Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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. 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. Mallios says noncommutative geometry can be considered a special case of ADG, and that ADG is similar to synthetic differential geometry.

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