List of coordinate charts
Encyclopedia
This article attempts to conveniently list articles on some of the most useful coordinate charts in some of the most useful examples of Riemannian manifold
s.
The notion of a coordinate chart is fundamental to various notions of a manifold which are used in mathematics.
In order of increasing level of structure:
For our purposes, the key feature of the last two examples is that we have defined a metric tensor
which we can use to integrate along a curve, such as a geodesic
curve. The key difference between Riemannian metrics and semi-Riemannian metrics is that the former arise from bundling
positive-definite quadratic form
s, whereas the latter arise from bundling indefinite
quadratic forms.
A four-dimensional semi-Riemannian manifold is often called a Lorentzian manifold, because these provide the mathematical setting for metric theories of gravitation such as general relativity
.
For many topics in applied mathematics
, mathematical physics
, and engineering
, it is important to be able to write the most important partial differential equations of mathematical physics
(as well as variants of this basic triad) in various coordinate systems which are adapted to any symmetries which may be present. While this may be how many students first encounter a non-Cartesian coordinate chart, such as the cylindrical chart on E3 (three dimensional Euclidean space), it turns out that these charts are useful for many other purposes, such as writing down interesting vector fields, congruences of curves, or frame fields in a convenient way.
Listing commonly encountered coordinate charts unavoidably involves some real and apparent overlap, for at least two reasons:
Therefore, seemingly any attempt to organize them into a list involves multiple overlaps, which we have accepted in this list in order to be able to offer a convenient if messy reference.
We emphasize that this list is far from exhaustive.
Here are some charts on some of the most useful Riemannian surfaces (note that there is some overlap, since many charts of S2 have closely analogous charts on H2; in such cases, both are discussed in the same article):
Favorite semi-Riemannian surface:
Note: the difference between these two surfaces is in a sense merely a matter of convention, according to whether we consider either the cyclic or the non-cyclic coordinate to be timelike; in higher dimensions the distinction is less trivial.
(Note: not every three manifold admits an isothermal chart.)
Here are some charts which can be used on some of the most useful Riemannian three-manifolds:
In addition, one can certainly consider coordinate charts on complex manifolds, perhaps with metrics which arise from bundling Hermitian forms. Indeed, this natural generalization is just the tip of iceberg. However, these generalizations are best dealt with in more specialized lists.
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
s.
The notion of a coordinate chart is fundamental to various notions of a manifold which are used in mathematics.
In order of increasing level of structure:
- topological manifoldTopological manifoldIn mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
- smooth manifold
- Riemannian manifoldRiemannian manifoldIn Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
and semi-Riemannian manifoldPseudo-Riemannian manifoldIn differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...
For our purposes, the key feature of the last two examples is that we have defined a metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
which we can use to integrate along a curve, such as a geodesic
Geodesic
In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
curve. The key difference between Riemannian metrics and semi-Riemannian metrics is that the former arise from bundling
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
positive-definite quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
s, whereas the latter arise from bundling indefinite
Indefinite
Indefinite may refer to:*In mathematics:**When talking about positive or negative indefinite forms in multilinear algebra, see definite bilinear form.**"Indefinite integral" refers to the antiderivative....
quadratic forms.
A four-dimensional semi-Riemannian manifold is often called a Lorentzian manifold, because these provide the mathematical setting for metric theories of gravitation such as general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
.
For many topics in applied mathematics
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
, mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
, and engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
, it is important to be able to write the most important partial differential equations of mathematical physics
- heat equationHeat equationThe heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...
- Laplace equation
- wave equationWave equationThe wave equation is an important second-order linear partial differential equation for the description of waves – as they occur in physics – such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics...
(as well as variants of this basic triad) in various coordinate systems which are adapted to any symmetries which may be present. While this may be how many students first encounter a non-Cartesian coordinate chart, such as the cylindrical chart on E3 (three dimensional Euclidean space), it turns out that these charts are useful for many other purposes, such as writing down interesting vector fields, congruences of curves, or frame fields in a convenient way.
Listing commonly encountered coordinate charts unavoidably involves some real and apparent overlap, for at least two reasons:
- many charts exist in all (sufficiently large) dimensions, but perhaps only for certain families of manifolds such as spheres,
- many charts most commonly encountered for specific manifolds, such as spheres ,actually can be used (with an appropriate metric tensor) for more general manifolds, such as spherically symmetric manifolds.
