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.

Contents 1 Applications 1.1 ADG Gravity 2 References 3 Further reading 4 See also

[edit] Applications

[edit] ADG Gravity

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

[edit] References

[edit] Further reading

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

[edit] See also

• Discrete differential geometry
• Analysis on fractals

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