Mannigfaltigen

Mannigfaltigen, the plural form of Mannigfaltigkeit in German, corresponds to the mathematical concept known in English as manifolds. In mathematics, a manifold is a topological space that locally resembles Euclidean space. More precisely, each point has a neighborhood that is homeomorphic to an open subset of R^n for some fixed n, which is called the dimension of the manifold. An atlas is a collection of charts that map the open parts of the manifold to open subsets of R^n, and the charts in an atlas are compatible on overlaps via transition maps.

A manifold may carry additional structures. A topological manifold is simply a manifold with this local Euclidean property. A differentiable (or smooth) manifold requires the transition maps to be differentiable. Real-analytic and complex manifolds impose stronger analyticity or holomorphicity conditions. Riemannian manifolds are equipped with a metric tensor that defines lengths and angles.

Common examples include the circle S^1, the sphere S^2, the torus T^2, and Euclidean space R^n. More advanced examples include projective spaces and complex manifolds such as complex projective spaces CP^n. In many contexts, separability by second countability and the Hausdorff property is assumed to avoid pathological cases.

Manifolds form a central object of study in differential geometry, topology, and mathematical physics, providing an intrinsic language for geometry independent of any ambient space. They also underpin theories in general relativity and gauge theories, where spacetime and field configurations are modeled as manifolds with extra structure.