A manifold is a mathematical space which is constructed, like a patchwork, by gluing and bending together copies of simple spaces. For example, a circle can be constructed by bending two line segments into arcs which overlap at their ends and gluing them together where they overlap. The motivation for working with manifolds is that you begin with a relatively simple space which is well understood, and build up a manifold, which may be very complicated, from copies of that simple space. By choosing different spaces as base material, di ...

Including:

• Manifold - Introduction
• Manifold - Motivational example: the circle
• Manifold - Charts atlases and transition maps
• Manifold - Construction
• Manifold - Charts
• Manifold - Patchwork
• Manifold - Zeros of a function
• Manifold - Identifying points of a manifold
• Manifold - Cartesian products
• Manifold - Gluing along boundaries
• Manifold - Topological manifolds
• Manifold - Differentiable manifolds
• Manifold - Orientability
• Manifold - Möbius strip
• Manifold - Klein bottle
• Manifold - Real projective plane
• Manifold - Other types and generalizations of manifolds
• Manifold - History

Read more here: » Manifold: Encyclopedia - Manifold

See also:

Manifold, Manifold - Introduction, Manifold - Motivational example: the circle, Manifold - Charts atlases and transition maps, Manifold - Construction, Manifold - Charts, Manifold - Patchwork, Manifold - Zeros of a function, Manifold - Identifying points of a manifold, Manifold - Cartesian products, Manifold - Gluing along boundaries, Manifold - Topological manifolds, Manifold - Differentiable manifolds, Manifold - Orientability, Manifold - Möbius strip, Manifold - Klein bottle, Manifold - Real projective plane, Manifold - Other types and generalizations of manifolds, Manifold - History

Read more here: » Manifold: Encyclopedia II - Manifold - Differentiable manifolds

For more details on this topic, see topological manifold. The simplest kind of manifold to define is the topological manifold, which looks locally like some "ordinary" Euclidean space Rn. Formally, a topological manifold is a topological space locally homeomorphic to a Euclidean space. This means that every point has a neighbourhood for which there exists a homeomorphism (a bijective continuous function whose inverse is also continuous) mapping that neighbourhood to Rn. Th ...

See also:

Manifold, Manifold - Introduction, Manifold - Motivational example: the circle, Manifold - Charts atlases and transition maps, Manifold - Construction, Manifold - Charts, Manifold - Patchwork, Manifold - Zeros of a function, Manifold - Identifying points of a manifold, Manifold - Cartesian products, Manifold - Gluing along boundaries, Manifold - Topological manifolds, Manifold - Differentiable manifolds, Manifold - Orientability, Manifold - Möbius strip, Manifold - Klein bottle, Manifold - Real projective plane, Manifold - Other types and generalizations of manifolds, Manifold - History

Read more here: » Manifold: Encyclopedia II - Manifold - Topological manifolds

A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly different viewpoint. Manifold - Charts. Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of R2 is identified, and then an atlas covering this subset is constructed. The concept of manifold grew historically from constructions like this. Here is another example, applying this method ...

See also:

Manifold, Manifold - Introduction, Manifold - Motivational example: the circle, Manifold - Charts atlases and transition maps, Manifold - Construction, Manifold - Charts, Manifold - Patchwork, Manifold - Zeros of a function, Manifold - Identifying points of a manifold, Manifold - Cartesian products, Manifold - Gluing along boundaries, Manifold - Topological manifolds, Manifold - Differentiable manifolds, Manifold - Orientability, Manifold - Möbius strip, Manifold - Klein bottle, Manifold - Real projective plane, Manifold - Other types and generalizations of manifolds, Manifold - History

Read more here: » Manifold: Encyclopedia II - Manifold - Construction

A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly different viewpoint. Manifold - Charts. Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of R2 is identified, and then an atlas covering this subset is constructed. The concept of manifold grew historically from constructions like this. Here is another example, applying this method ...

