Intrinsic definition of the Colombeau algebra of generalized functions

Sheaves of nonlinear generalized functions and manifold-valued distributions

“…[19], Sec. 6.3) we may directly generalize the transformation behavior (3), (4) by allowing f (2) Ψ β Φ α to be a matrix with distributional entries and by replacing compositions with distributional pullbacks. One restriction, however, immediately becomes apparent: f (1) Ψ β Φ α has to be supposed smooth in order for the right hand side of (4) to be well defined (i.e., to avoid ill-defined products).…”

“…A key step in this program was the construction of a diffeomorphism invariant scalar theory in the so-called 'full' setting where one has a canonical embedding of distributions into the algebra [15,17,21,22]. This built upon the pioneering work of [2,7,20]. However in order to develop a theory of generalized differential geometry one needs to go beyond this and have a description of generalized tensor fields, a topic of ongoing research.…”

Generalized Functions Valued in a Smooth Manifold

“…A solution to this problem can be based on generalizing the space G[Ω, Ω ′ ] introduced in [1], 7.3 (denoted there by G * (Ω, Ω ′ )) to the manifold setting. We shortly recall its definition.…”

Geometrical embeddings of distributions into algebras of generalized functions

“…Thus we do not need to work with local coordinates and use the sheaf structure of the algebra G. In addition the global construction makes it compatible with the geometry of the Riemannian manifold as these embeddings commute with the Laplace operator and hence are preserved under isometries. The embeddings introduced so far in the literature ( [1,7,16]) are 'nongeometric' in that they depend on choices of partitions of unity, cut-offs, etc. (cf.…”