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In mathematics, a manifold is a ... other examples of manifolds ... one often studies solution to partial differential equations, an important example of ...
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.
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.
In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor. In the case of hyperbolic dynamics, the corresponding notion is that of the hyperbolic set.
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.
The dot products on every tangent plane, packaged together into one mathematical object, are a Riemannian metric. In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined.
The notion of a differentiable manifold refines that of a manifold by requiring the functions that transform between charts to be differentiable. In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus.
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3- manifold can be thought of as a possible shape of the universe . Just as a sphere looks like a plane (a tangent plane ) to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer.