Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to po ...

, a Bryant surface is a 2-dimensional surface embedded in 3-dimensional

hyperbolic space In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. ...

with constant

mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. T ...

equal to 1. These surfaces take their name from the geometer Robert Bryant, who proved that every

simply-connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space ...

minimal surface in 3-dimensional

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...

isometric The term ''isometric'' comes from the Greek for "having equal measurement". isometric may mean: * Cubic crystal system, also called isometric crystal system * Isometre, a rhythmic technique in music. * "Isometric (Intro)", a song by Madeon from ...

to a Bryant surface by a holomorphic parameterization analogous to the (Euclidean) Weierstrass–Enneper parameterization..

References

Hyperbolic geometry Riemannian geometry Minimal surfaces {{differential-geometry-stub