Manifold

This is a proposed rewrite of manifold. Of course, all the deleted parts have to be found a home in some other article, such as Manifold/rewrite/topological manifold or Manifold/rewrite/differentiable manifold

Target audience (roughly):

• For lead section and examples: high school.
• For Topological manifolds and Smooth manifolds: first-year or second-year undergraduates.
• For the rest: graduates.

You are welcome to edit it.

---

For other meanings of this term, see manifold (disambiguation).

In mathematics, a manifold is a space which looks in a close-up view like a specific simple space. For instance, the earth looks flat when you are standing on its surface. So, a sphere looks locally like a plane, which makes it a manifold. However, the global structure is quite different: if you walk over the surface of a sphere in a fixed direction, you eventually return to your starting point.

We navigate on the earth using flat maps, collected in an atlas. Similarly, we can describe a manifold using mathematical maps, called coordinate charts, collected in a mathematical atlas. Unfortunately, it is generally not possible to describe the manifold with just one chart. When using multiple charts which together cover the manifold, we need to pay attention to the regions where they overlap. A point in such a region will be represented by more than one chart, and this is where the name manifold comes from.

There are many different kinds of manifolds. The simplest are topological manifolds, which look locally like some Euclidean space. If the work manifold is used without qualification, then most likely a differentiable manifold is meant. Other types include algebraic varieties and schemes.

The circle is the simplest example of a topological manifold after the Euclidean space itself. We can view the circle as a subset of the plane:

${\displaystyle \Gamma :=\{(x,y)\in \mathbf {R} ^{2}|x^{2}+y^{2}=1\}.}$

But locally, the circle looks just like a line, which is one-dimensional. In other words, we need only one coordinate to describe the circle locally. Consider for instance the part of the circle with positive y-coordinate, the red part in the figure to the right. Any point in this part can be described by the x-coordinate. So, there is a bijection χ_red, which maps the red part of the circle to the interval ]-1,1[ by simply projecting onto the second coordinate. Such a function is called a coordinate chart, a coordinate map or simply a chart. Similarly, there are charts for the blue, green and brown parts of the circle. Together, these parts cover the whole circle and we say that the four charts form an atlas for the circle. Thus we can define a manifold by specifying an atlas. Different atlases can define the same manifold. Such atlases are called compatible.

Note that the red and the blue part overlap. Their intersection is the part Γ₊₊ of the circle with positive coordinates. The two charts χ_red and χ_blue map Γ₊₊ bijectively to the interval ]0, 1[. Thus we can form a function from ]0, 1[ to itself by first following the red chart to the circle and then the blue chart back to the interval. Such a function is called a transition function, a transition map, a change of coordinates or a coordinate transformation. Since we know what it means for functions between subsets of Euclidean space to be differentiable, we know what it means for transition maps to be differentiable. We can define special kinds of topological manifolds by putting restrictions on the transition maps of an atlas.

[Explain coordinate transformations]

[Other examples need to be added, e.g., sphere, torus. Perhaps mention manifold as configuration space or phase space as application? Perhaps include a simple pathological example, like the line with two origins?]

The n-sphere Sⁿ is the set

${\displaystyle \mathbf {S} ^{n}:=\{x\in \mathbf {R} ^{n+1}:\|x\|=1\}.}$

We can turn it into a topological manifold by covering it with the patches {U_i}_{0 ≤ i ≤ n}.

${\displaystyle U_{i}:=\{x\in \mathbf {S} ^{n}:x_{i}>0\}\quad V_{i}:=\{x\in \mathbf {S} ^{n}:x_{i}<0\}}$

Then we can simply project out one direction to get the following atlas. Define

${\displaystyle \{\chi _{i}:U_{i}\to \mathbf {R} ^{n}:x=(x_{0},\ldots ,x_{n})\mapsto (x_{0},\ldots ,{\hat {x}}_{i},\ldots ,x_{n})\}}$

The hat means omit that component

Alternatively it may be constructed by gluing: The n-sphere Sⁿ is constructed by gluing together two copies of Rⁿ. Define

${\displaystyle \mathbf {R} ^{n}\setminus \{0\}\to \mathbf {R} ^{n}\setminus \{0\}:x\mapsto 1/x}$

The cylinder S×R is constructed by gluing together two copies of R². Define

${\displaystyle (\mathbf {R} ^{2})_{\mathrm {L} }:=\{x\in \mathbf {R} ^{2}:x_{1}<0\}\quad (\mathbf {R} ^{2})_{\mathrm {R} }:=\{x\in \mathbf {R} ^{2}:x_{1}>0\}}$

${\displaystyle (\mathbf {R} ^{2})_{\mathrm {L} }\to (\mathbf {R} ^{2})_{\mathrm {L} }:x=(x_{1},x_{2})\mapsto (1/x_{1},x_{2})}$

${\displaystyle (\mathbf {R} ^{2})_{\mathrm {R} }\to (\mathbf {R} ^{2})_{\mathrm {R} }:x=(x_{1},x_{2})\mapsto (1/x_{1},x_{2})}$

The n-torus Tⁿ is the product of n circles S¹.

