1325:
737:
765:
723:
437:
751:
589:
38:
1721:
Non-compact surfaces are more difficult to classify. As a simple example, a non-compact surface can be obtained by puncturing (removing a finite set of points from) a closed manifold. On the other hand, any open subset of a compact surface is itself a non-compact surface; consider, for example, the
980:
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
2059:
viewed as a complex manifold is a
Riemann surface. In fact, every compact orientable surface is realizable as a Riemann surface. Thus compact Riemann surfaces are characterized topologically by their genus: 0, 1, 2, .... On the other hand, the genus does not characterize the complex structure. For
777:
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
522:
In the previous section, a surface is defined as a topological space with certain properties, namely
Hausdorff and locally Euclidean. This topological space is not considered a subspace of another space. In this sense, the definition given above, which is the definition that mathematicians use at
1631:
surfaces, possibly with boundary, are simply closed surfaces with a finite number of holes (open discs that have been removed). Thus, a connected compact surface is classified by the number of boundary components and the genus of the corresponding closed surface – equivalently, by the number of
404:
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 the real projective plane with one point removed is homeomorphic to the open Möbius strip).
1434:
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.
384:
used without qualification refers to surfaces without boundary. In particular, a surface with empty boundary is a surface in the usual sense. A surface with empty boundary which is compact is known as a 'closed' surface. The two-dimensional sphere, the two-dimensional
1846:
If one removes the assumption of second-countability from the definition of a surface, there exist (necessarily non-compact) topological surfaces having no countable base for their topology. Perhaps the simplest example is the
Cartesian product of the
530:
A surface defined as intrinsic is not required to satisfy the added constraint of being a subspace of
Euclidean space. It may seem possible for some surfaces defined intrinsically to not be surfaces in the extrinsic sense. However, the
1829:
is infinite, then the topological type of M depends not only on these two numbers but also on how the infinite one(s) approach the space of ends. In general the topological type of M is determined by the four subspaces of
1635:
This classification follows almost immediately from the classification of closed surfaces: removing an open disc from a closed surface yields a compact surface with a circle for boundary component, and removing
1213:. Thus, the connected sum of three real projective planes is homeomorphic to the connected sum of the real projective plane with the torus. Any connected sum involving a real projective plane is nonorientable.
1119:
1457:
under the operation of connected sum, as indeed do manifolds of any fixed dimension. The identity is the sphere, while the real projective plane and the torus generate this monoid, with a single relation
1969:, which describes how curved or bent the surface is at each point. Curvature is a rigid, geometric property, in that it is not preserved by general diffeomorphisms of the surface. However, the famous
420:(making it possible to define length and angles on the surface), a complex structure (making it possible to define holomorphic maps to and from the surface—in which case the surface is called a
1658:
Conversely, the boundary of a compact surface is a closed 1-manifold, and is therefore the disjoint union of a finite number of circles; filling these circles with disks (formally, taking the
1009:, is obtained by removing a disk from each of them and gluing them along the boundary components that result. The boundary of a disk is a circle, so these boundary components are circles. The
416:, extra structure is added upon the topology of the surface. This added structure can be a smoothness structure (making it possible to define differentiable maps to and from the surface), a
1838:) that are limit points of infinitely many handles and infinitely many projective planes, limit points of only handles, limit points of only projective planes, and limit points of neither.
2043:
596:
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
2226:: "Recall that the genus of a compact surface S with boundary is defined to be the genus of the associated closed surface obtained ... by sewing a disc onto each boundary circle"
1420:
The surfaces in the third family are nonorientable. The Euler characteristic of the real projective plane is 1, and in general the Euler characteristic of the connected sum of
1314:
1574:), the surface is not orientable (there is no notion of side), so there is no difference between attaching a torus and attaching a Klein bottle, which explains the relation.
1707:
911:
829:
1328:
Some examples of orientable closed surfaces (left) and surfaces with boundary (right). Left: Some orientable closed surfaces are the surface of a sphere, the surface of a
952:
1195:. 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; in a formula,
1030:
860:
1632:
boundary components, the orientability, and Euler characteristic. The genus of a compact surface is defined as the genus of the corresponding closed surface.
1406:
of the surface. The sphere and the torus have Euler characteristics 2 and 0, respectively, and in general the Euler characteristic of the connected sum of
515:
of certain functions, usually polynomial functions. Such a definition considered the surface as part of a larger (Euclidean) space, and as such was termed
400:
is a surface on which the distinction between clockwise and counterclockwise can be defined locally, but not globally. In general, a surface is said to be
2051:
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
2724:
958:
Note that the sphere and the projective plane can both be realized as quotients of the 2-gon, while the torus and Klein bottle require a 4-gon (square).
1938:
Two smooth surfaces are diffeomorphic if and only if they are homeomorphic. (The analogous result does not hold for higher-dimensional manifolds.) Thus
2098:
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 defined over
2048:
This result exemplifies the deep relationship between the geometry and topology of surfaces (and, to a lesser extent, higher-dimensional manifolds).
1891:
425:
173:
1332:, and the surface of a cube. (The cube and the sphere are topologically equivalent to each other.) Right: Some surfaces with boundary are the
2208:
301:
The rest of this article will assume, unless specified otherwise, that a surface is nonempty, Hausdorff, second-countable, and connected.
279:. It is through this chart that the neighborhood inherits the standard coordinates on the Euclidean plane. These coordinates are known as
1336:, square surface, and hemisphere surface. The boundaries are shown in red. All three of these are topologically equivalent to each other.
357:
of the surface which is necessarily a one-manifold, that is, the union of closed curves. On the other hand, a point mapped to above the
290:
In most writings on the subject, it is often assumed, explicitly or implicitly, that as a topological space a surface is also nonempty,
700:
of the surface, by pairwise identification of its edges. For example, in each polygon below, attaching the sides with matching labels (
1048:
2662:
2557:
2500:
2481:
2402:
2380:
1907:
1593:), which brings every triangulated surface to a standard form. A simplified proof, which avoids a standard form, was discovered by
1442:
are classified by the class of each of their connected components, and thus one generally assumes that the surface is connected.
1585:
Topological and combinatorial proofs in general rely on the difficult result that every compact 2-manifold is homeomorphic to a
546:; on the other hand, the real projective plane, which is compact, non-orientable and without boundary, cannot be embedded into
1898:, one obtains a two-dimensional complex manifold (which is necessarily a 4-dimensional real manifold) with no countable base.
1567:) adds a handle with the two ends attached to opposite sides of an orientable surface; in the presence of a projective plane (
970:
2646:
2533:
2463:
2421:
2361:
2332:
1439:
2191:
Altınok, Selma; Bhupal, Mohan (2008), "Minimal page-genus of Milnor open books on links of rational surface singularities",
2576:
778:
opposite to the direction of traversal. The four models above, when traversed clockwise starting at the upper left, yield
2507:
1731:
236:, and many other disciplines, primarily in representing the surfaces of physical objects. For example, in analyzing the
2719:
736:
92:
of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any
2091:, which provides a finer classification of Riemann surfaces than the topological one by Euler characteristic alone.
2729:
1987:
582:
1730:. However, not every non-compact surface is a subset of a compact surface; two canonical counterexamples are the
1398:. It is convenient to combine the two families by regarding the sphere as the connected sum of 0 tori. The number
1735:
977:
532:
316:
261:
1970:
1560:) adds a handle with both ends attached to the same side of the surface, while connect-sum with a Klein bottle (
507:
Historically, surfaces were initially defined as subspaces of
Euclidean spaces. Often, these surfaces were the
2223:
1866:
surface. The PrĂĽfer manifold may be thought of as the upper half plane together with one additional "tongue"
428:, such as self-intersections and cusps, that cannot be described solely in terms of the underlying topology).
1175:. The connected sum is associative, so the connected sum of a finite collection of surfaces is well-defined.
1146:. This is because deleting a disk from the sphere leaves a disk, which simply replaces the disk deleted from
1272:
962:
620:
578:
571:
961:
The expression thus derived from a fundamental polygon of a surface turns out to be the sole relation in a
2072:
2068:
1644:
disjoint circles for boundary components. The precise locations of the holes are irrelevant, because the
1605:
1582:
The classification of closed surfaces has been known since the 1860s, and today a number of proofs exist.
1237:
291:
89:
1950:
1257:
764:
742:
688:. If the condition of non-vanishing gradient is dropped, then the zero locus may develop singularities.
648:
409:
390:
181:
140:
124:
116:
31:
2687:
2088:
696:
Each closed surface can be constructed from an oriented polygon with an even number of sides, called a
2693:
1290:
647:
are 2 variables that parametrize the image. A parametric surface need not be a topological surface. A
1848:
1345:
1010:
1807:
are finite, then these two numbers, and the topological type of space of ends, classify the surface
1756:), which informally speaking describes the ways that the surface "goes off to infinity". The space
1676:
867:
785:
2666:
2428:
2147:
2115:
2103:
2099:
1727:
1710:
1645:
1261:
697:
671:
616:
508:
1894:). By contrast, if one replaces the real numbers in the construction of the PrĂĽfer surface by the
1324:
535:
asserts every surface can in fact be embedded homeomorphically into
Euclidean space, in fact into
2625:
2593:
2514:
2301:
2259:
2084:
1966:
1586:
1333:
1280:
1253:
675:
632:
605:
512:
413:
374:
192:
169:
85:
2131:
1859:
157:. The most familiar examples arise as boundaries of solid objects in ordinary three-dimensional
2608:
Thomassen, Carsten (1992), "The Jordan-Schönflies theorem and the classification of surfaces",
1450:
Relating this classification to connected sums, the closed surfaces up to homeomorphism form a
1316:
that contains its boundary is a surface that is topologically closed but not a closed surface.
918:
2714:
2642:
2553:
2527:
2496:
2477:
2459:
2417:
2398:
2376:
2357:
2328:
2204:
2080:
2076:
1946:
1887:
1649:
1617:
966:
417:
312:
257:
233:
202:
112:
78:
1597:
circa 1992, which he called the "Zero
Irrelevancy Proof" or "ZIP proof" and is presented in (
2617:
2585:
2293:
2281:
2249:
2196:
2142:
2083:
on an oriented, closed surface is conformally equivalent to an essentially unique metric of
2060:
example, there are uncountably many non-isomorphic compact
Riemann surfaces of genus 1 (the
1786:
1628:
1521:
1383:
1129:
684:
555:
542:
In fact, any compact surface that is either orientable or has a boundary can be embedded in
327:
197:
154:
120:
115:
or a complex structure, that connects them to other disciplines within mathematics, such as
2218:
1015:
722:
2214:
2056:
2052:
1659:
1276:
836:
551:
421:
309:
295:
158:
105:
2683:
Math
Surfaces Animation, with JavaScript (Canvas HTML) for tens surfaces rotation viewing
2458:, Monographs and Textbooks in Pure and Applied Mathematics, vol. 72, Marcel Dekker,
2155:, a non-differentiable surface obtained by deforming (crumpling) a differentiable surface
1589:, which is of interest in its own right. The most common proof of the classification is (
1265:
750:
397:
969:
of the surface with the polygon edge labels as generators. This is a consequence of the
205:
is defined. For example, the surface of the Earth resembles (ideally) a two-dimensional
2452:
2321:
2061:
1924:
1895:
1746:
1594:
218:
1919:, are among the first surfaces encountered in geometry. It is also possible to define
377:
is a simple example of a surface with boundary. The boundary of the disc is a circle.
2708:
2305:
2137:
1942:
are classified up to diffeomorphism by their Euler characteristic and orientability.
1863:
1395:
1375:
1360:
1279:, a surface embedded in Euclidean space that is closed with respect to the inherited
1233:
990:
559:
319:
264:
191:; this means that a moving point on a surface may move in two directions (it has two
93:
275:. Such a neighborhood, together with the corresponding homeomorphism, is known as a
1249:
1222:
1189:
770:
237:
111:
Topological surfaces are sometimes equipped with additional information, such as a
97:
81:
42:
2678:
Math
Surfaces Gallery, with 60 ~surfaces and Java Applet for live rotation viewing
2195:, Contemp. Math., vol. 475, Amer. Math. Soc., Providence, RI, pp. 1–10,
2568:
1608:. This was originally proven only for Riemann surfaces in the 1880s and 1900s by
716:), so that the arrows point in the same direction, yields the indicated surface.
298:. It is also often assumed that the surfaces under consideration are connected.
2641:, Graduate Studies in Mathematics, vol. 74, American Mathematical Society,
2390:
2120:
1609:
1451:
229:
146:
2200:
436:
2698:
1912:
1765:
1723:
1613:
1348:
closed surface is homeomorphic to some member of one of these three families:
217:
provide two-dimensional coordinates on it (except at the poles and along the
168:. The exact definition of a surface may depend on the context. Typically, in
2508:(Original 1969-70 Orsay course notes in French for "Topologie des Surfaces")
2152:
1886:
In 1925, Tibor RadĂł proved that all Riemann surfaces (i.e., one-dimensional
563:
370:
214:
188:
101:
17:
574:. All these models are singular at points where they intersect themselves.
2682:
588:
37:
2677:
2671:
2346:, Princeton Mathematical Series, vol. 26, Princeton University Press
2102:
other than the complex numbers, nor are algebraic surfaces defined over
1958:
1954:
1932:
667:
609:
323:
268:
241:
210:
177:
74:
70:
62:
2629:
2597:
2297:
2263:
1973:
for closed surfaces states that the integral of the Gaussian curvature
225:
128:
2602:
2414:
Compact Riemann surfaces: an introduction to contemporary mathematics
1454:
1353:
1241:
728:
608:
manner (see figure). The two embedded tori are homeomorphic, but not
206:
165:
2621:
2589:
2563:, similar to Morse theoretic proof using sliding of attached handles
2254:
2237:
1604:
A geometric proof, which yields a stronger geometric result, is the
1275:
is closed if and only if it is the boundary of a solid. As with any
1042:
is the sum of the Euler characteristics of the summands, minus two:
539:: The extrinsic and intrinsic approaches turn out to be equivalent.
612:: They are topologically equivalent, but their embeddings are not.
1962:
1862:, which can be described by simple equations that show it to be a
1329:
1323:
1245:
756:
601:
587:
435:
386:
244:, the central consideration is the flow of air along its surface.
36:
1764:) is always topologically equivalent to a closed subspace of the
1916:
1854:
Another surface having no countable base for its topology, but
1157:
is also described as attaching a "handle" to the other summand
1114:{\displaystyle \chi (M{\mathbin {\#}}N)=\chi (M)+\chi (N)-2.\,}
283:
and these homeomorphisms lead us to describe surfaces as being
2327:, Pure and Applied Mathematics, vol. 89, Academic Press,
2087:. This provides a starting point for one of the approaches to
2067:
Complex structures on a closed oriented surface correspond to
1287:
necessarily a closed surface; for example, a disk embedded in
2435:
Morse theoretic proofs of classification up to diffeomorphism
1858:
requiring the Axiom of Choice to prove its existence, is the
1776:
of handles, as well as a finite or countably infinite number
2672:
The Classification of Surfaces and the Jordan Curve Theorem
424:), or an algebraic structure (making it possible to detect
2694:
History and Art of Surfaces and their Mathematical Models
2071:
of Riemannian metrics on the surface. One version of the
195:). In other words, around almost every point, there is a
345:-axis. A point on the surface mapped via a chart to the
2313:
Simplicial proofs of classification up to homeomorphism
1953:. A Riemannian metric endows a surface with notions of
1221:"Open surface" redirects here. Not to be confused with
651:
can be viewed as a special kind of parametric surface.
84:; for example, the sphere is the boundary of the solid
1738:, which are non-compact surfaces with infinite genus.
2338:, English translation of 1934 classic German textbook
1990:
1811:
up to topological equivalence. If either or both of
1679:
1293:
1051:
1018:
921:
870:
839:
788:
2639:
Elements of combinatorial and differential topology
1655:on any connected manifold of dimension at least 2.
585:embedding of the two-sphere into the three-sphere.
2567:Francis, George K.; Weeks, Jeffrey R. (May 1999),
2451:
2320:
2037:
1701:
1308:
1113:
1024:
946:
905:
854:
823:
1935:to be applied to surfaces to prove many results.
1178:The connected sum of two real projective planes,
172:, a surface may cross itself (and may have other
153:is a geometrical shape that resembles a deformed
1772:may have a finite or countably infinite number N
1590:
1264:(which is a sphere with two punctures), and the
353:. The collection of such points is known as the
1875:hanging down from it directly below the point (
1665:The unique compact orientable surface of genus
2633:, short elementary proof using spanning graphs
2238:"On the classification of noncompact surfaces"
976:Gluing edges of polygons is a special kind of
341:. The boundary of the upper half-plane is the
2652:, contains short account of Thomassen's proof
2319:Seifert, Herbert; Threlfall, William (1980),
2038:{\displaystyle \int _{S}K\;dA=2\pi \chi (S).}
1252:. Examples of non-closed surfaces include an
600:in the "standard" manner (which looks like a
566:, are models of the real projective plane in
432:Extrinsically defined surfaces and embeddings
8:
2178:
1598:
2516:Classification of surfaces via Morse Theory
2134:, for metric properties of Riemann surfaces
1981:is determined by the Euler characteristic:
1394:The surfaces in the first two families are
440:A sphere can be defined parametrically (by
365:. The collection of interior points is the
2004:
1240:. Examples of closed surfaces include the
61:In the part of mathematics referred to as
2284:(1888), "Beiträge zur Analysis situs I",
2253:
1995:
1989:
1923:, in which each point has a neighborhood
1684:
1678:
1640:open discs yields a compact surface with
1553:Geometrically, connect-sum with a torus (
1342:classification theorem of closed surfaces
1300:
1296:
1295:
1292:
1110:
1062:
1061:
1050:
1017:
935:
920:
894:
881:
869:
838:
812:
799:
787:
635:. Such an image is so-called because the
337:. These homeomorphisms are also known as
224:The concept of surface is widely used in
30:For broader coverage of this topic, see
2487:, careful proof aimed at undergraduates
2165:
718:
2525:
2454:Differential topology: an introduction
1726:in the sphere, otherwise known as the
1512:. This relation is sometimes known as
2342:Ahlfors, Lars V.; Sario, Leo (1960),
1673:boundary components is often denoted
7:
2373:A Basic Course in Algebraic Topology
2173:
2171:
2169:
1890:) are necessarily second-countable (
1525:
1132:for the connected sum, meaning that
88:. Other surfaces arise as graphs of
1939:
1153:Connected summation with the torus
682:does define a surface, known as an
369:of the surface which is always non-
306:(topological) surface with boundary
2663:Classification of Compact Surfaces
2522:, an exposition of Gramain's notes
1965:, and area. It also gives rise to
1949:are of foundational importance in
1681:
1063:
25:
2725:Differential geometry of surfaces
2474:Elements of differential topology
1908:Differential geometry of surfaces
1842:Assumption of second-countability
1320:Classification of closed surfaces
570:, but only the Boy surface is an
393:are examples of closed surfaces.
315:in which every point has an open
260:in which every point has an open
2367:, Cambridge undergraduate course
1851:with the space of real numbers.
1709:for example in the study of the
1528:), and the triple cross surface
1309:{\displaystyle \mathbb {R} ^{3}}
763:
749:
735:
721:
1402:of tori involved is called the
125:mathematical notions of surface
2690:Lecture Notes by Z.Fiedorowicz
2688:The Classification of Surfaces
2674:in Home page of Andrew Ranicki
2550:Topology: a geometric approach
2513:A. Champanerkar; et al.,
2029:
2023:
1945:Smooth surfaces equipped with
1702:{\displaystyle \Sigma _{g,k},}
1438:Closed surfaces with multiple
1101:
1095:
1086:
1080:
1071:
1055:
906:{\displaystyle ABA^{-1}B^{-1}}
824:{\displaystyle ABB^{-1}A^{-1}}
248:Definitions and first examples
1:
2601:; page discussing the paper:
2577:American Mathematical Monthly
2123:, for volumes of surfaces in
2106:other than the real numbers.
2069:conformal equivalence classes
1480:, which may also be written
73:. Some surfaces arise as the
1915:, such as the boundary of a
1591:Seifert & Threlfall 1980
100:is a surface that cannot be
2552:, Oxford University Press,
2371:Massey, William S. (1991).
1662:) yields a closed surface.
643:- directions of the domain
201:on which a two-dimensional
2746:
2532:: CS1 maint: postscript (
2472:Shastri, Anant R. (2011),
2416:(3rd ed.), Springer,
2352:Maunder, C. R. F. (1996),
2055:. Any complex nonsingular
1931:. This elaboration allows
1905:
1256:(which is a sphere with a
1220:
1165:is orientable, then so is
971:Seifert–van Kampen theorem
692:Construction from polygons
670:is nowhere zero, then the
658:is a smooth function from
138:
29:
947:{\displaystyle ABAB^{-1}}
533:Whitney embedding theorem
2450:Gauld, David B. (1982),
2445:(2nd ed.), Springer
2179:Francis & Weeks 1999
1977:over the entire surface
1879:,0), for each real
1599:Francis & Weeks 1999
27:Two-dimensional manifold
2637:Prasolov, V.V. (2006),
2491:Gramain, André (1984).
1273:three-dimensional space
833:real projective plane:
579:Alexander horned sphere
271:of the Euclidean plane
131:in the physical world.
2548:Lawson, Terry (2003),
2427:, for closed oriented
2356:, Dover Publications,
2323:A textbook of topology
2242:Trans. Amer. Math. Soc
2236:Richards, Ian (1963).
2201:10.1090/conm/475/09272
2073:uniformization theorem
2039:
1741:A non-compact surface
1703:
1624:Surfaces with boundary
1606:uniformization theorem
1542:is accordingly called
1337:
1310:
1271:A surface embedded in
1115:
1026:
948:
907:
856:
825:
627:to higher-dimensional
593:
504:
326:of the closure of the
58:
2701:at the Manifold Atlas
2603:On Conway's ZIP Proof
2443:Differential topology
2412:Jost, JĂĽrgen (2006),
2395:Topology and Geometry
2040:
1951:differential geometry
1704:
1327:
1311:
1232:is a surface that is
1116:
1027:
1025:{\displaystyle \chi }
949:
908:
857:
826:
743:real projective plane
649:surface of revolution
591:
439:
391:real projective plane
254:(topological) surface
189:two-dimensional space
182:differential geometry
141:Surface (mathematics)
139:Further information:
127:can be used to model
117:differential geometry
104:in three-dimensional
69:is a two-dimensional
40:
32:Surface (mathematics)
2569:"Conway's ZIP Proof"
2493:Topology of Surfaces
2429:Riemannian manifolds
1988:
1971:Gauss–Bonnet theorem
1927:to some open set in
1902:Surfaces in geometry
1717:Non-compact surfaces
1677:
1524:, who proved it in (
1440:connected components
1291:
1049:
1016:
1011:Euler characteristic
919:
868:
855:{\displaystyle ABAB}
837:
786:
484:) or implicitly (by
2610:Amer. Math. Monthly
2441:Hirsch, M. (1994),
2397:. Springer-Verlag.
