Riemann surface

A connected one-dimensional complex manifold that provides a domain on which complex-valued functions can be studied as single-valued functions.

Complex manifold: A space locally modeled on complex coordinate spaces, with holomorphic changes of coordinates. A Riemann surface is the one-dimensional case of this broader geometric structure.
Riemann surface of the square root: The two-sheeted complex surface defined by the relation w²=z. Its sheets join at the branch point and make the square root single-valued.
Algebraic curve: A one-dimensional algebraic variety defined by polynomial equations. A nonsingular complex algebraic curve naturally carries the structure of a compact Riemann surface.
Riemannian manifold: A smooth manifold equipped with a smoothly varying inner product on its tangent spaces. Its metric geometry differs from the conformal structure that defines a Riemann surface.
Holomorphic function: A complex-valued function that is complex differentiable at every point of its domain. Holomorphic functions are the central objects defined and studied on Riemann surfaces.
Branched covering: A map that is locally a covering except at isolated branch points. Projection to the complex plane often presents a Riemann surface as a branched covering.
Elliptic function: A meromorphic function with two independent complex periods. Its periods identify opposite edges of a parallelogram, producing a torus-shaped Riemann surface.
Topological surface: A two-dimensional topological manifold, locally resembling the plane. A Riemann surface adds complex coordinates to a topological surface rather than merely its topology.
Analytic continuation: The extension of a holomorphic function beyond its initial domain while preserving its local values. Continuing a function around loops reveals why its natural domain may need multiple sheets.
Branch point: A point around which analytic continuation can permute the local branches of a function. Branch points mark where sheets of a function's surface meet under projection.

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