Manifold

Additional structures and generalizations

In order to do geometry on manifolds it is usually necessary to adorn these spaces with additional structures, such as the differential structure discussed above. There are numerous other possibilities, depending on the kind of geometry one is interested in:

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• A Riemannian manifold is a differentiable manifold on which the tangent spaces are equipped with inner products in a differentiable fashion. The inner product structure is given in the form of a symmetric 2-tensor called the Riemannian metric. On a Riemannian manifold one has notions of length, volume, and angle.
• A pseudo-Riemannian manifold is a variant of Riemannian manifold where the metric tensor is allowed to have an indefinite signature (as opposed to a positive-definite one). Pseudo-Riemannian manifolds of signature (3, 1) are important in general relativity.
• A symplectic manifold is a manifold equipped with a closed, nondegenerate, alternating 2-form. Such manifolds arise in the study of Hamiltonian mechanics.
• A complex manifold is a manifold modeled on Cⁿ with holomorphic transition functions on chart overlaps. These manifolds are the basic objects of study in complex geometry.
• A Kähler manifold is a manifold which simultaneously carries a Riemannian structure, a symplectic structure, and a complex structure which are all compatible in some suitable sense.
• A Calabi-Yau manifold is a compact Ricci-flat Kähler manifold. In string theory the extra dimensions are curled up into a Calabi-Yau manifold.
• A Finsler manifold is a generalization of a Riemannian manifold.
• A Lie group is C^∞ manifold which also carries a smooth group structure. These are the proper objects for describing symmetries of analytical structures.

Manifolds "locally look like" Euclidean space Rn and are therefore inherently finite-dimensional objects. To allow for infinite dimensions, one may consider Banach manifolds which locally look like Banach spaces, or Fréchet manifolds, which locally look like Fréchet spaces.

Related Topics:
Euclidean space - Banach space - Fréchet space

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Another generalization of manifold allows one to omit the requirement that a manifold be Hausdorff. It still must be second-countable and locally Euclidean, however. Such spaces are called non-Hausdorff manifolds and are used in the study of codimension-1 foliations.

Related Topics:
Hausdorff - Second-countable - Codimension-1 foliations

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An orbifold is yet an another generalization of manifold, one that allows certain kinds of "singularities" in the topology. Roughly speaking, it is a space which locally looks like the quotient of Euclidean space by a finite group. The singularities correspond to fixed points of the group action.

Related Topics:
Orbifold - Singularities - Euclidean space - Finite group - Group action

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The category of smooth manifolds with smooth maps lacks certain desirable properties, and people have tried to generalize smooth manifolds in order to rectify this. The diffeological spaces, (differential spaces) use a different notion of chart known as "plot". Frölicher spaces are another attempt.

Related Topics:
Category - Diffeological space - Frölicher space

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