See also:

Manifold, Manifold - Introduction, Manifold - Motivational example: the circle, Manifold - Charts atlases and transition maps, Manifold - Construction, Manifold - Charts, Manifold - Patchwork, Manifold - Zeros of a function, Manifold - Identifying points of a manifold, Manifold - Cartesian products, Manifold - Manifold with boundary, Manifold - Gluing along boundaries, Manifold - Topological manifolds, Manifold - Differentiable manifolds, Manifold - Orientability, Manifold - Möbius strip, Manifold - Klein bottle, Manifold - Real projective plane, Manifold - Other types and generalizations of manifolds, Manifold - History

Read more here: » Manifold: Encyclopedia II - Manifold - Construction

In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is usually made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. The study of 3-manifolds is considered a field of mathematics, unlike, for example, the study of 167-dimensional manifolds. There are close connections to other fields, such as 4-manifolds, surfaces, knot theory, topological quantum field theory, and gauge theory. 3-manifold theory is a part o ...

Including:

• 3-manifold - Some famous examples of 3-manifolds
• 3-manifold - Some important classes of 3-manifolds
• 3-manifold - Some important structures on 3-manifolds
• 3-manifold - Foundational results
• 3-manifold - Some famous conjectures

Read more here: » 3-manifold: Encyclopedia - 3-manifold

In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. More precisely, a complex manifold has an atlas of charts to Cn, such that the change of coordinates between charts are holomorphic. Complex manifolds can be regarded as a special case of differentiable manifolds. For example, a 1-dimensional complex manifold is geometrically a surface, known as a Riemann surface. The requirement that the transition functions b ...

Including:

• Complex manifold - Examples of complex manifolds
• Complex manifold - Integrable almost-complex structures
• Complex manifold - Kähler and Calabi-Yau manifolds

Read more here: » Complex manifold: Encyclopedia - Complex manifold

In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, the topological and smooth categories are not equivalent. There exist some topological 4-manifolds which admit no smooth structure and even if there exists a smooth structure it need not be unique (i.e. there are smooth 4-manifol ...

Including:

• 4-manifold - Handle decomposition

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manifold: Varied. Having many forms, aspects, parts.

(See also: Manifold, Hinduism, Body Mind and Soul)

In mathematics, a Calabi-Yau manifold is a compact Kähler manifold with a vanishing first Chern class. A Calabi-Yau manifold of complex dimension n is also called a Calabi-Yau n-fold. The mathematician Eugenio Calabi conjectured in 1957 that all such manifolds admit a Ricci-flat metric (one in each Kähler class), and this conjecture was proved by Shing-Tung Yau in 1977 and became Yau's theorem. Consequently, a Calabi-Yau manifold can ...

Including:

• Calabi-Yau manifold - Examples
• Calabi-Yau manifold - Application in string theory

Read more here: » Calabi-Yau manifold: Encyclopedia - Calabi-Yau manifold

In mathematics and physics, the canonical coordinates are a special set of coordinates on the cotangent bundle of a manifold. They are usually written as a set of (qi,pj) or (xi,pj) with the x 's or q 's denoting the coordinates on the underlying manifold and the p 's denoting the conjugate momentum, which are 1-forms in the cotangent bundle at point < ...

Including:

• Canonical coordinates - Definition
• Canonical coordinates - Generalized coordinates

Read more here: » Canonical coordinates: Encyclopedia - Canonical coordinates

Topological manifold - Piecewise linear manifold. Merge from piecewise linear Topological manifold - Differentiable manifold. ...

See also:

Topological manifold, Topological manifold - Charts and transition maps, Topological manifold - Topological manifolds, Topological manifold - Topological manifold without boundary, Topological manifold - Topological manifold with boundary, Topological manifold - Sheaf of continuous functions, Topological manifold - Properties, Topological manifold - Subtypes, Topological manifold - Piecewise linear manifold, Topological manifold - Differentiable manifold, Topological manifold - Technical details, Topological manifold - The Hausdorff assumption

Read more here: » Topological manifold: Encyclopedia II - Topological manifold - Subtypes

Topological manifold - Topological manifold without boundary. The prototypical example of a topological manifold without boundary is Euclidean space. A general manifold without boundary looks locally, as a topological space, like Euclidean space. This is formalized by requiring that a manifold without boundary is a non-empty topological space in which every point has an open neighbourhood homeomorphic to (an open subset of) Rn (Euclidean n-space). Another way of saying this, ...

