An open surface with X-, Y-, and Z-contours shown. In Mathematics, the differential geometry of surfaces deals with smooth Surfaces with various additional structures most often a Riemannian metric

In mathematics, specifically in topology, a surface is a two-dimensional manifold. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space, E³. On the other hand, there are also more exotic surfaces, that are so "contorted" that they cannot be embedded in three-dimensional space at all. In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group

To say that a surface is "two-dimensional" means that, about each point, there is a coordinate patch on which a two-dimensional coordinate system is defined. In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point For example, the surface of the Earth is (ideally) a two-dimensional sphere, and latitude and longitude provide coordinates on it — except at the International Date Line and the poles, where longitude is undefined. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe Latitude, usually denoted symbolically by the Greek letter phi ( Φ) gives the location of a place on Earth (or other planetary body north or south of the Longitude (ˈlɒndʒɪˌtjuːd or ˈlɒŋgɪˌtjuːd symbolized by the Greek character Lambda (λ is the east-west Geographic coordinate measurement This example illustrates that in general it is not possible to extend any one coordinate patch to the entire surface; surfaces, like manifolds of all dimensions, are usually constructed by patching together multiple coordinate systems.

Surfaces find application in physics, engineering, computer graphics, and many other disciplines, primarily when they represent the surfaces of physical objects. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data For example, in analyzing the aerodynamic properties of an airplane, the central consideration is the flow of air along its surface. Overview Fixed-wing aircraft range from small training and recreational aircraft to Wide-body aircraft and military cargo aircraft.

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). In Topology and related branches of Mathematics, a Hausdorff space, separated space or T2 space is a Topological space Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Topology and related areas of Mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a Topological space. Topological equivalence redirects here see also Topological equivalence (dynamical systems. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in 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. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members The complement of the interior is called the boundary; it is a one-manifold, or union of closed curves. In Discrete mathematics and predominantly in Set theory, a complement is a concept used in comparisons of sets to refer to the unique values of one set in relation The simplest example of a surface with boundary is the closed disk in E²; its boundary is a circle. In Geometry, a disk (also spelled disc) is the region in a plane bounded by 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. In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar Construction Consider a Sphere, and let the Great circles of the sphere be "lines" and let pairs of Antipodal points be "points"

The Möbius strip is a surface with only one "side". This article is about the mathematical object See Mobius Band (music group for the music group 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. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Mathematics, a singular point of an Algebraic variety V is a point P that is 'special' (so singular in the geometric sense that V

Extrinsically defined surfaces and embeddings

A sphere can be defined parametrically (by x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ) or implicitly (by x² + y² + z² − r² = 0. )

Historically, surfaces were originally defined and constructed not using the abstract, intrinsic definition given above, but extrinsically, as subsets of Euclidean spaces such as E³.

Let f be a continuous, injective function from R² to R³. Then the image of f is said to be a parametric surface. In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage A parametric surface is a Surface in the Euclidean space R 3 which is defined by a Parametric equation with two parameters A surface of revolution can be viewed as a special kind of parametric surface. A surface of revolution is a Surface created by rotating a Curve lying on some plane (the Generatrix) around a Straight line (the Axis

On the other hand, suppose that f is a smooth function from R³ to R whose gradient is nowhere zero. In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar Then the locus of zeros of f is said to be an implicit surface. In Mathematics, a locus ( Latin for "place" plural loci) is a collection of points which share a property This article is about the zeros of a function which should not be confused with the value at zero. In Mathematics, a level set of a real -valued function f of n variables is a set of the form { ( x 1 If the condition of non-vanishing gradient is dropped then the zero locus may develop singularities.

One can also define parametric and implicit surfaces in higher-dimensional Euclidean spaces Eⁿ. It is natural to ask whether all surfaces (defined abstractly, as in the preceding section) arise as subsets of some Eⁿ. The answer is yes; the Whitney embedding theorem, in the case of surfaces, states that any surface can be embedded homeomorphically into E⁴. In Mathematics, particularly in Differential topology,there are two Whitney embedding theorems The strong Whitney embedding theorem states that any Therefore the extrinsic and intrinsic approaches turn out to be equivalent.

In fact, any compact surface that is either orientable or has a boundary can be embedded in E³; on the other hand, the real projective plane, which is compact, non-orientable and without boundary, cannot be embedded into E³ (see Gramain). Steiner surfaces, including Boy's surface, the Roman surface and the cross-cap, are immersions of the real projective plane into E³. In Geometry, the Steiner surfaces, discovered by Jakob Steiner, are representations of the real Projective plane in three-dimensional space In Geometry, Boy's surface is an immersion of the Real projective plane in 3-dimensional space found by Werner Boy in 1901 (he discovered it The Roman surface (so called because Jakob Steiner was in Rome when he thought of it is a self-intersecting mapping of the Real projective plane into In Mathematics, a cross-cap is a two-dimensional surface that is topologically equivalent to a Möbius strip. In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group These surfaces are singular where the immersions intersect themselves.

The Alexander horned sphere is a well-known pathological embedding of the two-sphere into the three-sphere. The Alexander horned sphere is one of the most famous pathological examples in Mathematics discovered in 1924 by J In Mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive

A knotted torus.

The chosen embedding (if any) of a surface into another space is regarded as extrinsic information; it is not essential to the surface itself. For example, a torus can be embedded into E³ in the "standard" manner (that looks like a bagel) or in a knotted manner (see figure). A bagel is a bread product, traditionally made of Yeasted Wheat dough in the form of a roughly hand-sized ring which is first boiled in water and then baked In Mathematics, a knot is an Embedding of a Circle in 3-dimensional Euclidean space, R 3 considered up to continuous deformations The two embedded tori are homeomorphic but not isotopic; they are topologically equivalent, but their embeddings are not. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical

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. In Mathematics, each closed Surface in the sense of Geometric topology can be constructed from an even-sided oriented Polygon, called a fundamental 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.

sphere

real projective plane

torus

Klein bottle

Any fundamental polygon can be written symbolically as follows. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe Construction Consider a Sphere, and let the Great circles of the sphere be "lines" and let pairs of Antipodal points be "points" In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar In Mathematics, the Klein bottle is a certain non- orientable Surface, i 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

• sphere: ABB ^{− 1}A ^{− 1}
• real projective plane: ABAB
• torus: ABA ^{− 1}B ^{− 1}
• Klein bottle: ABAB ^{− 1}.

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. In Mathematics, one method of defining a group is by a presentation. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. This is a consequence of the Seifert–van Kampen theorem. In Mathematics, the Seifert – van Kampen theorem of Algebraic topology, sometimes just called van Kampen's theorem, expresses the

Quotients and connected summation

Gluing edges of polygons is a special kind of quotient space process. In Topology and related areas of Mathematics, a quotient space (also called an identification space) is intuitively speaking the result of identifying 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 # N, is obtained by removing a disk from each of them and gluing them along the boundary components that result. In Mathematics, specifically in Topology, the operation of connected sum is a geometric modification on Manifolds Its effect is to join two given manifolds The Euler characteristic χ of M # N is the sum of the Euler characteristics of the summands, minus two:

The sphere S is an identity element for the connected sum, meaning that S # M = M. In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that This is because deleting a disk from the sphere leaves a disk, which simply replaces the disk deleted from M upon gluing.

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 # M. The connected sum can be iterated to attach any number g of handles to M.

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:

1. the sphere;
2. the connected sum of g tori, for ;
3. the connected sum of k real projective planes, for .