2375:. Springer-Verlag.
2148:Tetrahemihexahedron
2116:Boundary (topology)
1728:Cantor tree surface
1711:mapping class group
1646:homeomorphism group
981:the connected sum.
698:fundamental polygon
339:(coordinate) charts
96:. For example, the
2720:Geometric topology
2495:. BCS Associates.
2354:Algebraic topology
2298:10.1007/bf01443580
2089:TeichmĂĽller theory
2085:constant curvature
2079:) states that any
2035:
1967:Gaussian curvature
1947:Riemannian metrics
1699:
1587:simplicial complex
1338:
1306:
1281:Euclidean topology
1111:
1022:
944:
903:
852:
821:
633:parametric surface
594:
505:
414:algebraic geometry
349:-axis is termed a
304:More generally, a
277:(coordinate) chart
193:degrees of freedom
170:algebraic geometry
59:
2730:Analytic geometry
2667:Mathifold Project
2210:978-0-8218-4717-6
2081:Riemannian metric
1888:complex manifolds
1787:projective planes
1736:Loch Ness monster
1384:projective planes
967:fundamental group
619:of a continuous,
418:Riemannian metric
313:topological space
285:locally Euclidean
281:local coordinates
258:topological space
240:properties of an
234:computer graphics
203:coordinate system
113:Riemannian metric
79:three-dimensional
16:(Redirected from
2737:
2651:
2632:
2600:
2573:
2562:
2537:
2531:
2523:
2521:
2506:
2486:
2468:
2457:
2446:
2426:
2408:
2386:
2366:
2347:
2344:Riemann surfaces
2337:
2326:
2308:
2268:
2267:
2257:
2233:
2227:
2221:
2193:Singularities II
2188:
2182:
2175:
2153:Crumpled surface
2044:
2042:
2041:
2036:
2000:
1999:
1745:has a non-empty
1722:complement of a
1708:
1706:
1705:
1700:
1695:
1694:
1573:
1566:
1559:
1548:
1547:
1541:
1522:Walther von Dyck
1518:
1517:
1511:
1497:
1479:
1446:Monoid structure
1430:
1416:
1344:states that any
1315:
1313:
1312:
1307:
1305:
1304:
1299:
1212:
1187:
1174:
1145:
1130:identity element
1120:
1118:
1117:
1112:
1067:
1066:
1041:
1031:
1029:
1028:
1023:
993:of two surfaces
953:
951:
950:
945:
943:
942:
912:
910:
909:
904:
902:
901:
889:
888:
861:
859:
858:
853:
830:
828:
827:
822:
820:
819:
807:
806:
767:
753:
739:
725:
685:implicit surface
631:is said to be a
592:A knotted torus.
581:is a well-known
572:immersed surface
552:Steiner surfaces
502:
328:upper half-plane
292:second-countable
198:coordinate patch
187:A surface is a
121:complex analysis
57:-contours shown.
21:
2745:
2744:
2740:
2739:
2738:
2736:
2735:
2734:
2705:
2704:
2659:
2649:
2636:
2622:10.2307/2324180
2607:
2590:10.2307/2589143
2571:
2566:
2560:
2547:
2544:
2524:
2519:
2512:
2503:
2490:
2484:
2471:
2466:
2449:
2440:
2437:
2424:
2411:
2405:
2391:Bredon, Glen E.
2389:
2383:
2370:
2364:
2351:
2341:
2335:
2318:
2315:
2280:
2277:
2272:
2271:
2255:10.2307/1993768
2235:
2234:
2230:
2211:
2190:
2189:
2185:
2176:
2167:
2162:
2132:Poincaré metric
2112:
2096:complex surface
2062:elliptic curves
2057:algebraic curve
2053:Riemann surface
1991:
1986:
1985:
1940:closed surfaces
1921:smooth surfaces
1910:
1904:
1896:complex numbers
1874:
1860:PrĂĽfer manifold
1844:
1828:
1819:
1806:
1797:
1784:
1775:
1719:
1680:
1675:
1674:
1626:
1580:
1568:
1561:
1554:
1545:
1544:
1529:
1515:
1514:
1499:
1481:
1459:
1448:
1425:
1411:
1322:
1294:
1289:
1288:
1277:closed manifold
1226:
1219:
1217:Closed surfaces
1196:
1179:
1166:
1133:
1047:
1046:
1033:
1014:
1013:
987:
931:
917:
916:
890:
877:
866:
865:
835:
834:
808:
795:
784:
783:
773:
768:
759:
754:
745:
740:
731:
726:
694:
550:(see Gramain).
485:
434:
422:Riemann surface
250:
159:Euclidean space
143:
137:
106:Euclidean space
35:
28:
23:
22:
15:
12:
11:
5:
2743:
2741:
2733:
2732:
2727:
2722:
2717:
2707:
2706:
2703:
2702:
2696:
2691:
2685:
2680:
2675:
2669:
2658:
2657:External links
2655:
2654:
2653:
2647:
2634:
2605:
2564:
2558:
2543:
2540:
2539:
2538:
2510:
2501:
2488:
2482:
2469:
2464:
2447:
2436:
2433:
2432:
2431:
2422:
2409:
2403:
2387:
2381:
2368:
2362:
2349:
2339:
2333:
2314:
2311:
2310:
2309:
2292:(4): 459–512,
2276:
2273:
2270:
2269:
2248:(2): 259–269.
2228:
2209:
2183:
2164:
2163:
2161:
2158:
2157:
2156:
2150:
2145:
2140:
2135:
2129:
2118:
2111:
2108:
2046:
2045:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2003:
1998:
1994:
1906:Main article:
1903:
1900:
1892:RadĂł's theorem
1870:
1843:
1840:
1824:
1815:
1802:
1793:
1780:
1773:
1732:Jacob's ladder
1718:
1715:
1698:
1693:
1690:
1687:
1683:
1625:
1622:
1618:Henri Poincaré
1595:John H. Conway
1579:
1576:
1546:Dyck's surface
1516:Dyck's theorem
1447:
1444:
1392:
1391:
1372:
1357:
1321:
1318:
1303:
1298:
1230:closed surface
1218:
1215:
1122:
1121:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1065:
1060:
1057:
1054:
1021:
986:
985:Connected sums
983:
978:quotient space
956:
955:
941:
938:
934:
930:
927:
924:
915:Klein bottle:
913:
900:
897:
893:
887:
884:
880:
876:
873:
862:
851:
848:
845:
842:
831:
818:
815:
811:
805:
802:
798:
794:
791:
775:
774:
769:
762:
760:
755:
748:
746:
741:
734:
732:
727:
720:
693:
690:
623:function from
433:
430:
363:interior point
351:boundary point
249:
246:
219:180th meridian
184:, it may not.
136:
133:
123:. The various
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2742:
2731:
2728:
2726:
2723:
2721:
2718:
2716:
2713:
2712:
2710:
2700:
2697:
2695:
2692:
2689:
2686:
2684:
2681:
2679:
2676:
2673:
2670:
2668:
2664:
2661:
2660:
2656:
2650:
2644:
2640:
2635:
2631:
2627:
2623:
2619:
2616:(2): 116–13,
2615:
2611:
2606:
2604:
2599:
2595:
2591:
2587:
2583:
2579:
2578:
2570:
2565:
2561:
2559:0-19-851597-9
2555:
2551:
2546:
2545:
2541:
2535:
2529:
2518:
2517:
2511:
2509:
2504:
2502:0-914351-01-X
2498:
2494:
2489:
2485:
2483:9781439831601
2479:
2476:, CRC Press,
2475:
2470:
2467:
2461:
2456:
2455:
2448:
2444:
2439:
2438:
2434:
2430:
2425:
2419:
2415:
2410:
2406:
2404:0-387-97926-3
2400:
2396:
2392:
2388:
2384:
2382:0-387-97430-X
2378:
2374:
2369:
2365:
2359:
2355:
2350:
2345:
2340:
2336:
2330:
2325:
2324:
2317:
2316:
2312:
2307:
2303:
2299:
2295:
2291:
2287:
2283:
2282:Dyck, Walther
2279:
2278:
2274:
2265:
2261:
2256:
2251:
2247:
2243:
2239:
2232:
2229:
2225:
2220:
2216:
2212:
2206:
2202:
2198:
2194:
2187:
2184:
2180:
2174:
2172:
2170:
2166:
2159:
2154:
2151:
2149:
2146:
2144:
2143:Boy's surface
2141:
2139:
2138:Roman surface
2136:
2133:
2130:
2128:
2126:
2122:
2119:
2117:
2114:
2113:
2109:
2107:
2105:
2101:
2097:
2092:
2090:
2086:
2082:
2078:
2074:
2070:
2065:
2063:
2058:
2054:
2049:
2032:
2026:
2020:
2017:
2014:
2011:
2008:
2005:
2001:
1996:
1992:
1984:
1983:
1982:
1980:
1976:
1972:
1968:
1964:
1960:
1956:
1952:
1948:
1943:
1941:
1936:
1934:
1930:
1926:
1925:diffeomorphic
1922:
1918:
1914:
1909:
1901:
1899:
1897:
1893:
1889:
1884:
1882:
1878:
1873:
1869:
1865:
1864:real-analytic
1861:
1857:
1852:
1850:
1841:
1839:
1837:
1833:
1827:
1823:
1818:
1814:
1810:
1805:
1801:
1796:
1792:
1788:
1783:
1779:
1771:
1767:
1763:
1759:
1755:
1751:
1748:
1747:space of ends
1744:
1739:
1737:
1733:
1729:
1725:
1716:
1714:
1712:
1696:
1691:
1688:
1685:
1672:
1668:
1663:
1661:
1656:
1654:
1653:-transitively
1652:
1647:
1643:
1639:
1633:
1630:
1623:
1621:
1619:
1615:
1611:
1607:
1602:
1600:
1596:
1592:
1588:
1583:
1577:
1575:
1572:
1565:
1558:
1551:
1549:
1540:
1536:
1532:
1527:
1523:
1519:
1510:
1506:
1502:
1496:
1492:
1488:
1484:
1478:
1474:
1470:
1466:
1462:
1456:
1453:
1445:
1443:
1441:
1436:
1432:
1429:
1423:
1418:
1415:
1409:
1405:
1401:
1397:
1389:
1385:
1381:
1377:
1376:connected sum
1373:
1370:
1366:
1362:
1361:connected sum
1358:
1355:
1351:
1350:
1349:
1347:
1343:
1335:
1331:
1326:
1319:
1317:
1301:
1286:
1282:
1278:
1274:
1269:
1267:
1263:
1259:
1255:
1251:
1247:
1243:
1239:
1235:
1231:
1224:
1216:
1214:
1211:
1207:
1203:
1199:
1194:
1191:
1186:
1182:
1176:
1173:
1169:
1164:
1160:
1156:
1151:
1150:upon gluing.
1149:
1144:
1140:
1136:
1131:
1127:
1107:
1104:
1098:
1092:
1089:
1083:
1077:
1074:
1068:
1058:
1052:
1045:
1044:
1043:
1040:
1036:
1019:
1012:
1008:
1004:
1000:
996:
992:
991:connected sum
984:
982:
979:
974:
972:
968:
964:
959:
939:
936:
932:
928:
925:
922:
914:
898:
895:
891:
885:
882:
878:
874:
871:
863:
849:
846:
843:
840:
832:
816:
813:
809:
803:
800:
796:
792:
789:
781:
780:
779:
772:
766:
761:
758:
752:
747:
744:
738:
733:
730:
724:
719:
717:
715:
711:
707:
703:
699:
691:
689:
687:
686:
681:
677:
673:
669:
665:
661:
657:
652:
650:
646:
642:
638:
634:
630:
626:
622:
618:
613:
611:
607:
603:
599:
590:
586:
584:
580:
575:
573:
569:
565:
561:
560:Roman surface
557:
556:Boy's surface
553:
549:
545:
540:
538:
534:
528:
526:
520:
518:
514:
510:
500:
496:
492:
488:
483:
479:
475:
471:
467:
463:
459:
455:
451:
447:
443:
438:
431:
429:
427:
426:singularities
423:
419:
415:
411:
406:
403:
399:
394:
392:
388:
383:
378:
376:
373:. The closed
372:
368:
364:
360:
356:
352:
348:
344:
340:
336:
332:
329:
325:
321:
318:
317:neighbourhood
314:
311:
307:
302:
299:
297:
293:
288:
286:
282:
278:
274:
270:
266:
263:
262:neighbourhood
259:
255:
247:
245:
243:
239:
235:
231:
227:
222:
220:
216:
212:
208:
204:
200:
199:
194:
190:
185:
183:
179:
176:), while, in
175:
174:singularities
171:
167:
163:
160:
156:
152:
148:
142:
134:
132:
130:
126:
122:
118:
114:
109:
107:
103:
99:
95:
94:ambient space
91:
87:
83:
82:solid figures
80:
76:
72:
68:
64:
56:
52:
48:
44:
39:
33:
19:
2638:
2613:
2609:
2581:
2575:
2549:
2542:Other proofs
2515:
2492:
2473:
2453:
2442:
2413:
2394:
2372:
2353:
2343:
2322:
2289:
2285:
2245:
2241:
2231:
2192:
2186:
2127:
2124:
2095:
2093:
2066:
2050:
2047:
1978:
1974:
1944:
1937:
1928:
1920:
1911:
1885:
1880:
1876:
1871:
1867:
1855:
1853:
1845:
1835:
1831:
1825:
1821:
1816:
1812:
1808:
1803:
1799:
1794:
1790:
1781:
1777:
1769:
1761:
1757:
1753:
1749:
1742:
1740:
1720:
1670:
1666:
1664:
1657:
1650:
1641:
1637:
1634:
1627:
1603:
1584:
1581:
1570:
1563:
1556:
1552:
1543:
1538:
1534:
1530:
1513:
1508:
1504:
1500:
1494:
1490:
1486:
1482:
1476:
1472:
1468:
1464:
1460:
1449:
1437:
1433:
1427:
1421:
1419:
1413:
1407:
1403:
1399:
1393:
1387:
1379:
1368:
1364:
1341:
1339:
1334:disk surface
1284:
1270:
1266:Möbius strip
1250:Klein bottle
1236:and without
1229:
1227:
1223:Free surface
1209:
1205:
1201:
1197:
1192:
1190:Klein bottle
1184:
1180:
1177:
1171:
1167:
1162:
1158:
1154:
1152:
1147:
1142:
1138:
1134:
1125:
1123:
1038:
1034:
1006:
1002:
998:
994:
988:
975:
963:presentation
960:
957:
776:
771:Klein bottle
713:
709:
705:
701:
695:
683:
679:
663:
659:
655:
653:
644:
640:
636:
628:
624:
614:
597:
595:
583:pathological
576:
567:
554:, including
547:
543:
541:
536:
529:
524:
523:present, is
521:
516:
506:
498:
494:
490:
486:
481:
477:
473:
469:
465:
461:
457:
453:
449:
445:
441:
410:differential
407:
401:
398:Möbius strip
395:
381:
379:
366:
362:
361:-axis is an
358:
354:
350:
346:
342:
338:
334:
330:
320:homeomorphic
305:
303:
300:
289:
284:
280:
276:
272:
265:homeomorphic
253:
251:
223:
196:
186:
161:
150:
144:
110:
98:Klein bottle
66:
60:
54:
50:
46:
43:open surface
18:Open surface
2699:2-manifolds
2348:, Chapter I
2121:Volume form
1610:Felix Klein
1452:commutative
1424:of them is
1412:2 − 2
1124:The sphere
324:open subset
269:open subset
238:aerodynamic
230:engineering
147:mathematics
2709:Categories
2648:0821838091
2584:(5): 393,
2465:0824717090
2423:3540330658
2363:0486691314
2334:0126348502
2286:Math. Ann.
2275:References
1789:. If both
1766:Cantor set
1724:Cantor set
1614:Paul Koebe
1426:2 −
1396:orientable
1001:, denoted
604:) or in a
402:orientable
389:, and the
164:, such as
135:In general
75:boundaries
2306:118123073
2021:χ
2018:π
1993:∫
1913:Polyhedra
1849:long line
1682:Σ
1669:and with
1526:Dyck 1888
1367:tori for
1346:connected
1254:open disk
1188:, is the
1105:−
1093:χ
1078:χ
1064:#
1053:χ
1020:χ
937:−
896:−
883:−
814:−
801:−
621:injective
564:cross-cap
525:intrinsic
517:extrinsic
380:The term
310:Hausdorff
296:Hausdorff
215:longitude
90:functions
2715:Surfaces
2528:citation
2393:(1993).
