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It is distinct from "geometric topology", which more narrowly involves applications of topology to geometry. It includes: Differential geometry and topology; Geometric topology (including low-dimensional topology and surgery theory) It does not include such parts of algebraic topology as homotopy theory, but some areas of geometry and topology ...
In mathematics, differential topology is the field dealing with the topological properties and smooth properties [a] of smooth manifolds.In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the geometric properties of smooth manifolds, including notions of size, distance, and rigid shape.
Differential geometry is closely related to, and is sometimes taken to include, differential topology, which concerns itself with properties of differentiable manifolds that do not rely on any additional geometric structure (see that article for more discussion on the distinction between the two subjects).
They are one of the unifying geometric concepts in algebraic topology, differential geometry and algebraic geometry. Surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by Milnor . Surgery refers to cutting out parts of the manifold and replacing it with a part of another ...
A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...
Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of linear elliptic PDEs dates at least as far back as Hodge theory.
Typically, in algebraic geometry, a surface may cross itself (and may have other singularities), while, in topology and differential geometry, it may not. A surface is a two-dimensional space ; this means that a moving point on a surface may move in two directions (it has two degrees of freedom ).
Hence, it is a more primitive definition of the structure (see synthetic differential geometry). A final advantage of this approach is that it allows for natural direct descriptions of many of the fundamental objects of study to differential geometry and topology.