The Möbius band M is constructed by gluing together two copies of R². Define

${\displaystyle (\mathbf {R} ^{2})_{\mathrm {L} }:=\{x\in \mathbf {R} ^{2}:x_{1}<0\}\quad (\mathbf {R} ^{2})_{\mathrm {R} }:=\{x\in \mathbf {R} ^{2}:x_{1}>0\}}$

${\displaystyle (\mathbf {R} ^{2})_{\mathrm {L} }\to (\mathbf {R} ^{2})_{\mathrm {L} }:x=(x_{1},x_{2})\mapsto (1/x_{1},x_{2})}$

${\displaystyle (\mathbf {R} ^{2})_{\mathrm {R} }\to (\mathbf {R} ^{2})_{\mathrm {R} }:x=(x_{1},x_{2})\mapsto (1/x_{1},-x_{2})}$

The Klein bottle K. Compared with torus an extra twist while gluing a cylinder.

Every real manifold can be embedded in some Euclidean space. This is the extrinsic view. When a manifold is viewed in this way, it is easy to use intuition from Euclidean spaces to define additional structure. For example, in an Euclidean space it is always clear whether a vector at some point is tangential or normal to some surface through that point. When we view a manifold simply as a topological space without any embedding, then it is much harder to imagine what a tangent vector might be. This is the intrinsic view.

If you imagine yourself, or an ant, within a certain manifold, say the surface of Earth, you have the intrinsic view. When you step outside of the manifold, say by getting into a rocket and flying into space, and then look back at the ant, you have the extrinsic view.

The circle can be defined intrisically by gluing together two copies of the line. We do this by identifying non-zero points in the first copy by their multiplicative inverse in the second copy. This circle is not embedded in anything.

[Refer back to examples]

The simplest kind of manifold to define is the topological manifold. A topological manifold is a topological space that looks locally like the "ordinary" Euclidean space Rⁿ. Usually additional technical requirements are made, such as Hausdorffness and second countability. To make precise the notion of "looks locally like" one uses local coordinate systems or charts. A connected manifold has a definite topological dimension, which equals the number of coordinates needed in each local coordinate system.

...

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 smooth manifold, works better. In particular it is possible to apply "calculus" on a smooth manifold.

We start with a topological manifold M. ...

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. These manifolds are called differentiable.

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.

To allow for infinite dimensions, one may consider Banach manifolds which locally look like Banach spaces.

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.

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.

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

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.

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 really required a generalization of manifolds to higher dimensions. Abelian varieties were at that time already implicitly known, as complex manifolds. Lagrangian mechanics and Hamiltonian mechanics, when considered geometrically, are also naturally manifold theories.

...

The foundational aspects of the subject were clarified during the 1930s, making precise intuitions dating back to the latter half of the 19th century, and developed through differential geometry and Lie group theory.

• algebraic variety
• scheme
• list of manifolds
• surface
• 3-manifold
• 4-manifold

• Guillemin, Victor and Anton Pollack, Differential Topology, Prentice-Hall (1974) ISBN 0132126052. This text was inspired by Milnor, and is commonly used for undergraduate courses.
• Hirsch, Morris, Differential Topology, Springer (1997) ISBN 0387901485. Hirsch gives the most complete account with historical insights and excellent, but difficult problems. This is the best reference for those wishing to have a deep understanding of the subject.
• Kirby, Robion C.; Siebenmann, Laurence C. Foundational Essays on Topological Manifolds. Smoothings, and Triangulations. Princeton, New Jersey: Princeton University Press (1977). ISBN 0-691-08190-5. A detailed study of the category of topological manifolds.
• Lee, John M. Introduction to Topological Manifolds, Springer-Verlag, New York (2000). ISBN 0-387-98759-2. Introduction to Smooth Manifolds, Springer-Verlag, New York (2003). ISBN 0-387-95495-3. Graduate-level textbooks on topological and smooth manifolds.
• Milnor, John, Topology from the Differentiable Viewpoint, Princeton University Press, (revised, 1997) ISBN 0691048339. This short text may be the best math book ever written.
• Spivak, Michael, Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. HarperCollins Publishers (1965). ISBN 0805390219. This is the standard text used in most graduate courses.