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The study of calculus on differentiable manifolds is known as differential geometry. "Differentiability" of a manifold has been given several meanings, including: continuously differentiable, k-times differentiable, smooth (which itself has many meanings), and analytic.
In differential geometry, a G-structure on an n-manifold M, for a given structure group [1] G, is a principal G-subbundle of the tangent frame bundle FM (or GL(M)) of M.. The notion of G-structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields.
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus , integral calculus , linear algebra and multilinear algebra .
In vector calculus and differential geometry the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, [1] is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.
Manifolds in contemporary mathematics come in a number of types. These include: smooth manifolds, which are basic in calculus in several variables, mathematical analysis and differential geometry; piecewise-linear manifolds; topological manifolds. There are also related classes, such as homology manifolds and orbifolds, that resemble manifolds.
In differential geometry, a -structure is an important type of G-structure that can be defined on a smooth manifold. If M is a smooth manifold of dimension seven, then a G 2 -structure is a reduction of structure group of the frame bundle of M to the compact , exceptional Lie group G 2 .
For compact manifolds, results depend on the complexity of the manifold as measured by the second Betti number b 2. For large Betti numbers b 2 > 18 in a simply connected 4-manifold, one can use a surgery along a knot or link to produce a new differential structure. With the help of this procedure one can produce countably infinite many ...
In mathematics, a CR manifold, or Cauchy–Riemann manifold, [1] is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge.