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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.
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.
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 ...
Every closed manifold is a Euclidean neighborhood retract and thus has finitely generated homology groups. [2] If is a closed connected n-manifold, the n-th homology group is or 0 depending on whether is orientable or not. [3] Moreover, the torsion subgroup of the (n-1)-th homology group is 0 or depending on whether is orientable or not.
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 ...
Topological manifold. In topology, a topological manifold is a topological space that locally resembles real n - dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by ...
Einstein manifold. In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field ...
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 ...