General Riemannian Manifolds Research Articles

Interest rate models on Lie groups

This paper examines an alternative approach to interest rate modeling, in which the nonlinear and random behavior of interest rates is captured by a stochastic differential equation evolving on a curved state space. We consider as candidate state spaces the matrix Lie groups; these offer not only a rich geometric structure, but—unlike general Riemannian manifolds—also allow for diffusion processes to be constructed easily without invoking the machinery of stochastic calculus on manifolds. After formulating bilinear stochastic differential equations on general matrix Lie groups, we then consider interest rate models in which the short rate is defined as linear or quadratic functions of the state. Stochastic volatility is also augmented to these models in a way that respects the Riemannian manifold structure of symmetric positive-definite matrices. Methods for numerical integration, parameter identification, pricing, and other practical issues are addressed through examples.

Isoperimetric inequalities for wave fronts and a generalization of Menzin's conjecture for bicycle monodromy on surfaces of constant curvature

Abstract The classical isoperimetric inequality relates the lengths of curves to the areas that they bound. More specifically, we have that for a smooth, simple closed curve of length L bounding area A on a surface of constant curvature c, L 2 ≥ 4πA – cA 2 with equality holding only if the curve is a geodesic circle. We prove generalizations of the isoperimetric inequality for both spherical and hyperbolic wave fronts (i.e. piecewise smooth curves which may have cusps). We then discuss “bicycle curves” using the generalized isoperimetric inequalities. The euclidean model of a bicycle is a unit segment AB that can move so that it remains tangent to the trajectory of point A (the rear wheel is fixed on the bicycle frame), as discussed in [Finn, The College Mathematics Journal, 33, 2002], [Tabachnikov, Israel J. Math. 151: 1–28, 2006], and [Levi, Tabachnikov, Experiment. Math. 18: 173–186, 2009]. We extend this definition to a general Riemannian manifold, and concern ourselves in particular with bicycle curves in the hyperbolic plane H 2 and on the sphere S 2. We prove results along the lines of those in [Levi, Tabachnikov, Experiment. Math. 18: 173–186, 2009] and resolve both spherical and hyperbolic versions of Menzin's conjecture, which relates the area bounded by a curve to its associated monodromy map.

完备非紧黎曼流形上的射线的平行性 The Parallelism of the Rays in a Complete Noncompact Riemannian Manifold

将欧式空间平行射线的概念推广到一般完备非紧的黎曼流形上，证明了只有符号相反的Busemann函数的完备非紧的黎曼流形具有完全平行性质。部分地回答了Wu-Chen问题。 The paper generalizes the concept of the parallelism for the rays in Eucliden space to a general noncompact Riemannian manifold, proves that if the manifold has only two Busemann function with adverse signs, then the manifold is with complete parallel property. This conclusion answers Wu-Chen Problem partly.

Higher-order smoothing splines versus least squares problems on Riemannian manifolds

In this paper, we present a generalization of the classical least squares problem on Euclidean spaces, introduced by Lagrange, to more general Riemannian manifolds. Using the variational definition of Riemannian polynomials, we formulate a higher-order variational problem on a manifold equipped with a Riemannian metric, which depends on a smoothing parameter and gives rise to what we call smoothing geometric splines. These are curves with a certain degree of smoothness that best fit a given set of points at given instants of time and reduce to Riemannian polynomials when restricted to each subinterval. We show that the Riemannian mean of the given points is achieved as a limiting process of the above. Also, when the Riemannian manifold is an Euclidean space, our approach generates, in the limit, the unique polynomial curve which is the solution of the classical least squares problem. These results support our belief that the approach presented in this paper is the natural generalization of the classical least squares problem to Riemannian manifolds.

Nonparametric Regression between General Riemannian Manifolds

We study nonparametric regression between Riemannian manifolds based on regularized empirical risk minimization. Regularization functionals for mappings between manifolds should respect the geometry of input and output manifold and be independent of the chosen parametrization of the manifolds. We define and analyze the three most simple regularization functionals with these properties and present a rather general scheme for solving the resulting optimization problem. As application examples we discuss interpolation on the sphere, fingerprint processing, and correspondence computations between three-dimensional surfaces. We conclude with characterizing interesting and sometimes counterintuitive implications and new open problems that are specific to learning between Riemannian manifolds and are not encountered in multivariate regression in Euclidean space.

Geodesic finite elements for Cosserat rods

AbstractWe introduce geodesic finite elements as a new way to discretize the non‐linear configuration space of a geometrically exact Cosserat rod. These geodesic finite elements naturally generalize standard one‐dimensional finite elements to spaces of functions with values in a Riemannian manifold. For the special orthogonal group, our approach reproduces the interpolation formulas of Crisfield and Jelenić. Geodesic finite elements are conforming and lead to objective and path‐independent problem formulations. We introduce geodesic finite elements for general Riemannian manifolds, discuss the relationship between geodesic finite elements and coefficient vectors, and estimate the interpolation error. Then we use them to find static equilibria of hyperelastic Cosserat rods. Using the Riemannian trust‐region algorithm of Absil et al. we show numerically that the discretization error depends optimally on the mesh size. Copyright © 2009 John Wiley & Sons, Ltd.

Contact Geometry of Curves

Cartan's method of moving frames is briefly recalled in the context of immersed curves in the homogeneous space of a Lie group G. The contact geometry of curves in low dimensional equi-affine geometry is then made explicit. This delivers the complete set of invariant data which solves the G-equivalence problem via a straightforward procedure, and which is, in some sense a supplement to the equivariant method of Fels and Olver. Next, the contact geometry of curves in general Riemannian manifolds (M,g) is described. For the special case in which the isometries of (M,g) act transitively, it is shown that the contact geometry provides an explicit algorithmic construction of the differential invariants for curves in M. The inputs required for the construction consist only of the metric g and a parametrisation of structure group SO(n); the group action is not required and no integration is involved. To illustrate the algorithm we explicitly construct complete sets of differential invariants for curves in the Poincare half-space H 3 and in a family of constant curvature 3-metrics. It is conjectured that similar results are possible in other Cartan geometries.

Some results about the existence of critical points for the Willmore functional

Using a perturbative approach, it is shown existence and multiplicity of critical points for the Willmore functional in ambient manifold \({(\mathbb{R}^3, g_\epsilon)}\) —where \({g_\epsilon}\) is a metric close and asymptotic to the Euclidean one. With the same technique it is proved a non-existence result in general Riemannian manifold (M, g) of dimension three.

Remarks on the Cwikel–Lieb–Rozenblum and Lieb–Thirring Estimates for Schrödinger Operators on Riemannian Manifolds

Let M be a general complete Riemannian manifold and consider a Schrodinger operator −Δ+V on L2(M). We prove Cwikel–Lieb–Rozenblum as well as Lieb–Thirring type estimates for −Δ+V. These estimates are given in terms of the potential and the heat kernel of the Laplacian on the manifold. Some of our results hold also for Schrodinger operators with complex-valued potentials.

Log-Sobolev inequalities with potential functions on pinned path groups

We establish a refined version of Gross's log-Sobolev inequalities on pinned path groups. We explain the reason why it is useful in a lower bound estimate of Schrodinger operators on path spaces. semi-group and the equivalent notion of the validity of the log-Sobolev inequality are used to give a lower bound on the bottom of spectrum of a Schrodinger operator which is given by the sum of the generator of the semi-group and a potential function. We ap- plied this lower bound estimate to study the semi-classical behavior of the bottom of spectrum of Schrodinger operators on path spaces over compact Riemannian manifolds ((3, 4)) partly motivated by an application to P(`)-type Hamiltonian and an extension of (19) to infinite dimensional curved spaces. In this paper, we establish a refined version of Gross's log-Sobolev inequalities on a pinned path group. Pinned path group Pe,a(G) is a space of continuous paths with values in a compact Lie group G over the time interval (0,1) with a fixed starting point e (unit element) and the fixed end point a. We will apply the log- Sobolev inequalities with potential functions to study the semi-classical behavior of the low lying spectrum of Schrodinger operators on pinned path groups in a separate paper. The structure of the paper is as follows. In Section 2, we consider smooth pinned path spaces over a general compact Riemannian manifold M. We intro- duce a Riemannian structure on the pinned path space using a metric connection on M. Next we calculate the gradient of the energy function E(∞) = 1 R 1 0 |˙ ∞(t)| 2 dt. This calculation and a formal argument show that an LSI with a special potential function may be useful for the study of semi-classical behavior of low lying spec- trum of Schrodinger operators over a pinned path space. This kind of log-Sobolev inequality with special potential function already appeared in (13, 10). In Section 3, we consider a pinned path group and introduce an H-derivative on Pe,a(G) and the Dirichlet form in L 2 -space with respect to the (scaled) pinned Brownian motion measure with scaling (semi-classical) parameter ‚. The H-derivative is considered as the gradient operator on the path space which is defined by the right invariant

What is Applied Harmonic Analysis?

The purpose of this paper is to serve as an introduction into the new field of Applied Harmonic Analysis, which is nowadays already one of the major research area in Applied Mathematics. 1. Data, Data, Data,... Today we are living in a data-drenched world, in which we are challenged to not only provide the methodology to process various different types of data, but – especially as mathematicians – to also analyze the accuracy of such methods and to provide a deeper understanding of the underlying structures. There is a pressing need for those tasks coming from various fields as diverse as air traffic control, digital communications, seismology, medical imaging, and cosmology. As diverse as those fields are the characteristics of the data themselves, where data are usually modeled as functions f : X → Y or just collections of points in X. Here X can, for instance, be Z or R for arbitrarily large n, a compact subset Ω ⊂ R, or a general Riemannian manifold, and Y can be similarly diverse concerning its mathematical structure. Let us take a quick look at some intriguing examples of such modern data. • High-dimensional data can be found in various applications where the most prominent one might be the internet. Here usually the task of search engines is to organize webpages in the widest sense. One common model for such data is to regard a collection of attributes (characterizing words, etc.) for each single webpage as one point in R usually with n > 10.000. • New technology also generates new types of data not encountered before, such as manifold-valued data. Here we would like to mention an example coming from air plane control, where a time series of aircraft orientations (pitch, roll, yaw) can be modeled as ranging over SO(3). • Certainly, also ‘common’ types of data such as 2-D images appear in new fashions, for instance, astronomical images from galaxies, medical images from MRI machines, surveillance images from military aircrafts, and so forth, each having its own specifics in, for instance, how the data are measured and what their main features are. The task which we now face is manifold, starting from how to measure data in the most efficient way, especially where time is a factor such as when collecting MRI data. Since data are never ‘clean’, usually the next task is to denoise the data, which requires a suitable model for the noise. If data are missing, we face the task of inpainting, which certainly calls The author would like to thank the Department of Statistics at Stanford University and the Department of Mathematics at Yale University for their hospitality and support during her visits. She acknowledges support by Deutsche Forschungsgemeinschaft (DFG) Heisenberg-Fellowship KU 1446/8-1.

Globally Convergent Optimization Algorithms on Riemannian Manifolds: Uniform Framework for Unconstrained and Constrained Optimization

This paper proposes several globally convergent geometric optimization algorithms on Riemannian manifolds, which extend some existing geometric optimization techniques. Since any set of smooth constraints in the Euclidean space Rn (corresponding to constrained optimization) and the Rn space itself (corresponding to unconstrained optimization) are both special Riemannian manifolds, and since these algorithms are developed on general Riemannian manifolds, the techniques discussed in this paper provide a uniform framework for constrained and unconstrained optimization problems. Unlike some earlier works, the new algorithms have less restrictions in both convergence results and in practice. For example, global minimization in the one-dimensional search is not required. All the algorithms addressed in this paper are globally convergent. For some special Riemannian manifold other than Rn, the new algorithms are very efficient. Convergence rates are obtained. Applications are discussed.

Curve shortening in a Riemannian manifold

In this paper, we study the curve shortening flow in a general Riemannian manifold. We have many results for the global behavior of the flow. In particular, we show the following results: let M be a compact Riemannian manifold. (1) If the curve shortening flow exists for infinite time, and \(\lim_{t\rightarrow\infty}L(\gamma_{t})>0\), then for every n > 0, \(\lim_{t\rightarrow\infty}\sup\left(\left|\frac{D^{n}T}{\partial s^{n}}\right|\right)=0\). Furthermore, the limiting curve exists and is a closed geodesic in M. (2) In M × S1, if γ0 is a ramp, then we have a global flow which converges to a closed geodesic in C∞ norm. As an application, we prove the theorem of Lyusternik and Fet.

Hidden Variables as Computational Tools: The Construction of a Relativistic Spinor Field

Hidden variables are usually presented as potential completions of the quantum description. We describe an alternative role for these entities, as aids to calculation in quantum mechanics. This is illustrated by the computation of the time-dependence of a massless relativistic spinor field obeying Weyl’s equation from a single-valued continuum of deterministic trajectories (the “hidden variables”). This is achieved by generalizing the exact method of state construction proposed previously for spin 0 systems to a general Riemannian manifold from which the spinor construction is extracted as a special case. The trajectories form a non-covariant structure and the Lorentz covariance of the spinor field theory emerges as a kind of collective effect. The method makes manifest the spin 1/2 analogue of the quantum potential that is tacit in Weyl’s equation. This implies a novel definition of the “classical limit” of Weyl’s equation.

Hydrodynamic construction of the electromagnetic field

We present an alternative Eulerian hydrodynamic model for the electromagnetic field in which the discrete vector indices in Maxwell's equations are replaced by continuous angular freedoms, and develop the corresponding Lagrangian picture in which the fluid particles have rotational and translational freedoms. This enables us to extend to the electromagnetic field the exact method of state construction proposed previously for spin 0 systems, in which the time-dependent wavefunction is computed from a single-valued continuum of deterministic trajectories where two spacetime points are linked by at most a single orbit. The deduction of Maxwell's equations from continuum mechanics is achieved by generalizing the spin 0 theory to a general Riemannian manifold from which the electromagnetic construction is extracted as a special case. In particular, the flat-space Maxwell equations are represented as a curved-space Schrödinger equation for a massive system. The Lorentz covariance of the Eulerian field theory is obtained from the non-covariant Lagrangian-coordinate model as a kind of collective effect. The method makes manifest the electromagnetic analogue of the quantum potential that is tacit in Maxwell's equations. This implies a novel definition of the ‘classical limit’ of Maxwell's equations that differs from geometrical optics. It is shown that Maxwell's equations may be obtained by canonical quantization of the classical model. Using the classical trajectories a novel expression is derived for the propagator of the electromagnetic field in the Eulerian picture. The trajectory and propagator methods of solution are illustrated for the case of a light wave.

Discrete quantum gravity in the Regge calculus formalism

An approach to the discrete quantum gravity based on the Regge calculus is discussed which was developed in a number of our papers. Regge calculus is general relativity for the subclass of general Riemannian manifolds called piecewise flat ones. Regge calculus deals with the discrete set of variables, triangulation lengths, and contains continuous general relativity as a particular limiting case when the lengths tend to zero. In our approach the quantum length expectations are nonzero and of the order of Plank scale $10^{-33}cm$. This means the discrete spacetime structure on these scales.

Cartan–Weyl Dirac and Laplacian Operators, Brownian Motions: The Quantum Potential and Scalar Curvature, Maxwell’s and Dirac-Hestenes Equations, and Supersymmetric Systems

We present the Dirac and Laplacian operators on Clifford bundles over space–time, associated to metric compatible linear connections of Cartan–Weyl, with trace-torsion, Q. In the case of nondegenerate metrics, we obtain a theory of generalized Brownian motions whose drift is the metric conjugate of Q. We give the constitutive equations for Q. We find that it contains Maxwell’s equations, characterized by two potentials, an harmonic one which has a zero field (Bohm-Aharonov potential) and a coexact term that generalizes the Hertz potential of Maxwell’s equations in Minkowski space.We develop the theory of the Hertz potential for a general Riemannian manifold. We study the invariant state for the theory, and determine the decomposition of Q in this state which has an invariant Born measure. In addition to the logarithmic potential derivative term, we have the previous Maxwellian potentials normalized by the invariant density. We characterize the time-evolution irreversibility of the Brownian motions generated by the Cartan–Weyl laplacians, in terms of these normalized Maxwell’s potentials. We prove the equivalence of the sourceless Maxwell equation on Minkowski space, and the Dirac-Hestenes equation for a Dirac-Hestenes spinor field written on Minkowski space provided with a Cartan–Weyl connection. If Q is characterized by the invariant state of the diffusion process generated on Euclidean space, then the Maxwell’s potentials appearing in Q can be seen alternatively as derived from the internal rotational degrees of freedom of the Dirac-Hestenes spinor field, yet the equivalence between Maxwell’s equation and Dirac-Hestenes equations is valid if we have that these potentials have only two components corresponding to the spin-plane. We present Lorentz-invariant diffusion representations for the Cartan–Weyl connections that sustain the equivalence of these equations, and furthermore, the diffusion of differential forms along these Brownian motions. We prove that the construction of the relativistic Brownian motion theory for the flat Minkowski metric, follows from the choices of the degenerate Clifford structure and the Oron and Horwitz relativistic Gaussian, instead of the Euclidean structure and the orthogonal invariant Gaussian. We further indicate the random Poincare–Cartan invariants of phase-space provided with the canonical symplectic structure. We introduce the energy-form of the exact terms of Q and derive the relativistic quantum potential from the groundstate representation. We derive the field equations corresponding to these exact terms from an average on the invariant state Cartan scalar curvature, and find that the quantum potential can be identified with 1 / 12R(g), where R(g) is the metric scalar curvature. We establish a link between an anisotropic noise tensor and the genesis of a gravitational field in terms of the generalized Brownian motions. Thus, when we have a nontrivial curvature, we can identify the quantum nonlocal correlations with the gravitational field. We discuss the relations of this work with the heat kernel approach in quantum gravity. We finally present for the case of Q restricted to this exact term a supersymmetric system, in the classical sense due to E.Witten, and discuss the possible extensions to include the electromagnetic potential terms of Q

Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory

This paper is the first in a series that lays the groundwork for a structure and classification theory of second-order superintegrable systems, both classical and quantum, in conformally flat spaces. Many examples of such systems are known, and lists of possible systems have been determined for constant curvature spaces in two and three dimensions, as well as few other spaces. Observed features of these systems are multiseparability, closure of the quadratic algebra of second-order symmetries at order 6, use of representation theory of the quadratic algebra to derive spectral properties of the quantum Schrödinger operator, and a close relationship with exactly solvable and quasi-exactly solvable systems. Our approach is, rather than focus on particular spaces and systems, to use a general theoretical method based on integrability conditions to derive structure common to all systems. In this first paper we consider classical superintegrable systems on a general two-dimensional Riemannian manifold and uncover their common structure. We show that for superintegrable systems with nondegenerate potentials there exists a standard structure based on the algebra of 2×2 symmetric matrices, that such systems are necessarily multiseparable and that the quadratic algebra closes at level 6. Superintegrable systems with degenerate potentials are also analyzed. This is all done without making use of lists of systems, so that generalization to higher dimensions, where relatively few examples are known, is much easier.

Symmetry and Bayesian Function Estimation1

Summary This paper develops Bayesian function estimation on compact Riemannian manifolds. The approach is to combine Bayesian methods along with aspects of spectral geometry associated with the Laplace-Beltrami operator on Riemannian manifolds. Although frequentist nonparametric function estimation in Euclidean space abound, to date, no attempt has been made with respect to Bayesian function estimation on a general Riemannian manifold. The Bayesian approach to function estimation is very natural for manifolds because one can elicit very specific prior information on the possible symmetries in question . One can then establish Bayes estimators that possess built in symmetries. A detailed analysis for the 2–sphere is provided.

The Dirichlet-to-Neumann map, viscosity solutions to Eikonal equations, and the self-dual equations of pattern formation

We study the limiting behavior as ε ↘ 0 of solutions u ε to the Dirichlet problem ε 2 Δ u ε − u ε = 0 on Ω , u ε | ∂ Ω = e − θ / ε , where Ω ¯ is a bounded domain and θ a given smooth function on its boundary ∂ Ω . We provide a natural criterion on θ in order to obtain an estimate ε ∂ ν u ε ( x ) u ε ( x ) ≤ C < ∞ , x ∈ ∂ Ω , independent of ε as ε ↘ 0 , where ∂ ν u ε denotes the normal derivative of u ε . The results of this paper serve to significantly strengthen the analysis of asymptotic minimizers for a Ginzburg–Landau variational problem for irrotational vector fields (gradient vector fields) known as the regularized Cross–Newell variational problem in the pattern formation literature. In particular, this yields estimates on the asymptotic energy of these minimizers for general Dirichlet viscosity boundary conditions. The class of boundary conditions for this variational problem to which our methods apply is quite general (even including domains which are general Riemannian manifolds with boundary). This, for instance, provides a first step for extending the Ginzburg–Landau type model we consider to the larger class of vector fields that are locally gradient (often called director fields).