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
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
notion of smoothness, developed by
Anastasios Mallios and
Ioannis Raptis from 1998 onwards.
Instead of calculus, an axiomatic treatment of differential geometry is built via
sheaf theory
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
and
sheaf cohomology using
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
sheaves in place of
bundles based on arbitrary
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s. Mallios says
noncommutative geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some g ...
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
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
and propose this as a route to
quantum gravity.
See also
*
Discrete differential geometry
Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics, g ...
*
Analysis on fractals
References
Further reading
Space-time foam dense singularities and de Rham cohomology A Mallios, EE Rosinger, Acta Applicandae Mathematicae, 2001
Differential geometry
Sheaf theory
General relativity
Quantum gravity
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