Manifold - Introduction

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

Charts A coordinate map, a coordinate chart, or simply a chart of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure. For a topological manifold, the simple space is some Euclidean space Rn and we are interested in the topological structure. This structure is preserved by homeomo ...

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 - Charts, atlases and transition maps

A manifold is a space that looks, locally, like a Euclidean space of some fixed dimension. This may be one of the familiar one, two, or three dimensional spaces: a line, a plane, or the three-dimensional space which we inhabit; or, it may be an abstract space of some higher dimension or even of infinite dimension. Some authors allow manifolds to have separate pieces of different dimensions, but all authors require all pieces of a connected manifold to have the same dimension. A manifold with all pieces of dimension n is called an n-manifold. By contrast, gluing a one-dimensional "string" to three dimensional "ball" makes an ob ...

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 - Introduction

A manifold is a space that looks, locally, like a Euclidean space of some fixed dimension. This may be one of the familiar one, two, or three dimensional spaces: a line, a plane, or the three-dimensional space in which we live. Or, it may be an abstract space of some higher dimension or even of infinite dimension. Some authors allow manifolds to have separate pieces of different dimensions, but all authors require all pieces of a connected manifold to have the same dimension. A manifold with all pieces of dimension n is called an n-manifold. By contrast, gluing a one-dimensional string to three dimensional ball makes an ob ...

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 - Introduction

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

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

The circle is the simplest example of a topological manifold after Euclidean space itself. Consider, for instance, the circle of radius 1 with its centre at the origin. If x and y are the coordinates of a point on the circle, then we have x² + y² = 1. Locally, the circle resembles a line, which is one-dimensional. In other words, we need only one coordinate to describe the circle locally. Consider, for instance, the top part of the circle, for which the y-coordinate is positive (this is the yellow ...

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 - Motivational example: the circle

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. These homeomorphisms are the charts of the manifold. Usually additional t ...

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 - Topological manifolds

Consider a topological manifold with charts mapping to Rn. Given an ordered basis for Rn, a chart causes its piece of the manifold to itself acquire a sense of ordering, which we can think of as either right-handed or left-handed. Overlapping charts are not required to agree in their sense of ordering, which gives manifolds an important freedom. For some manifolds, like the sphere, we can choose charts so that overlapping regions agree on their "handedness"; these are orientable manifo ...

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 - Orientability

The first to have conceived clearly of curves and surfaces as spaces by themselves was possibly Carl Friedrich Gauss, the founder of intrinsic differential geometry with his theorema egregium. Bernhard Riemann was the first to do extensive work that required a generalization of manifolds to higher dimensions. The name manifold comes from Riemann's original German term, Mannigfaltigkeit, which William Kingdon Clifford translates as "manifoldness". In his Göttingen inaugural lecture, Riemann states the possible values a p ...

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 - History

Charts A coordinate map, a coordinate chart, or simply a chart of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure. For a topological manifold, the simple space is some Euclidean space Rn and we are interested in the topological structure. This structure is preserved by homeomor ...

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 - Charts atlases and transition maps

The circle is the simplest example of a topological manifold after Euclidean space itself. Consider, for instance, the circle of radius 1 with its centre at the origin. If x and y are the coordinates of a point on the circle, then we have x² + y² = 1. Locally, the circle resembles a line, which is one-dimensional. In other words, only one coordinate is needed to describe the circle locally. Consider, for instance, the top part of the circle, for which the y-coordinate is positive (the yellow part i ...

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 - Motivational example: the circle

Consider a topological manifold with charts mapping to Rn. Given an ordered basis for Rn, a chart causes its piece of the manifold to itself acquire a sense of ordering, which we can think of as either right-handed or left-handed. Overlapping charts are not required to agree in their sense of ordering, which gives manifolds an important freedom. For some manifolds, like the sphere, we can choose charts so that overlapping regions agree on their "handedness"; these are orientable manifo ...

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 - Orientability

The first to have conceived clearly of curves and surfaces as spaces by themselves was possibly Carl Friedrich Gauss, the founder of intrinsic differential geometry with his theorema egregium. Bernhard Riemann was the first to do extensive work that required a generalization of manifolds to higher dimensions. The name manifold comes from Riemann's original German term, Mannigfaltigkeit, which William Kingdon Clifford translates as "manifoldness". In his Göttingen inaugural lecture, Riemann states the possible values a p ...

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 - History

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

Charts A coordinate map, a coordinate chart, or simply a chart of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure. For a topological manifold, the simple space is some Euclidean space Rn and we are interested in the topological structure. This structure is preserved by homeomor ...

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 - Charts atlases and transition maps

For more details on this topic, see differentiable manifold. It is easy to define topological manifolds, but it is very hard to work with them. For most applications a special kind of topological manifold, a differentiable manifold, works better. If the local charts on a manifold are compatible in a certain sense, one can talk about directions, tangent spaces, and differentiable functions on that manifold. In particular it is possi ...

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