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Definitions and first examples A (topological) surface with boundary is a Hausdorff topological space in which every point has an open neighbourhood homeomorphic to some open subset of the closed half space of E2 (Euclidean 2-space). The neighborhood, along with the homeomorphism to Euclidean space, is called a (coordinate) chart. The set of points that have an open neighbourhood homeomorphic to E2 is called the interior of the surface; it is always non-empty. The complement of the interior is called the boundary; it is a one-manifold, or union of closed curves. The simplest example of a surface with boundary is the closed disk in E2; its boundary is a circle. A surface with an empty boundary is called boundaryless. (Sometimes the word surface, used alone, refers only to boundaryless surfaces.) A closed surface is one that is boundaryless and compact. The two-dimensional sphere, the two-dimensional torus, and the real projective plane are examples of closed surfaces. The Möbius strip is a surface with only one "side". In general, a surface is said to be orientable if it does not contain a homeomorphic copy of the Möbius strip; intuitively, it has two distinct "sides". For example, the sphere and torus are orientable, while the real projective plane is not (because deleting a point or disk from the real projective plane produces the Möbius strip). More generally, it is common in differential and algebraic geometry to study surfaces with singularities, such as self-intersections, cusps, etc. Extrinsically defined surfaces and embeddings
Construction from polygons Each closed surface can be constructed from an oriented polygon with an even number of sides, called a fundamental polygon of the surface, by pairwise identification of its edges. For example, in each polygon below, attaching the sides with matching labels (A with A, B with B), so that the arrows point in the same direction, yields the indicated surface. Image:SphereAsSquare.svg|sphere Image:ProjectivePlaneAsSquare.svg|real projective plane Image:TorusAsSquare.svg|torus Image:KleinBottleAsSquare.svg|Klein bottle Any fundamental polygon can be written symbolically as follows. Begin at any vertex, and proceed around the perimeter of the polygon in either direction until returning to the starting vertex. During this traversal, record the label on each edge in order, with an exponent of -1 if the edge points opposite to the direction of traversal. The four models above, when traversed clockwise starting at the upper left, yield The expression thus derived from a fundamental polygon of a surface turns out to be the sole relation in a presentation of the fundamental group of the surface with the polygon edge labels as generators. This is a consequence of the Seifert–van Kampen theorem. Quotients and connected summation Gluing edges of polygons is a special kind of quotient space process. The quotient concept can be applied in greater generality to produce new or alternative constructions of surfaces. For example, the real projective plane can be obtained as the quotient of the sphere by identifying all pairs of opposite points on the sphere. Another example of a quotient is the connected sum. The connected sum of two surfaces M and N, denoted M . The sphere S is an identity element for the connected sum, meaning that S Connected summation with the torus T has the effect of attaching a "handle" to the other summand M. If M is orientable, then so is T The connected sum of two real projective planes is the Klein bottle. The connected sum of the real projective plane and the Klein bottle is homeomorphic to the connected sum of the real projective plane with the torus. Any connected sum involving a real projective plane is nonorientable. Classification of closed surfaces The classification of closed surfaces states that any closed surface is homeomorphic to some member of one of these three families: The surfaces in the first two families are orientable. It is convenient to combine the two families by regarding the sphere as the connected sum of 0 tori. The number g of tori involved is called the genus of the surface. Since the sphere and the torus have Euler characteristics 2 and 0, respectively, it follows that the Euler characteristic of the connected sum of g tori is 2 - 2g. The surfaces in the third family are nonorientable. Since the Euler characteristic of the real projective plane is 1, the Euler characteristic of the connected sum of k of them is 2 - k. It follows that a closed surface is determined, up to homeomorphism, by two pieces of information: its Euler characteristic, and whether it is orientable or not. In other words, Euler characteristic and orientability completely classify closed surfaces up to homeomorphism. Surfaces in differential geometry It is also possible to define smooth surfaces, in which each point has a neighborhood diffeomorphic to some open set in E2. This elaboration allows calculus to be applied to surfaces to prove many results. Smooth surfaces equipped with Riemannian metrics are of foundational importance in differential geometry. Minimal surfaces are surfaces that minimize the surface area for given boundary conditions. Examples include soap films stretched across a wire frame, catenoids and helicoids. Developable surfaces are surfaces that can be flattened to a plane without stretching; examples include the cylinder, the cone, and the torus. Ruled surfaces are surfaces that have at least one straight line running through every point. Examples include the cylinder and the hyperboloid of one sheet. Surfaces in complex and algebraic geometry Any n-dimensional complex manifold is, at the same time, a real (2n)-dimensional real manifold. Thus any complex one-manifold (also called a Riemann surface) is a topological surface. Any complex algebraic curve or real algebraic surface is also a topological surface, possibly with singularities. A complex surface is a complex two-manifold and thus a real four-manifold; it is not a surface in the sense of this article. Neither are algebraic curves or surfaces defined over fields other than the complex numbers. See also | |||||||||||
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