We discuss some basic concepts of semi-Riemannian geometry in low-regularity situations. In particular, we compare the settings of (linear) distributional geometry in the sense of L. Schwartz and nonlinear distributional geometry in the sense of J.F. Colombeau. AMS Mathematics Subject Classification (2000): Primary: 83C75; secondary: 46T30, 53B30, 46F10, 46F30.

This paper is devoted to the study of nonlinear geometric optics in Colombeau algebras of generalized functions in the case of Cauchy problems for semilinear hyperbolic systems in one space variable. Extending classical results, we establish a generalized variant of nonlinear geometric optics. As an application, a nonlinear superposition principle is obtained when distributional initial data ar...

The concept of generalised (in the sense of Colombeau) connection on a principal fibre bundle is introduced. This definition is then used to extend results concerning the geometry of principal fibre bundles to those that only have a generalised connection. Some applications to singular solutions of Yang-Mills theory are given. Mathematics Subject Classification (2000): Primary: 46T30; secondary...

The first part of my thesis lays the foundations to generalized Lorentz geometry. The basic algebraic structure of finite-dimensional modules over the ring of generalized numbers is investigated. This includes a new characterization of invertibility in the ring of generalized numbers as well as a characterization of free elements inside the n-dimensional module e R. The index of symmetric bilin...

It is briefly shown that, due to the growth conditions in their definition, the Colombeau algebras cannot handle arbitrary Lie groups, and in particular, cannot allow the formulation, let alone, solution of Hilbert’s Fifth Problem. “History is written with the feet ...” Ex-Chairman Mao, of the Long March fame ... Science is not done scientifically, since it is mostly done by non-scientists ...

We study invariance properties of Colombeau generalized functions under actions of smooth Lie transformation groups. Several characterization results analogous to the smooth setting are derived and applications to generalized rotational invariance are given. Mathematics Subject Classification (2000): Primary: 46F30; secondary: 58E40,46T30, 35A30

We consider the Cauchy problem for a hyperbolic pseudodifferential operator whose symbol is a Colombeau generalized function. Such equations arise, for example, after a reduction-decoupling of second-order model systems of differential equations in seismology. We prove existence of a unique generalized solution under log-type growth conditions on the symbol, thereby extending known results for ...