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2 results

  • Appendix E - Differential Forms on Infinite-Dimensional Manifolds

  • Alexander Schmeding, Nord Universitet, Norway
  • Book:
    An Introduction to Infinite-Dimensional Differential Geometry
    Published online:
    08 December 2022
    Print publication:
    22 December 2022, pp 231-243
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  • Summary

    In this appendix, we give a short introduction to differential forms on infinite-dimensional manifolds. The main difference between the finite dimensional (or Banach) and our setting, is that it is in general impossible to interprete differential forms as (smooth) sections into certain bundles of linear forms. The reason for this is again that the topology on spaces of linear forms breaks down beyond the Banach setting. Even worse, the many equivalent ways to define differential forms in finite dimensions become inequivalent in the infinite-dimensional setting. Most notably, there is no useful way to describe differential forms as a sum of differential forms coming from a local coordinate system. We begin with the definition of a differential form. This definition is geared towards avoiding any reference to topologies on spaces of linear mappings. Then, we shall discuss differential forms on a Lie group and in particular the Maurer–Cartan form.

  • EIGENVALUES AND EIGENFUNCTIONS OF METRIC MEASURE MANIFOLDS

  • YUXIN GE, ZHONGMIN SHEN
  • Journal:
    Proceedings of the London Mathematical Society / Volume 82 / Issue 3 / May 2000
    Published online by Cambridge University Press:
    20 August 2001, pp. 725-746
    Print publication:
    May 2000

  • In this paper we study the eigenvalues and eigenfunctions of metric measure manifolds. We prove that any eigenfunction is $C^{1,\alpha}$ at its critical points and $C^{\infty}$ elsewhere. Moreover, the eigenfunction corresponding to the first eigenvalue in the Dirichlet problem does not change sign. We also discuss the first eigenvalue, the Sobolev constants and their relationship with the isoperimetric constants. 2000 Mathematics Subject Classification: 47J05, 47J10, 53C60, 58E05, 58C40.