No single map can cover a whole circle without tearing it
A manifold is a space that looks like ordinary flat space if you zoom in close enough on any point. Even the humble circle shows the catch: every patch can be flattened onto a stretch of line, but it can be proved that no single chart covers the whole circle, so you always need an atlas.
Formally, every point of a manifold has a neighbourhood that can be matched continuously and reversibly with an open region of n-dimensional Euclidean space. In one dimension, lines and circles pass the test, but a figure-eight fails at its crossing point. Two-dimensional manifolds are surfaces, among them the plane, the sphere, the torus, the Klein bottle and the real projective plane.
Because topology ignores bending, a small arc of circle counts the same as a short segment. Take the upper half of the unit circle: each point there has a unique x-coordinate, so projecting onto x maps the arc onto the interval from minus 1 to 1 continuously and invertibly. Such a map plus its region is a chart. Similar charts for the bottom, left and right halves together form an atlas, and where two charts overlap, a transition map converts one set of coordinates into the other; from the top chart to the right chart it sends a to the square root of 1 minus a squared.
Atlases are not unique. Two charts based on slopes of lines through the points (minus 1, 0) and (plus 1, 0) also cover the circle, each missing just its own pivot point. Gluing the ends of a single interval into a loop does not give a chart, because part of the circle then lands on both ends at once and the map stops being invertible.
The payoff is that complicated shapes can be handled through the familiar topology of simple pieces. Manifolds arise naturally as solution sets of equations and as graphs of functions, and computer graphics uses them to attach coordinates to images such as CT scans. Adding structure makes them more powerful: differentiable manifolds support calculus, a Riemannian metric allows lengths and angles, symplectic manifolds are the phase spaces of Hamiltonian mechanics, and four-dimensional Lorentzian manifolds model spacetime in general relativity. Working with them calls for a grounding in both calculus and topology.
Source: Manifold
