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Some illustrative examples of non-orientable manifolds include: (1) the Möbius strip, which is a manifold with boundary, (2) the Klein bottle, which must intersect itself in its 3-space representation, and (3) the real projective plane, which arises naturally in geometry.
Flag manifold; Grassmann manifold; Stiefel manifold; Lie groups provide several interesting families. See Table of Lie groups for examples. See also: List of simple Lie groups and List of Lie group topics.
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.
Formally, classifying manifolds is classifying objects up to isomorphism.There are many different notions of "manifold", and corresponding notions of "map between manifolds", each of which yields a different category and a different classification question.
A smooth manifold endowed with a Riemannian metric is a Riemannian manifold, denoted (,). [3] A Riemannian metric is a special case of a metric tensor. A Riemannian metric is not to be confused with the distance function of a metric space, which is also called a metric.
Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure. [1] However, not every topological manifold can be endowed with a particular additional structure. For example, the E8 manifold is a topological manifold which cannot be endowed with a differentiable structure.
Manifold theory has come to focus exclusively on these intrinsic properties (or invariants), while largely ignoring the extrinsic properties of the ambient space. Another, more topological example of an intrinsic property of a manifold is the Euler characteristic.
Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a ...