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- This article attempts to conveniently list articles on some of the most useful coordinate charts in some of the most useful examples of Riemannian manifolds. 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 manifold
* smooth manifold
* Riemannian manifold and semi-Riemannian manifold 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 forms, 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
* heat equation
* Laplace equation
* wave equation (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. (en)
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