Abstract differential geometry

The adjective abstract has often been applied to differential geometry before, but the abstract differential geometry (ADG) of this article is a form of differential geometry without the calculus notion of smoothness, developed by Anastasios Mallios and others from 1998 onwards.^[1]

Instead of calculus, an axiomatic treatment of differential geometry is built via sheaf theory and sheaf cohomology using vector sheaves in place of bundles based on arbitrary topological spaces.^[2] Mallios says noncommutative geometry can be considered a special case of ADG, and that ADG is similar to synthetic differential geometry.

Applications

ADG Gravity

Mallios and Raptis use ADG to avoid the singularities in general relativity and propose this as a route to quantum gravity.^[3]

References

↑ "Geometry of Vector Sheaves: An Axiomatic Approach to Differential Geometry", Anastasios Mallios, Springer, 1998, ISBN 978-0-7923-5005-7
↑ "Modern Differential Geometry in Gauge Theories: Maxwell fields", Anastasios Mallios, Springer, 2005, ISBN 978-0-8176-4378-2
↑ Smooth Singularities Exposed: Chimeras of the Differential Spacetime Manifold, Anastasios Mallios, Ioannis Raptis

Paradigms	Classical theories of gravitation Quantum gravity Theory of everything

Classical	Einstein–Cartan Bimetric theories Gauge theory gravity Teleparallelism Composite gravity f(R) gravity Massive gravity Modified Newtonian dynamics (MOND) Nonsymmetric gravitation Scalar–tensor theories Brans–Dicke Scalar–tensor–vector Conformal gravity Scalar theories Nordström Whitehead Geometrodynamics Induced gravity Tensor–vector–scalar Chameleon Pressuron

Quantisation	Euclidean quantum gravity Canonical quantum gravity Wheeler–DeWitt equation Loop quantum gravity Spin foam Causal dynamical triangulation Causal sets DGP model

Unification	Kaluza–Klein theory Dilaton Supergravity

Unification and quantisation	Noncommutative geometry Self-creation cosmology Semiclassical gravity Superfluid vacuum theory Logarithmic BEC vacuum String theory M-theory F-theory Heterotic string theory Type I string theory Type 0 string theory Bosonic string theory Type II string theory Little string theory Twistor theory Twistor string theory

Generalisations / Extensions of GR	Liouville gravity Lovelock theory (2+1)-dimensional topological gravity Gauss–Bonnet gravity Jackiw–Teitelboim gravity

Pre-Newtonian theories
and Toy models

Abstract differential geometry

Applications

ADG Gravity

See also

References

Further reading