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 geometry

Main article: Differential geometry of surfaces

Polyhedra, such as the boundary of a cube, are among the first surfaces encountered in geometry. In Mathematics, the differential geometry of surfaces deals with smooth Surfaces with various additional structures most often a Riemannian metric A cube is a three-dimensional solid object bounded by six square faces facets or sides with three meeting at each vertex. It is also possible to define smooth surfaces, in which each point has a neighborhood diffeomorphic to some open set in E². In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable This elaboration allows calculus to be applied to surfaces to prove many results. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives

Two smooth surfaces are diffeomorphic if and only if they are homeomorphic. (The analogous result does not hold for higher-dimensional manifolds. ) Thus closed surfaces are classified up to diffeomorphism by their Euler characteristic and orientability. In Mathematics a closed surface (2- Manifold) is a Closed manifold of dimension two with a single Connected component.

Smooth surfaces equipped with Riemannian metrics are of foundational importance in differential geometry. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry A Riemannian metric endows a surface with notions of geodesic, distance, angle, and area. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces Distance is a numerical description of how far apart objects are In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called It also gives rise to Gaussian curvature, which describes how curved or bent the surface is at each point. In Differential geometry, the Gaussian curvature or Gauss curvature of a point on a Surface is the product of the Principal curvatures Curvature is a rigid, geometric property, in that it is not preserved by general diffeomorphisms of the surface. However, the famous Gauss-Bonnet theorem for closed surfaces states that the integral of the Gaussian curvature K over the entire surface S is determined by the Euler characteristic:

This result exemplifies the deep relationship between the geometry and topology of surfaces (and, to a lesser extent, higher-dimensional manifolds). The Gauss–Bonnet theorem or Gauss–Bonnet formula in Differential geometry is an important statement about Surfaces which connects their geometry (in

Another way in which surfaces arise in geometry is by passing into the complex domain. A complex one-manifold is a smooth oriented surface, also called a Riemann surface. In Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional Any complex nonsingular algebraic curve viewed as a real manifold is a Riemann surface. In Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one

Every closed surface admits complex structures. Complex structures on a closed oriented surface correspond to conformal equivalence classes of Riemannian metrics on the surface. In Mathematics and Theoretical physics, two geometries are conformally equivalent if there exists a Conformal transformation (an angle-preserving transformation One version of the uniformization theorem (due to Poincaré) states that any Riemannian metric on an oriented, closed surface is conformally equivalent to an essentially unique metric of constant curvature. In Mathematics, the uniformization theorem for Surfaces says that any surface admits a Riemannian metric of constant Gaussian curvature. Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M See also Space form In Mathematics, constant curvature in Differential geometry is a concept most commonly applied to Surfaces For This provides a starting point for one of the approaches to Teichmüller theory, which provides a finer classification of Riemann surfaces than the topological one by Euler characteristic alone. In Mathematics, given a Riemann surface X, the Teichmüller space of X, notated TX or Teich( X) is a complex

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. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

See also

• Volume form, for volumes of surfaces in Eⁿ
• Poincaré metric, for metric properties of Riemann surfaces
• Area element, the area of a differential element of a surface

References

• Gramain, André (1984). In Mathematics, a volume form is a nowhere zero differential ''n''-form on an n - Manifold. In Mathematics, the Poincaré metric, named after Henri Poincaré, is the Metric tensor describing a two-dimensional surface of constant negative Curvature In Mathematics, a volume form is a nowhere zero differential ''n''-form on an n - Manifold. Topology of Surfaces. BCS Associates. ISBN 091435101X.   (Original 1969-70 Orsay course notes in French for "Topologie des Surfaces")
• Bredon, Glen E. (1993). Topology and Geometry. Springer-Verlag. ISBN 0-387-97926-3.  
• Massey, William S. (1991). A Basic Course in Algebraic Topology. Springer-Verlag. ISBN 0-387-97430-X.  

External links

• Math Surfaces Gallery, with 60 ~surfaces and Java Applet for live rotation viewing
• 3D Graph Explorer, a free java applet/application to explore mathematically defined surfaces