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Topological manifold, Topological manifold - Charts and transition maps, Topological manifold - Topological manifolds, Topological manifold - Topological manifold without boundary, Topological manifold - Topological manifold with boundary, Topological manifold - Sheaf of continuous functions, Topological manifold - Properties, Topological manifold - Subtypes, Topological manifold - Piecewise linear manifold, Topological manifold - Differentiable manifold, Topological manifold - Technical details, Topological manifold - The Hausdorff assumption

Read more here: » Topological manifold: Encyclopedia II - Topological manifold - Topological manifolds

Differentiable manifold - Smooth manifolds. A smooth manifold is a Differentiable manifold for which all the charts in the atlas are smooth. That is derivatives of all orders exist. Differentiable manifold - Analytic manifolds. An analytic manifold is a smooth manifold with the additional condition that each chart is analytic. That is the taylor ...

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Differentiable manifold, Differentiable manifold - History, Differentiable manifold - Definition, Differentiable manifold - Atlas, Differentiable manifold - Sheaf, Differentiable manifold - Differentiable functions, Differentiable manifold - Algebra of scalars, Differentiable manifold - Tangent bundle, Differentiable manifold - Cotangent bundle, Differentiable manifold - Jet bundle, Differentiable manifold - Tensor bundle, Differentiable manifold - Exterior calculus, Differentiable manifold - Exterior derivative, Differentiable manifold - Interior product, Differentiable manifold - Lie derivative, Differentiable manifold - Classification, Differentiable manifold - Subtypes, Differentiable manifold - Smooth manifolds, Differentiable manifold - Analytic manifolds, Differentiable manifold - pseudo-Riemannian manifolds, Differentiable manifold - Symplectic manifolds, Differentiable manifold - Lie groups, Differentiable manifold - Generalizations

Read more here: » Differentiable manifold: Encyclopedia II - Differentiable manifold - Subtypes

In order to measure lengths and angles, even more structure is needed: one defines Riemannian manifolds to recover these geometrical ideas. ... 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 Riemannia ...

See also:

Differentiable manifold, Differentiable manifold - History, Differentiable manifold - Definition, Differentiable manifold - Atlas, Differentiable manifold - Sheaf, Differentiable manifold - Differentiable functions, Differentiable manifold - Algebra of scalars, Differentiable manifold - Tangent bundle, Differentiable manifold - Cotangent bundle, Differentiable manifold - Jet bundle, Differentiable manifold - Tensor bundle, Differentiable manifold - Exterior calculus, Differentiable manifold - Exterior derivative, Differentiable manifold - Interior product, Differentiable manifold - Lie derivative, Differentiable manifold - Classification, Differentiable manifold - Subtypes, Differentiable manifold - pseudo-Riemannian manifolds, Differentiable manifold - Symplectic manifolds, Differentiable manifold - Lie groups, Differentiable manifold - Generalizations

Read more here: » Differentiable manifold: Encyclopedia II - Differentiable manifold - pseudo-Riemannian manifolds

In order to measure lengths and angles, even more structure is needed: one defines Riemannian manifolds to recover these geometrical ideas. ... 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 Riemannia ...

See also:

Differentiable manifold, Differentiable manifold - History, Differentiable manifold - Definition, Differentiable manifold - Atlas, Differentiable manifold - Sheaf, Differentiable manifold - Differentiable functions, Differentiable manifold - Algebra of scalars, Differentiable manifold - Tangent bundle, Differentiable manifold - Cotangent bundle, Differentiable manifold - Jet bundle, Differentiable manifold - Tensor bundle, Differentiable manifold - Exterior calculus, Differentiable manifold - Exterior derivative, Differentiable manifold - Interior product, Differentiable manifold - Lie derivative, Differentiable manifold - Classification, Differentiable manifold - Subtypes, Differentiable manifold - Smooth manifolds, Differentiable manifold - Analytic manifolds, Differentiable manifold - pseudo-Riemannian manifolds, Differentiable manifold - Symplectic manifolds, Differentiable manifold - Lie groups, Differentiable manifold - Generalizations

Read more here: » Differentiable manifold: Encyclopedia II - Differentiable manifold - pseudo-Riemannian manifolds

Every connected second-countable topological 1-manifold without boundary is homeomorphic to R or to S(the circle). The unconnected ones are disjoint unions of these two. For a classification of 2-manifolds, see surface. The 3-dimensional case may be solved. Thurston's geometrization conjecture, if true, together with current knowledge, would imply a classification of 3-manifolds. Grigori Perelman may have proven this conjecture; h ...

