Advertisement

Acta Mathematica Hungarica

, Volume 94, Issue 4, pp 307–320 | Cite as

Contributions to the Study of Monotone Vector Fields

  • J. X. Da Cruz Neto
  • O. P. Ferreira
  • L. R. Lucambio pérez
Article

Abstract

We introduce the concept of a strongly monotone vector field on a Riemannian manifold and give an example. We also demonstrate relationships between different kinds of monotonicity of vector fields and different kinds of definiteness of its differential operator. Some topological and metric consequences of the strict and strongly monotone vector fields" existence are shown.

monotone vector field Riemannian manifold convex function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc., 145 (1969), 1-49.Google Scholar
  2. [2]
    M. P. do Carmo, Riemannian Geometry, Birkhäuser (Boston, 1992).Google Scholar
  3. [3]
    J. X. da Cruz Neto, L. L. Lima, and P. R. Oliveira, Geodesic algorithm in Riemannian manifolds, Balkan Journal of Geometry its Applications, 3 (1998), 89-100.Google Scholar
  4. [4]
    J. X. da Cruz Neto, O. P. Ferreira and L. R. Lucambio Perez, A proximal regularization of the steepest descent method in Riemannian manifold, Balkan Journal of Geometry its Applications, 4 (1999), 1-8.Google Scholar
  5. [5]
    O. P. Ferreira and P.R. Oliveira, Subgradient algorithm on Riemannian manifolds, Journal of Optimization Theory and Applications, 97 (1998), 93-104.Google Scholar
  6. [6]
    D. Gabay, Minimizing a differentiable function over a differentiable manifolds, Journal of Optimization Theory and Applications, 37 (1982), 177-219.Google Scholar
  7. [7]
    J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms, I and II, Springer-Verlag (1993).Google Scholar
  8. [8]
    S. Z. Németh, Five kinds of monotone vector fields, Pure and Applied Mathematics, 9 (1998), 417-428.Google Scholar
  9. [9]
    S. Z. Németh, Geodesic monotone vector fields, Lobachevskii Journal of Mathematics, 5 (1999), 13-28.Google Scholar
  10. [10]
    S. Z. Németh, Monotonicity of the complementary vector field of a nonexpansive map, Acta Math. Hungar., 84 (1999), 189-197.Google Scholar
  11. [11]
    S. Z. Németh, Monotone vector fields, Publ. Math. Debrecen, 54 (1999), 437-449.Google Scholar
  12. [12]
    G. Perelman, Proof of the soul conjecture of Cheeger and Gromoll, J. Differential Geometry, 40 (1994), 209-212.Google Scholar
  13. [13]
    T. Rapcsák, Smooth Nonlinear Optimization in R n, Kluwer Academic Publishers (Dordrecht, 1997).Google Scholar
  14. [14]
    T. Sakai, Riemannian Geometry, Translations of Mathematical Monographs, Vol. 149, American Mathematical Society (Providence, R.I., 1996).Google Scholar
  15. [15]
    K. Shiohama, Topology of complete noncompact manifolds, Geometry of Geodesics and Related Topics, Advanced Studies in Pure Mathematics, 3 (1984), 423-450.Google Scholar
  16. [16]
    C. Udriste, Convex functions on Riemannian manifolds, St. Cerc. Mat., 28 (1976), 735-745.Google Scholar
  17. [17]
    C. Udriste, Convex functions and optimization methods on Riemannian manifolds, Mathematics and its Applications, Vol. 297, Kluwer Academic Publishers (1994).Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • J. X. Da Cruz Neto
    • 1
  • O. P. Ferreira
    • 1
  • L. R. Lucambio pérez
    • 1
  1. 1.DM/CCN/Universidade Federal Do PiauíTeresina, PiBrazil

Personalised recommendations