# 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 Ioannis Raptis 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[edit]

### ADG Gravity[edit]

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

## See also[edit]

## References[edit]

**^**"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**^**Mallios, Anastasios; Raptis, Ioannis (2004). "Smooth Singularities Exposed: Chimeras of the Differential Spacetime Manifold". arXiv:gr-qc/0411121.

## Further reading[edit]

- Space-time foam dense singularities and de Rham cohomology, A Mallios, EE Rosinger, Acta Applicandae Mathematicae, 2001