A nonlinear theory of distributional geometry.

Area of Science:

• Differential Geometry
• Nonlinear Generalized Functions
• Tensor Field Theory

Background:

• Builds upon the established theory of nonlinear generalized functions.
• Addresses the need for diffeomorphism-invariant nonlinear theories of generalized tensor fields.

Purpose of the Study:

• To extend the theory of nonlinear generalized functions to generalized tensor fields.
• To introduce and develop a non-smooth theory of differential geometry.
• To investigate the properties of generalized metrics and their associated curvature.

Main Methods:

• Introduced the generalized Lie derivative and demonstrated its commutation properties.
• Defined a generalized covariant derivative that commutes with embedding.
• Developed a generalized metric to formulate a non-smooth differential geometry.

Main Results:

• Established a diffeomorphism-invariant nonlinear theory of generalized tensor fields with the sheaf property.
• Showed that embedding continuous metrics results in generalized metrics with well-defined connection and curvature.
• Demonstrated that for C^2 metrics, curvature is preserved at the level of association, with a conical metric example yielding delta function curvature.

Conclusions:

• The developed theory provides a framework for non-smooth differential geometry.
• Generalized metrics offer a robust approach to defining curvature in non-smooth spaces.
• The findings have implications for understanding generalized tensor fields and their geometric properties.