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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.

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