A *manifold* is a generalization of the idea of a smooth surface embedded in Euclidean space. The critical feature of an n-dimensional manifold is that locally (near any point), it looks like n-dimensional Euclidean space.

## Parts of a Manifold

Around every point on the manifold, is a “simply-connected open set” called the *coordinate patch*, that maps every point to a tuple of n real numbers, and a one-to-one continuous function, the *coordinate function* or *chart*, mapping every point in that open set to a tuple of n real numbers, the *coordinates*.

A consistent system of coordinate patches and coordinate functions that covers the entire manifold is called an *atlas*.

## Examples

An example of a two-dimensional manifold is the surface of a sphere or of a coffee cup. The space of all configurations of a double pendulum is a more abstract example (and has the shape of a torus).

An example of a coordinate function is mapping of points on a sphere to the tuple of latitude and longitude. It is to be noted that the “simply connected open set” or “patch” here does not include either poles (as longitude is undefined at the poles ) or the 180 deg meridian (due to discontinuity in longitude). These places will have to be covered by other coordinate systems.

[1]: *Functional Differential Geometry*, Sussman and Wisdom