. We state well-posedness, stability, convergence and convergence rates of the regularization methods. Moreover, we study in detail the numerical analysis and the numerical implementation. Finally, we provide for three different inverse problems numerical experiments.

Weak Sharp Minima on Riemannian Manifolds

This is the first paper dealing with the study of weak sharp minima for constrained optimization problems on Riemannian manifolds, which are important in many applications. We consider the notions of local weak sharp minima, boundedly weak sharp minima, and global weak sharp minima for such problems and establish their complete characterizations in the case of convex problems on finite-dimensional Riemannian manifolds and Hadamard manifolds. A number of the results obtained in this paper are also new for the case of conventional problems in finite-dimensional Euclidean spaces. Our methods involve appropriate tools of variational analysis and generalized differentiation on Riemannian and Hadamard manifolds developed and efficiently implemented in this paper.

On Restricting Planar Curve Evolution to Finite Dimensional Implicit Subspaces with Non-Euclidean Metric

This paper deals with restricting curve evolution to a finite and not necessarily flat space of curves, obtained as a subspace of the infinite dimensional space of planar curves endowed with the usual but weak parametrization invariant curve L 2-metric. We first show how to solve differential equations on a finite dimensional Riemannian manifold defined implicitly as a submanifold of a parameterized one, which in turn may be a Riemannian submanifold of an infinite dimensional one, using some optimal control techniques. We give an elementary example of the technique on a spherical submanifold of a 3-sphere and then a series of examples on a highly non-linear subspace of the space of closed spline curves, where we have restricted mean curvature motion, Geodesic Active contours and compute geodesic between two curves.

Computing Fenchel-Nielsen Coordinates in Teichmüller Shape Space

Teichmuller shape space is a finite dimensional Riemannian manifold, where each point represents a class of surfaces, which are conformally equivalent, and a path represents a deformation process from one shape to the other. Two surfaces in the real world correspond to the same point in the Teichmuller space, only if they can be conformally mapped to each other. Teichmuller shape space can be used for surface classification purpose in shape modeling. This work focuses on the computation of the coordinates of high genus surfaces in the Teichmuller space. The coordinates are called as Fenchel-Nielsen coordinates. The main idea is to deform the surface conformally using surface Ricci flow, such that the Gaussian curvature is −1 everywhere. The surface is decomposed to several pairs of hyperbolic pants. Each pair of pants is a genus zero surface with three boundaries, equipped with hyperbolic metric. Furthermore, all the boundaries are geodesics. Each pair of hyperbolic pants can be uniquely described by the lengths of its boundaries. The way of gluing different pairs of pants can be represented by the twisting angles between two adjacent pairs of pants which share a common boundary. The algorithms are based on Teichmuller space theory in conformal geometry, and they utilize the discrete surface Ricci flow. Most computations are carried out using hyperbolic geometry. The method is automatic, rigorous and efficient. The Teichmuller shape space coordinates can be used for surface classification and indexing. Experimental results on surfaces acquired from real world showed the practical value of the method for geometric database indexing, shape comparison and classification.