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An example of a quotient space of a manifold that is also a manifold is the real projective space, identified as a quotient space of the corresponding sphere. One method of identifying points (gluing them together) is through a right (or left) action of a group , which acts on the manifold.
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.
The study of maps of 1-dimensional manifolds are a non-trivial area. For example: Groups of diffeomorphisms of 1-manifolds are quite difficult to understand finely [2] Maps from the circle into the 3-sphere (or more generally any 3-dimensional manifold) are studied as part of knot theory.
holds, and a (pseudo-)Riemannian manifold whose metric is Einstein is called an Einstein manifold. [32] Examples of Einstein manifolds include Euclidean space, the -sphere, hyperbolic space, and complex projective space with the Fubini-Study metric.
The only connected one-dimensional example is a circle. The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. The real projective space RP n is a closed n-dimensional manifold. The complex projective space CP n is a closed 2n-dimensional manifold. [1] A line is not closed because it is not compact.
Types of manifolds in engineering include: Exhaust manifold An engine part that collects the exhaust gases from multiple cylinders into one pipe. Also known as headers. Hydraulic manifold 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 ...
A 3-torus in this sense is an example of a 3-dimensional compact manifold. It is also an example of a compact abelian Lie group. This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication). Group multiplication on the torus is then defined by coordinate-wise ...
In geometry, if X is a manifold with an action of a topological group G by analytical diffeomorphisms, the notion of a (G, X)-structure on a topological space is a way to formalise it being locally isomorphic to X with its G-invariant structure; spaces with a (G, X)-structure are always manifolds and are called (G, X)-manifolds.
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