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Nonlinear generalized functions on manifolds.

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This study introduces a novel theory for generalized functions on manifolds using smoothing operators, extending concepts like the Lie derivative for differential geometry applications. It enables covariant derivatives for generalized scalar fields, paving the way for distributional geometry.

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Area of Science:

  • Differential Geometry
  • Analysis on Manifolds
  • Algebraic Structures

Background:

  • Existing theories of generalized functions on manifolds have limitations in applicability to differential geometry.
  • Schwartz distributions and their derivatives are foundational but require extensions for broader geometric applications.

Purpose of the Study:

  • To develop a global theory of algebras of generalized functions on manifolds.
  • To generalize existing theories for applications in differential geometry.
  • To lay the groundwork for a nonlinear theory of distributional geometry.

Main Methods:

  • Utilizing the concept of smoothing operators for constructing the theory.
  • Introducing and analyzing the generalized Lie derivative.
  • Defining a covariant derivative for generalized scalar fields.

Main Results:

  • A generalized theory of algebras of generalized functions on manifolds is established.
  • The generalized Lie derivative is shown to extend the Lie derivative of Schwartz distributions.
  • A covariant derivative for generalized scalar fields is defined, extending the covariant derivative of distributions.

Conclusions:

  • The developed theory provides a suitable framework for applications in differential geometry.
  • The work establishes foundations for nonlinear distributional geometry using Colombeau algebras.
  • The new approach offers enhanced capabilities for handling generalized functions and derivatives on manifolds.