Manifolds

Manifolds are spaces that, on small scales, look like ordinary Euclidean spaces. An n-dimensional manifold M is a topological space such that every point has a neighborhood that is homeomorphic to an open subset of R^n. When M is equipped with an atlas of charts whose transition maps are smooth, M is called a smooth (or C^k) manifold of dimension n; if the transition maps are merely continuous, M is a topological manifold. In the smooth setting, one requires the space to be Hausdorff and second countable, though conventions vary.

Examples include the circle S^1, the sphere S^n, the torus T^2, and real projective space RP^n. These spaces are locally like R^n but may have nontrivial global structure. A surface in R^3 defined by a smooth equation can be a 2-manifold if the gradient is nonzero at points of interest; spaces with self-intersections (such as a figure eight) fail to be manifolds at the intersection.

For smooth manifolds, one can define smooth maps f: M -> N; the derivative df_p maps T_pM to T_f(p)N. The tangent space T_pM consists of equivalence classes of curves or velocity vectors at p. The collection of all tangent spaces forms the tangent bundle TM.

Manifolds can be constructed as submanifolds of Euclidean spaces, or as quotients of manifolds by smooth group actions. Embedding and immersion theorems (Whitney) guarantee broad representability in Euclidean spaces.

Many manifolds carry additional structure, such as a Riemannian metric, which defines distances and angles, or a Lie group structure, where group operations are smooth. Manifolds provide the natural setting for differential geometry and are used throughout mathematics and theoretical physics, including general relativity and gauge theories.