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We present a geometric approach to defining an algebra Ĝ (M) (the Colombeau algebra) of generalized functions on a smooth manifold M containing the space DOE(M) of distributions on M. Based on differential calculus in convenient vector spaces we achieve an intrinsic construction of Ĝ (M). Ĝ (M) is a differential algebra, its elements possessing Lie derivatives with respect to arbitrary smooth vector fields. Moreover, we construct a canonical linear embedding of

This paper gives a comprehensive analysis of algebras of Colombeau-type generalized functions in the range between the diffeomorphism-invariant quotient algebra G d = E M /N introduced in part I and Colombeau's original algebra G e . Three main results are established: First, a simple criterion describing membership in N (applicable to all types of Colombeau algebras) is given. Second, two counterexamples demonstrate that G d is not injectively included in G e . Finally, it is shown that in the range "between" G d and G e only one more construction leads to a diffeomorphism invariant algebra. In analyzing the latter, several classification results essential for obtaining an intrinsic description of G d on manifolds are derived.2000 Mathematics Subject Classification. Primary 46F30; Secondary 26E15, 46E50, 35D05.

The geodesic as well as the geodesic deviation equation for impulsive gravitational waves involve highly singular products of distributions (␦, 2 ␦,␦ 2 ). A solution concept for these equations based on embedding the distributional metric into the Colombeau algebra of generalized functions is presented. Using a universal regularization procedure we prove existence and uniqueness results and calculate the distributional limits of these solutions explicitly. The obtained limits are regularization independent and display the physically expected behavior.

The geometry of impulsive pp-waves is explored via the analysis of the geodesic and geodesic deviation equation using the distributional form of the metric. The geodesic equation involves formally ill-defined products of distributions due to the nonlinearity of the equations and the presence of the Dirac δ-distribution in the space time metric. Thus, strictly speaking, it cannot be treated within Schwartz's linear theory of distributions. To cope with this difficulty we proceed by first regularizing the δ-singularity, then solving the regularized equation within classical smooth functions and, finally, obtaining a distributional limit as solution to the original problem. Furthermore it is shown that this limit is independent of the regularization without requiring any additional condition, thereby confirming earlier results in a mathematical rigorous fashion. We also treat the Jacobi equation which, despite being linear in the deviation vector field, involves even more delicate singular expressions, like the "square" of the Dirac δ-distribution. Again the same regularization procedure provides us with a perfectly well behaved smooth regularization and a regularization-independent distributional limit. Hence it is concluded that the geometry of impulsive pp-waves can be described consistently using distributions as long as careful regularization procedures are used to handle the ill-defined products.