See also:

Differentiable manifold, Differentiable manifold - History, Differentiable manifold - Definition, Differentiable manifold - Atlas, Differentiable manifold - Sheaf, Differentiable manifold - Differentiable functions, Differentiable manifold - Algebra of scalars, Differentiable manifold - Tangent bundle, Differentiable manifold - Cotangent bundle, Differentiable manifold - Jet bundle, Differentiable manifold - Tensor bundle, Differentiable manifold - Exterior calculus, Differentiable manifold - Exterior derivative, Differentiable manifold - Interior product, Differentiable manifold - Lie derivative, Differentiable manifold - Classification, Differentiable manifold - Subtypes, Differentiable manifold - Smooth manifolds, Differentiable manifold - Analytic manifolds, Differentiable manifold - pseudo-Riemannian manifolds, Differentiable manifold - Symplectic manifolds, Differentiable manifold - Lie groups, Differentiable manifold - Generalizations

Read more here: » Differentiable manifold: Encyclopedia II - Differentiable manifold - Classification

Topological manifolds are usually required to be Hausdorff and second-countable. 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. ...

See also:

Topological manifold, Topological manifold - Charts and transition maps, Topological manifold - Topological manifolds, Topological manifold - Topological manifold without boundary, Topological manifold - Topological manifold with boundary, Topological manifold - Sheaf of continuous functions, Topological manifold - Properties, Topological manifold - Subtypes, Topological manifold - Piecewise linear manifold, Topological manifold - Differentiable manifold, Topological manifold - Technical details, Topological manifold - The Hausdorff assumption

Read more here: » Topological manifold: Encyclopedia II - Topological manifold - Technical details

Every connected second-countable topological 1-manifold without boundary is homeomorphic to R or to S(the circle). The unconnected ones are disjoint unions of these two. For a classification of 2-manifolds, see surface. The 3-dimensional case may be solved. Thurston's Geometrization Conjecture, if true, together with current knowledge, would imply a classification of 3-manifolds. Grigori Perelman may have proven this conjecture; h ...

See also:

Differentiable manifold, Differentiable manifold - History, Differentiable manifold - Definition, Differentiable manifold - Atlas, Differentiable manifold - Sheaf, Differentiable manifold - Differentiable functions, Differentiable manifold - Algebra of scalars, Differentiable manifold - Tangent bundle, Differentiable manifold - Cotangent bundle, Differentiable manifold - Jet bundle, Differentiable manifold - Tensor bundle, Differentiable manifold - Exterior calculus, Differentiable manifold - Exterior derivative, Differentiable manifold - Interior product, Differentiable manifold - Lie derivative, Differentiable manifold - Classification, Differentiable manifold - Subtypes, Differentiable manifold - pseudo-Riemannian manifolds, Differentiable manifold - Symplectic manifolds, Differentiable manifold - Lie groups, Differentiable manifold - Generalizations

Read more here: » Differentiable manifold: Encyclopedia II - Differentiable manifold - Classification

The hierarchy makes proving certain kinds of theorems about Haken manifolds a matter of induction. One proves the theorem for 3-balls. Then one proves that if the theorem is true for pieces resulting from a cutting of a Haken manifold, that it is true for that Haken manifold. The key here is that the cutting takes place along a surface that was very "nice", i.e. incompressible. This makes proving the induction step feasible in many cases. Haken sketched out a proof of an algorithm to check if two Haken manifolds were homeomorphic or n ...

See also:

Haken manifold, Haken manifold - Haken Hierarchy, Haken manifold - Applications, Haken manifold - Examples of Haken manifolds

Read more here: » Haken manifold: Encyclopedia II - Haken manifold - Applications