Circle
After a line, a circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated the same as a small piece of a line.

The spherical Earth is navigated using flat maps or charts, collected in an atlas. Similarly, a differentiable manifold can be described using mathematical maps, called coordinate charts, collected in a mathematical atlas. It is not generally possible to describe a

A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly different viewpoint.
Charts
Perhaps the simplest way to construct a manifold is the one used

Topological manifolds
The simplest kind of manifold to define is the topological manifold, which looks locally like some "ordinary" Euclidean space . By

Informally, a manifold is a space that is "modeled on" Euclidean space.
There are many different kinds of manifolds. In

A manifold with boundary is a manifold with an edge. For example, a sheet of paper is a 2-manifold with a 1-dimensional boundary. The boundary

The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and