2110:See also
2077:Poincaré
2075:(due to
1959:distance
1955:geodesic
1933:calculus
1734:and the
1498:, since
1410:tori is
1262:cylinder
1258:puncture
1248:and the
1238:boundary
782:sphere:
668:gradient
610:isotopic
562:and the
497:−
367:interior
355:boundary
322:to some
267:to some
242:airplane
211:latitude
178:topology
129:surfaces
102:embedded
71:manifold
63:topology
2630:2324180
2598:2589143
2264:1993768
2219:2454357
1629:Compact
1234:compact
965:of the
864:torus:
606:knotted
382:surface
226:physics
166:spheres
151:surface
67:surface
53:-, and
2645:
2628:
2596:
2556:
2499:
2480:
2462:
2420:
2401:
2379:
2360:
2331:
2304:
2262:
2222:; see
2217:
2207:
2104:fields
2100:fields
1616:, and
1520:after
1455:monoid
1354:sphere
1244:, the
1242:sphere
1128:is an
729:sphere
666:whose
639:- and
558:, the
294:, and
209:, and
207:sphere
2626:JSTOR
2594:JSTOR
2572:(PDF)
2520:(PDF)
2302:S2CID
2260:JSTOR
2160:Notes
1963:angle
1648:acts
1578:Proof
1404:genus
1382:real
1330:torus
1260:), a
1246:torus
1161:. If
757:torus
712:with
704:with
676:zeros
672:locus
617:image
602:bagel
513:zeros
509:locus
387:torus
371:empty
308:is a
256:is a
155:plane
45:with
2643:ISBN
2554:ISBN
2534:link
2497:ISBN
2478:ISBN
2460:ISBN
2418:ISBN
2399:ISBN
2377:ISBN
2358:ISBN
2329:ISBN
2205:ISBN
1917:cube
1820:and
1798:and
1660:cone
1390:≥ 1.
1386:for
1374:the
1371:≥ 1,
1359:the
1352:the
1340:The
997:and
989:The
615:The
577:The
480:cos
468:sin
464:sin
452:cos
448:sin
412:and
396:The
375:disk
213:and
180:and
149:, a
119:and
86:ball
65:, a
2665:in
2618:doi
2586:doi
2582:106
2294:doi
2250:doi
2246:106
2224:p.2
2197:doi
2064:).
1856:not
1785:of
1768:.
1601:).
1378:of
1363:of
1285:not
1283:is
1032:of
678:of
674:of
662:to
654:If
511:of
501:= 0
408:In
333:in
221:).
145:In
77:of
49:-,
41:An
2711::
2624:,
2614:99
2612:,
2592:,
2580:,
2574:,
2530:}}
2526:{{
2300:,
2290:32
2288:,
2258:.
2244:.
2240:.
2215:MR
2213:,
2203:,
2168:^
2094:A
1961:,
1957:,
1883:.
1713:.
1620:.
1612:,
1569:#
1562:#
1555:#
1550:.
1537:#
1533:#
1507:#
1503:=
1493:#
1489:=
1485:#
1475:#
1471:=
1467:#
1463:#
1431:.
1417:.
1268:.
1228:A
1208:#
1204:=
1200:#
1183:#
1170:#
1141:=
1137:#
1108:2.
1037:#
1005:#
973:.
708:,
527:.
519:.
503:.)
493:+
489:+
476:=
472:,
460:=
456:,
444:=
287:.
252:A
232:,
228:,
108:.
2620::
2588::
2536:)
2505:.
2407:.
2385:.
2296::
2266:.
2252::
2199::
2181:)
2177:(
2125:E
2033:.
2030:)
2027:S
2024:(
2015:2
2012:=
2009:A
2006:d
2002:K
1997:S
1979:S
1975:K
1929:E
1881:x
1877:x
1872:x
1868:T
1836:M
1834:(
1832:E
1826:p
1822:N
1817:h
1813:N
1809:M
1804:p
1800:N
1795:h
1791:N
1782:p
1778:N
1774:h
1770:M
1762:M
1760:(
1758:E
1754:M
1752:(
1750:E
1743:M
1697:,
1692:k
1689:,
1686:g
1671:k
1667:g
1651:k
1642:k
1638:k
1571:P
1564:K
1557:T
1539:P
1535:P
1531:P
1509:P
1505:P
1501:K
1495:T
1491:P
1487:K
1483:P
1477:T
1473:P
1469:P
1465:P
1461:P
1428:k
1422:k
1414:g
1408:g
1400:g
1388:k
1380:k
1369:g
1365:g
1356:,
1302:3
1297:R
1225:.
1210:T
1206:P
1202:K
1198:P
1193:K
1185:P
1181:P
1172:M
1168:T
1163:M
1159:M
1155:T
1148:M
1143:M
1139:M
1135:S
1126:S
1102:)
1099:N
1096:(
1090:+
1087:)
1084:M
1081:(
1075:=
1072:)
1069:N
1059:M
1056:(
1039:N
1035:M
1007:N
1003:M
999:N
995:M
954:.
940:1
933:B
929:A
926:B
923:A
899:1
892:B
886:1
879:A
875:B
872:A
850:B
847:A
844:B
841:A
817:1
810:A
804:1
797:B
793:B
790:A
714:B
710:B
706:A
702:A
680:f
664:R
660:R
656:f
645:R
641:y
637:x
629:R
625:R
598:E
568:E
548:E
544:E
537:E
499:r
495:z
491:y
487:x
482:θ
478:r
474:z
470:φ
466:θ
462:r
458:y
454:φ
450:θ
446:r
442:x
359:x
347:x
343:x
335:C
331:H
273:E
162:R
55:z
51:y
47:x
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
