Riemannian Manifold (riemannian + manifold)

Distribution by Scientific Domains

Kinds of Riemannian Manifold

  • compact riemannian manifold


  • Selected Abstracts


    Estimation of the Euler method error on a Riemannian manifold

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 11 2002
    Andrzej Bielecki
    Abstract This article presents an estimation of the Euler method on a Riemannian manifold. A distance between the nth iteration of the cascade generated by the time- h map of a gradient flow and the nth iteration of the cascade generated by the Euler method of this flow is estimated. The application possibilities of the presented estimation are also discussed. Copyright © 2002 John Wiley & Sons, Ltd. [source]


    Geodesic finite elements for Cosserat rods

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 13 2010
    Oliver Sander
    Abstract We 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 Jeleni,. 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. [source]


    The steady states and convergence to equilibria for a 1-D chemotaxis model with volume-filling effect

    MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 1 2010
    Yanyan Zhang
    Abstract We consider a chemotaxis model with volume-filling effect introduced by Hillen and Painter. They also proved the existence of global solutions for a compact Riemannian manifold without boundary. Moreover, the existence of a global attractor in W1, p(,,,n), p>n, p,2, was proved by Wrzosek. He also proved that the ,-limit set consists of regular stationary solutions. In this paper, we prove that the 1-D stationary problem has at most an infinitely countable number of regular solutions. Furthermore, we show that as t,, the solution of the 1-D evolution problem converges to an equilibrium in W1, p, p,2. Copyright © 2009 John Wiley & Sons, Ltd. [source]


    Positivity and time behavior of a linear reaction,diffusion system, non-local in space and time

    MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2008
    Andrii Khrabustovskyi
    Abstract We consider a general linear reaction,diffusion system in three dimensions and time, containing diffusion (local interaction), jumps (nonlocal interaction) and memory effects. We prove a maximum principle and positivity of the solution and investigate its asymptotic behavior. Moreover, we give an explicit expression of the limit of the solution for large times. In order to obtain these results, we use the following method: We construct a Riemannian manifold with complicated microstructure depending on a small parameter. We study the asymptotic behavior of the solution to a simple diffusion equation on this manifold as the small parameter tends to zero. It turns out that the homogenized system coincides with the original reaction,diffusion system. Using this and the facts that the diffusion equation on manifolds satisfies the maximum principle and its solution converges to a easily calculated constant, we can obtain analogous properties for the original system. Copyright © 2008 John Wiley & Sons, Ltd. [source]


    The first Dirichlet eigenvalue of a compact manifold and the Yang conjecture

    MATHEMATISCHE NACHRICHTEN, Issue 12 2007
    Jun Ling
    Abstract We give a new estimate on the lower bound of the first Dirichlet eigenvalue of a compact Riemannian manifold with negative lower bound of Ricci curvature and provide a solution for a conjecture of H. C. Yang. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


    Area-minimizing projective planes in 3-manifolds

    COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 9 2010
    H. Bray
    Let (M,g) be a compact Riemannian manifold of dimension 3, and let , denote the collection of all embedded surfaces homeomorphic to \input amssym ${\Bbb R}{ \Bbb P}^2$. We study the infimum of the areas of all surfaces in ,. This quantity is related to the systole of (M,g). It makes sense whenever , is nonempty. In this paper, we give an upper bound for this quantity in terms of the minimum of the scalar curvature of (M,g). Moreover, we show that equality holds if and only if (M,g) is isometric to \input amssym ${\Bbb R}{ \Bbb P}^3$ up to scaling. The proof uses the formula for the second variation of area and Hamilton's Ricci flow. © 2010 Wiley Periodicals, Inc. [source]


    Density results relative to the Dirichlet energy of mappings into a manifold

    COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 12 2006
    Mariano Giaquinta
    Let ,, be a smooth, compact, oriented Riemannian manifold without boundary. Weak limits of graphs of smooth maps uk:Bn , ,, with an equibounded Dirichlet integral give rise to elements of the space cart2,1 (Bn × ,,). Assume that ,, is 1-connected and that its 2-homology group has no torsion. In any dimension n we prove that every element T in cart2,1 (Bn × ,,) with no singular vertical part can be approximated weakly in the sense of currents by a sequence of graphs of smooth maps uk:Bn , ,, with Dirichlet energies converging to the energy of T. © 2006 Wiley Periodicals, Inc. [source]


    Geodesic finite elements for Cosserat rods

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 13 2010
    Oliver Sander
    Abstract We 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 Jeleni,. 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. [source]


    Infima of universal energy functionals on homotopy classes

    MATHEMATISCHE NACHRICHTEN, Issue 15 2006
    Stefan Bechtluft-Sachs
    Abstract We consider the infima (f) on homotopy classes of energy functionals E defined on smooth maps f: Mn , Vk between compact connected Riemannian manifolds. If M contains a sub-manifold L of codimension greater than the degree of E then (f) is determined by the homotopy class of the restriction of f to M \ L. Conversely if the infimum on a homotopy class of a functional of at least conformal degree vanishes then the map is trivial in homology of high degrees. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]


    Proof of a decomposition theorem for symmetric tensors on spaces with constant curvature

    ANNALEN DER PHYSIK, Issue 8 2008
    N. Straumann
    Abstract In cosmological perturbation theory a first major step consists in the decomposition of the various perturbation amplitudes into scalar, vector and tensor perturbations, which mutually decouple. In performing this decomposition one uses , beside the Hodge decomposition for one-forms , an analogous decomposition of symmetric tensor fields of second rank on Riemannian manifolds with constant curvature. While the uniqueness of such a decomposition follows from Gauss' theorem, a rigorous existence proof is not obvious. In this note we establish this for smooth tensor fields, by making use of some important results for linear elliptic differential equations. [source]


    Heat flow on Finsler manifolds

    COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 10 2009
    Shin-ichi Ohta
    This paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F(x, ·) : TxM , ,+ on each tangent space. Mostly, we will require that this norm be strongly convex and smooth and that it depend smoothly on the base point x. The particular case of a Hilbert norm on each tangent space leads to the important subclasses of Riemannian manifolds where the heat flow is widely studied and well understood. We present two approaches to the heat flow on a Finsler manifold: as gradient flow on L2(M, m) for the energy as gradient flow on the reverse L2 -Wasserstein space ,,2(M) of probability measures on M for the relative entropy Both approaches depend on the choice of a measure m on M and then lead to the same nonlinear evolution semigroup. We prove ,,1, , regularity for solutions to the (nonlinear) heat equation on the Finsler space (M, F, m). Typically solutions to the heat equation will not be ,,2. Moreover, we derive pointwise comparison results à la Cheeger-Yau and integrated upper Gaussian estimates à la Davies. © 2008 Wiley Periodicals, Inc. [source]


    Nonstationary weak limit of a stationary harmonic map sequence

    COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 2 2003
    Weiyue Ding
    Let M and N be two compact Riemannian manifolds. Let uk be a sequence of stationary harmonic maps from M to N with bounded energies. We may assume that it converges weakly to a weakly harmonic map u in H1,2 (M, N) as k , ,. In this paper, we construct an example to show that the limit map u may not be stationary. © 2002 Wiley Periodicals, Inc. [source]