Abstract differential geometry

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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.


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]


  1. ^ "Geometry of Vector Sheaves: An Axiomatic Approach to Differential Geometry", Anastasios Mallios, Springer, 1998, ISBN 978-0-7923-5005-7
  2. ^ "Modern Differential Geometry in Gauge Theories: Maxwell fields", Anastasios Mallios, Springer, 2005, ISBN 978-0-8176-4378-2
  3. ^ Mallios, Anastasios; Raptis, Ioannis (2004). "Smooth Singularities Exposed: Chimeras of the Differential Spacetime Manifold". arXiv:gr-qc/0411121.

Further reading[edit]