# Topological manifold (manifold) The intuition for manifolds comes from the define coordinate on a curved earth using plane maps. The maps have intersection area so a point has different coordinate in each map. Doing calculus over the manifold is consistent if there is a smooth transition between the charts. More exact, an n-dimension manifold is a topological space with charts that maps open sets that cover the manifold to open sets in $\mathbb{R}^n$ homeomorphically. There is a transition map between charts that is smooth i.e., all its derivatives exist in $\mathbb{R}^n$ . Invented by Riemann, he distinguish between the concept of topology and geometry. First comes topology. Later comes the geometry which is defined using a metric over the topological space. Example: The sphere with the North-pole projection to the plane $\mathbb{R}^2$ and the South-pole projection to a plane $\mathbb{R}^2$ is a manifold. Reference: [[@The Poincare Conjecture]] ## Created 2024-09-08 12:54