Therefore, seemingly any attempt to organize them into a list involves multiple overlaps, which we have accepted in this list in order to be able to offer a convenient if messy reference.
We emphasize that this list is far from exhaustive.
Favorite surfaces
Here are some charts which (with appropriate metric tensors) can be used in the stated classes of Riemannian and semi-Riemannian surfaces:- isothermal chart
- Radially symmetric surfaces:
- polar chart
- Surfaces embedded in E3:
- Monge chart
- Certain minimal surfaceMinimal surfaceIn mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....
s:- asymptotic chart (see also asymptotic line)
Here are some charts on some of the most useful Riemannian surfaces (note that there is some overlap, since many charts of S2 have closely analogous charts on H2; in such cases, both are discussed in the same article):
- Euclidean plane E2:
- Cartesian chart
- Maxwell chart
- Sphere S2:
- polar chart (arc length radial chart)
- stereographic chart
- central projection chart
- axial projection chart
- Mercator chart
- Hyperbolic plane H2:
- polar chart
- stereographic chart (Poincaré model)
- upper half space chart (another Poincaré model)
- central projection chart (Klein model)
- Mercator chart
Favorite semi-Riemannian surface:
- AdS2 (or S1,1) and dS2 (or H1,1):
- central projection
- equatorial trig
Note: the difference between these two surfaces is in a sense merely a matter of convention, according to whether we consider either the cyclic or the non-cyclic coordinate to be timelike; in higher dimensions the distinction is less trivial.
Favorite Riemannian three-manifolds
Here are some charts which (with appropriate metric tensors) can be used in the stated classes of three-dimensional Riemannian manifolds:- Diagonalizable manifolds:
- isothermal chart
(Note: not every three manifold admits an isothermal chart.)
- Axially symmetric manifolds:
- cylindrical chart
- parabolic chart
- hyperbolic chart
- prolate spheroidal chart (rational and trigonometric forms)
- oblate spheroidal chart (rational and trigonometric forms)
- toroidal chart
Here are some charts which can be used on some of the most useful Riemannian three-manifolds:
- Three-dimensional Euclidean space E3:
- cartesian
- polar spherical chart
- cylindrical chart
- elliptical cylindrical, hyperbolic cylindrical, parabolic cylindrical charts
- parabolic chart
- hyperbolic chart
- prolate spheroidal chart (rational and trigonometric forms)
- oblate spheroidal chart (rational and trigonometric forms)
- toroidal chart
- Cassini toroidal chart and Cassini bipolar chart
- Three-sphere S3
- polar chart
- stereographic chart
- Hopf chart
- Hyperbolic three-space H3
- polar chart
- upper half space chart (Poincaré model)
- Hopf chart
A few higher dimensional examples
- Sn
- Hopf chart
- Hn
- upper half space chart (Poincaré model)
- Hopf chart
Omitted examples
There are of course many important and interesting examples of Riemannian and semi-Riemannian manifolds which are not even mentioned here, including:- Bianchi groupBianchi classificationIn mathematics, the Bianchi classification, named for Luigi Bianchi, is a classification of the 3-dimensional real Lie algebras into 11 classes, 9 of which are single groups and two of which have a continuum of isomorphism classes...
s: there is a short list (up to local isometry) of three-dimensional real Lie groups, which when considered as Riemannian-three manifolds give homogeneous but (usually) non-isotropic geometries. - other noteworthy real Lie groupLie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s, - Lorentzian manifolds which (perhaps with some added structure such as a scalar field) serve as solutions to the field equations of various metric theories of gravitation, in particular general relativityGeneral relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
. There is some overlap here; in particular: - axisymmetric spacetimes such as Weyl vacuums possess various charts discussed here; the prolate spheroidal chart turns out to be particularly useful,
- de Sitter models in cosmology are, as manifolds, nothing other than H1,3 and as such possess numerous interesting and useful charts modeled after ones listed here.
In addition, one can certainly consider coordinate charts on complex manifolds, perhaps with metrics which arise from bundling Hermitian forms. Indeed, this natural generalization is just the tip of iceberg. However, these generalizations are best dealt with in more specialized lists.
See also
- metric tensorMetric tensorIn the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
- List of mathematics lists, particularly: