Related papers: Generalized Differential Geometry
In this work, standard methods of the mixed thin-shell foramlism are refined using the framework of Colombeau's theory of generalized functions. To this end, systematic use is made of smooth generalized functions, in particular…
The higher dimensional Quantum General Relativity of a Riemannian manifold being an embedded space in a space-time being a Lorentzian manifold is investigated. The model of quantum geometrodynamics, based on the Wheeler-DeWitt equation…
A generalized Clifford manifold is proposed in which there are coordinates not only for the basis vector generators, but for each element of the Clifford group, including the identity scalar. These new quantities are physically interpreted…
This paper builds on the theory of generalised functions begun in [1]. The Colombeau theory of generalised scalar fields on manifolds is extended to a nonlinear theory of generalised tensor fields which is diffeomorphism invariant and has…
We present herewith certain thoughts on the important subject of nowadays physics, pertaining to the so-called ``singularities'', that emanated from looking at the theme in terms of ADG (: abstract differential geometry). Thus, according to…
In this paper, we introduce a generalization of a class of tilings which appear in the literature: the tilings over which a height function can be defined (for example, the famous tilings of polyominoes with dominoes). We show that many…
Generalized differential forms are used in discussions of metric geometries and Einstein's vacuum field equations. Cartan's structure equations are generalized and applied. In particular flat generalized connections are associated with any…
We introduce the notion of generalized function taking values in a smooth manifold into the setting of full Colombeau algebras. After deriving a number of characterization results we also introduce a corresponding concept of generalized…
Vector fields with components which are generalized zero-forms are constructed. Inner products with generalized forms, Lie derivatives and Lie brackets are computed. The results are shown to generalize previously reported results for…
Despite remarkable success in describing supergravity reductions and backgrounds, generalized geometry and the closely related exceptional field theory are still lacking a fundamental object of differential geometry, the Riemann tensor. We…
We give detailed exposition of modern differential geometry from global coordinate independent point of view as well as local coordinate description suited for actual computations. In introduction, we consider Euclidean spaces and different…
A pedagogical but concise overview of Riemannian geometry is provided, in the context of usage in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions,…
We reformulate the Hamiltonian form of bosonic eleven dimensional supergravity in terms of an object that unifies the three-form and the metric. For the case of four spatial dimensions, the duality group is manifest and the metric and…
We consider spaces for which there is a notion of harmonicity for complex valued functions defined on them. For instance, this is the case of Riemannian manifolds on one hand, and (metric) graphs on the other hand. We observe that it is…
Generalized uncertainty principles are able to serve as useful descriptions of some of the phenomenology of quantum gravity effects, providing an intuitive grasp on non-trivial space-time structures such as a fundamental discreteness of…
A local existence and uniqueness theorem for ODEs in the special algebra of generalized functions is established, as well as versions including parameters and dependence on initial values in the generalized sense. Finally, a Frobenius…
In this paper, we elaborate on the connection between leading singularities and canonical bases of Feynman integrals beyond polylogarithms. We start by discussing a notion of leading singularities in dimensional regularization, which can be…
Using the adjoint action of the infinitesimal translations (with respect to some (in)dependant variables) on specific finite-dimensional subspaces of the space of generalized symmetries of some system of partial differential equations, we…
The paper deals with a comprehensive theory of mappings, whose local behavior can be described by means of linear subspaces, contained in the graphs of two (primal and dual) generalized derivatives. This class of mappings includes the…
Real analytic generalized functions are investigated as well as the analytic singular support and analytic wave front of a generalized function in $\mathcal{G}(\Omega)$ are introduced and described.