Manifold

A manifold is an n-dimensional space whose points each have a neighbourhood homeomorphic to ℝⁿ — locally, the manifold looks like ordinary flat Euclidean space, even when its global shape is curved or topologically non-trivial. The 2-sphere is the prototypical example: a small patch around any point on the sphere can be flattened onto a piece of paper without much distortion, even though the sphere as a whole cannot. The notion of "locally Euclidean" is made precise by demanding that the manifold be covered by an atlas of overlapping coordinate charts whose pairwise transition maps are continuous; a smooth manifold strengthens this by requiring the transition maps to be infinitely differentiable, which is the condition that lets one do calculus on the manifold without specifying any particular embedding.

In general relativity the spacetime arena is a four-dimensional smooth manifold. Tensors live at each point on this manifold, and the metric tensor g_{μν}(x) — which determines lengths, angles, and proper times — varies smoothly across it. Crucially, the manifold is the abstract underlying object, defined without reference to any larger space it might be sitting inside; intrinsic curvature, parallel transport, and geodesic flow can all be defined without ever invoking an embedding. This intrinsic point of view, due to Riemann's 1854 habilitation lecture, is what allowed Einstein to formulate gravity as the geometry of spacetime rather than a force acting through some background space.