In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional...

below. Classification of Euclidean plane isometries Classification theorems of surfaces Classification of two-dimensional closed manifolds – Two-dimensional...

Enriques surfaces in characteristic 2, and quasi-hyperelliptic surfaces in characteristics 2 and 3. The Enriques–Kodaira classification of compact complex...

surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can...

uniformization theorem states that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk...

of notable theorems. Lists of theorems and similar statements include: List of algebras List of algorithms List of axioms List of conjectures List of...

mathematics, Thurston's classification theorem characterizes homeomorphisms of a compact orientable surface. William Thurston's theorem completes the work...

A Guide to the Classification Theorem for Compact Surfaces is a textbook in topology, on the classification of two-dimensional surfaces. It was written...

genus of such a surface is g. A genus g surface is a two-dimensional manifold. The classification theorem for surfaces states that every compact connected...

algebraic surfaces include (κ is the Kodaira dimension): κ = −∞: the projective plane, quadrics in P3, cubic surfaces, Veronese surface, del Pezzo surfaces, ruled...

torus are also occasionally used. The classification theorem for surfaces states that every compact connected surface is topologically equivalent to either...

a classification: explicitly, by an enumeration, or implicitly, in terms of invariants. For instance, for orientable surfaces, the classification of surfaces...

Other important examples of Kähler manifolds include Riemann surfaces, K3 surfaces, and Calabi–Yau manifolds. Serre's GAGA theorem asserts that projective...

bundle) of curves and surfaces help with the classification of complex manifolds, e.g. Enriques–Kodaira classification. Kawamata–Viehweg vanishing theorem Mumford...

self-intersections. The classification theorem of closed surfaces states that any connected closed surface is homeomorphic to some member of one of these three families:...

to find a measure-classification theorem similar to Ratner's theorems but for diagonalizable actions, motivated by conjectures of Furstenberg and Margulis...

embedding theorems. They state that every Riemannian manifold can be isometrically embedded in a Euclidean space Rn. In all of the following theorems we assume...

conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple...

quasi-elliptic surfaces in characteristics two and three. These are surfaces fibred over a curve where the general fibre is a curve of arithmetic genus...

recourse to the classification of compact complex surfaces, that every compact Kähler surface is a deformation of a projective Kähler surface. This was later...

invariants of smooth spaces. Famous theorems in differential topology include the Whitney embedding theorem, the hairy ball theorem, the Hopf theorem, the Poincaré–Hopf...

Smith, Corti (2004), Theorems 2.1 and 2.2. Bruce, J. W.; Wall, C. T. C. (1979). "On the Classification of Cubic Surfaces". Journal of the London Mathematical...

This is a list of named algebraic surfaces, compact complex surfaces, and families thereof, sorted according to their Kodaira dimension following Enriques–Kodaira...

Minimal surfaces of class VII (those with no rational curves with self-intersection −1) are called surfaces of class VII0. Every class VII surface is birational...

glued together. The main point of Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the...

surfaces are the simplest of the 10 or so classes of surface in the Enriques–Kodaira classification of complex surfaces, and were the first surfaces to...

birational geometry of surfaces studied by the Italian school, and is currently an active research area within algebraic geometry. The basic idea of the theory...

uniqueness of minimal surfaces of finite genus in R 3 {\displaystyle \mathbb {R} ^{3}} . The final topological classification of embedded minimal surfaces in...

(2d − 1)(g − 1) for all d ≥ 2. Compare with the Uniformization theorem for surfaces (real surfaces, since a complex curve has real dimension 2): Kodaira dimension...

defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids...