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Manifolds need not be closed; thus a line segment without its end points is a manifold. They are never countable, unless the dimension of the manifold is 0. Putting these freedoms together, other examples of manifolds are a parabola, a hyperbola, and the locus of points on a cubic curve y 2 = x 3 − x (a closed loop piece and an open, infinite ...
The most familiar non-Hausdorff manifold is the line with two origins, [1] or bug-eyed line. This is the quotient space of two copies of the real line, and (with ), obtained by identifying points and whenever. An equivalent description of the space is to take the real line and replace the origin with two origins and The subspace retains its ...
A manifold is metrizable if and only if it is paracompact. The long line is an example a normal Hausdorff 1-dimensional topological manifold that is not metrizable nor paracompact. Since metrizability is such a desirable property for a topological space, it is common to add paracompactness to the definition of a manifold.
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. Euclidean space, the -sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifolds.
A Kähler manifold is a complex manifold with a Hermitian metric whose associated 2-form is closed. In more detail, gives a positive definite Hermitian form on the tangent space at each point of , and the 2-form is defined by. for tangent vectors and (where is the complex number ). For a Kähler manifold , the Kähler form is a real closed (1,1 ...
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. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies ...
Real line, R; Real projective line, RP 1 ≅ S 1; 2-manifolds. ... For more examples see 4-manifold. Special types of manifolds. Manifolds related to spheres
A component used to regulate fluid flow in a hydraulic system, thus controlling the transfer of power between actuators and pumps. Inlet manifold (or "intake manifold") An engine part that supplies the air or fuel/air mixture to the cylinders. Scuba manifold. In a scuba set, connects two or more diving cylinders. Vacuum gas manifold.