
In mathematics (topology), a surface is a two-dimensional manifold. Examples arise in three-dimensional space as the boundaries of three-dimensional solid objects. The surface of a fluid object, such as a rain drop or soap bubble, is an idealisation. To speak of the surface of a snowflake, which has a great deal of fine structure, is to go beyond the simple mathematical definition. For the nature of real surfaces see surface tension, surface chemistry, surface energy. ==Definition== In what follows, all surfaces are considered to be second-countable 2-dimensional manifolds. More precisely: a topological surface (with boundary) is a Hausdorff space in which every point has an open topological neighbourhood homeomorphism to either an open set of ''E2'' (Euclidean space) or an open subset of the closed half of ''E2''. The set of points which have an open neighbourhood homeomorphic to ''En'' is called the interior of the manifold; it is always non-empty. The complement of the interior, is called the boundary; it is a (''1'')-manifold, or union of closed curves. A surface with empty boundary is said to be closed if it is compact space, and open if it is not compact. ==Classification of closed surfaces== There is a complete classification of closed (i.e Compact space without boundary) connected, surfaces up to homeomorphism. Any such surface falls into one of three infinite collections: * sphere with ''n'' handles attached (called ''n-tori''). These are orientable surfaces with Euler characteristic 2-2''n'', also called surfaces of genus ''n''. * Real projective planes with ''n'' handles attached. These are non-orientable surfaces with Euler characteristic 1-2''n''. * Klein bottles with ''n'' handles attached. These are non-orientable surfaces with Euler characteristic -2''n''. Therefore Euler characteristic and orientability describe a compact surfaces up to homeomorphism (and if surfaces are smooth then up to diffeomorphism). ==Compact surfaces== Compact surfaces with boundary are just these with one or more removed open disks whose closures are disjoint. ==Embeddings in R3== A compact surface can be embedded in R3 if it is orientable or if it has nonempty boundary. It is a consequence of the Whitney embedding theorem that any surface can be embedded in R4. ==Differential geometry== A simple review of the embedding of a surface in ''n'' dimensions, and a computation of the area of such a surface, is provided in the article volume form. Metric properties of Riemann surfaces are briefly reviewed in the the article Poincaré metric. ==Some models== To make some models, attach the sides of these (and remove the corners to puncture): * * B B v v v ^ *>>>>>* *>>>>>* v v v ^ v v v v A v v A A v ^ A A v v A A v v A v v v ^ v v v v v v v ^ *<<<<<* *>>>>>* * * B B sphere real projective plane Klein bottle torus (punctured Möbius band) (donut) ==Fundamental polygon== Each closed surface can be constructed from an even sided oriented polygon, called a fundamental polygon by pairwise identification of its edges. This construction can be represented as a string of length 2n of n distinct symbols where each symbol appears twice with exponent either +1 or -1. The exponent -1 signifies that the corresponding edge has the orientation opposing the one of the fundamental polygon. The above models can be described as follows: * sphere: * projective plane: * Klein bottle: * torus: ''(See the main article ''fundamental polygon'' for details.)'' ==Connected sum of surfaces== Given two surfaces M and M', their connected sum M # M' is obtained by removing a disk in each of them and gluing them along the newly formed boundary components. We use the following notation. * sphere: S * torus: T * Klein bottle: K * Projective plane: P Facts: * S # S = S * S # M = M * P # P = K * P # K = P # T We use a shorthand natation: nM = M # M # ... # M (n-times) with 0M = S. Closed surfaces are classified as follows: * gT (g-fold torus): orientable surface of genus g. * gP (g-fold projective plane): non-orientable surface of genus g. ==Algebraic surface== This notion of a surface is distinct from the notion of an ''algebraic surface''. A non-singular complex projective algebraic curve is a smooth surface. Algebraic surfaces over the complex number field have dimension 4 when considered as a manifold. ==See also== *minimal surface *Riemann surface *algebraic surface *Klein bottle *Torus *Sphere *Cylinder *Projective plane ==External links== *[http://xahlee.org/surface/gallery.html Math Surfaces Gallery, with 60 ~surfaces and Java Applet for live rotation viewing] Surfaces Geometric topology Articles with ASCII art
See other meanings of words starting from letter:
Words begining with Surface:
Surface
Surface-conduction_Electron-emitter_Displays
Surface-mount
Surface-mount_technology
Surface-to-Air_missile
Surface-to-air_missile
Surface-to-air_missiles
Surface-to-air_missiles
Surface-to-surface_missile
Surface-to-surface_missiles
Surface-to-surface_piercing
Surface-to-surface_piercing
Surface-water_hydrology
Surface-water_hydrology
Surface-wave-sustained_mode
Surfaces
Surfaces
SurfaceWarrior
Surface_(disambiguation)
Surface_(magazine)
Surface_(Myst)
Surface_(physics)
Surface_Acoustic_Wave
Surface_acoustic_wave
Surface_air_temperature
Surface_air_temperature
Surface_area
Surface_area_of_earth
Surface_brightness
Surface_bundle
Surface_bundles_over_the_circle
Surface_bundle_over_the_circle
Surface_caching
Surface_caching
Surface_chemistry
Surface_combatant
Surface_core_level_shift
Surface_current_collection
Surface_effect_ship
Surface_energy
Surface_energy
Surface_epithelial-stromal_tumour
Surface_feature_nomenclature_of_solar_system_bodies
Surface_forces_apparatus
Surface_functionalization
Surface_gravity
Surface_hardening
Surface_integral
Surface_layer
Surface_Marker_Buoy
Surface_Marker_Buoy
Surface_marker_buoy
Surface_Marshal
Surface_micromachining
Surface_Mining
Surface_mining
Surface_mount
Surface_mount_technology
Surface_normal
Surface_normal
Surface_normals
Surface_of_revolution
Surface_Patterns
Surface_Patterns
Surface_physics
Surface_piercing
Surface_piercing
Surface_piercings
Surface_plasmon
Surface_Plasmon_Resonance
Surface_plasmon_resonance
Surface_plate
Surface_pressure
Surface_run_off
Surface_selection_rule
Surface_ship
Surface_structure
Surface_supplied_diver
Surface_supplied_diving
Surface_target
Surface_temperature
Surface_Temperature_Inversion
Surface_temperature_inversion
Surface_temperature_record
Surface_temperature_record
Surface_Tension
Surface_tension
Surface_tension
Surface_texel
Surface_to_air_missile
Surface_Transportation_Act
Surface_Transportation_Board
Surface_Transportation_Board
Surface_warfare
Surface_Warfare_Badge
Surface_Warfare_insignia
Surface_Warfare_Officer_Pin
Surface_Warfare_Pin
Surface_Water
Surface_water
Surface_wave
Surface_waves
Surface
In mathematics (topology), a surface is a two-dimensional manifold. Examples arise in three-dimensional space as the boundaries of three-dimensional solid objects. The surface of a fluid object, such as a rain drop or soap bubble, is an idealisation. To speak of the surface of a snowflake, which has a great deal of fine structure, is to go beyond the simple mathematical definition. For the nature of real surfaces see surface tension, surface chemistry, surface energy. ==Definition== In what follows, all surfaces are considered to be second-countable 2-dimensional manifolds. More precisely: a topological surface (with boundary) is a Hausdorff space in which every point has an open topological neighbourhood homeomorphism to either an open set of ''E2'' (Euclidean space) or an open subset of the closed half of ''E2''. The set of points which have an open neighbourhood homeomorphic to ''En'' is called the interior of the manifold; it is always non-empty. The complement of the interior, is called the boundary; it is a (''1'')-manifold, or union of closed curves. A surface with empty boundary is said to be closed if it is compact space, and open if it is not compact. ==Classification of closed surfaces== There is a complete classification of closed (i.e Compact space without boundary) connected, surfaces up to homeomorphism. Any such surface falls into one of three infinite collections: * sphere with ''n'' handles attached (called ''n-tori''). These are orientable surfaces with Euler characteristic 2-2''n'', also called surfaces of genus ''n''. * Real projective planes with ''n'' handles attached. These are non-orientable surfaces with Euler characteristic 1-2''n''. * Klein bottles with ''n'' handles attached. These are non-orientable surfaces with Euler characteristic -2''n''. Therefore Euler characteristic and orientability describe a compact surfaces up to homeomorphism (and if surfaces are smooth then up to diffeomorphism). ==Compact surfaces== Compact surfaces with boundary are just these with one or more removed open disks whose closures are disjoint. ==Embeddings in R3== A compact surface can be embedded in R3 if it is orientable or if it has nonempty boundary. It is a consequence of the Whitney embedding theorem that any surface can be embedded in R4. ==Differential geometry== A simple review of the embedding of a surface in ''n'' dimensions, and a computation of the area of such a surface, is provided in the article volume form. Metric properties of Riemann surfaces are briefly reviewed in the the article Poincaré metric. ==Some models== To make some models, attach the sides of these (and remove the corners to puncture): * * B B v v v ^ *>>>>>* *>>>>>* v v v ^ v v v v A v v A A v ^ A A v v A A v v A v v v ^ v v v v v v v ^ *<<<<<* *>>>>>* * * B B sphere real projective plane Klein bottle torus (punctured Möbius band) (donut) ==Fundamental polygon== Each closed surface can be constructed from an even sided oriented polygon, called a fundamental polygon by pairwise identification of its edges. This construction can be represented as a string of length 2n of n distinct symbols where each symbol appears twice with exponent either +1 or -1. The exponent -1 signifies that the corresponding edge has the orientation opposing the one of the fundamental polygon. The above models can be described as follows: * sphere: * projective plane: * Klein bottle: * torus: ''(See the main article ''fundamental polygon'' for details.)'' ==Connected sum of surfaces== Given two surfaces M and M', their connected sum M # M' is obtained by removing a disk in each of them and gluing them along the newly formed boundary components. We use the following notation. * sphere: S * torus: T * Klein bottle: K * Projective plane: P Facts: * S # S = S * S # M = M * P # P = K * P # K = P # T We use a shorthand natation: nM = M # M # ... # M (n-times) with 0M = S. Closed surfaces are classified as follows: * gT (g-fold torus): orientable surface of genus g. * gP (g-fold projective plane): non-orientable surface of genus g. ==Algebraic surface== This notion of a surface is distinct from the notion of an ''algebraic surface''. A non-singular complex projective algebraic curve is a smooth surface. Algebraic surfaces over the complex number field have dimension 4 when considered as a manifold. ==See also== *minimal surface *Riemann surface *algebraic surface *Klein bottle *Torus *Sphere *Cylinder *Projective plane ==External links== *[http://xahlee.org/surface/gallery.html Math Surfaces Gallery, with 60 ~surfaces and Java Applet for live rotation viewing] Surfaces Geometric topology Articles with ASCII art
See other meanings of words starting from letter:
S
SB | SC | SD | SE | SF | SG | SH | SI | SJ | SK | SL | SM | SN | SO | SP | SR | SS | ST | SU | SW | SX | SY | SZ |Words begining with Surface:
Surface
Surface-conduction_Electron-emitter_Displays
Surface-mount
Surface-mount_technology
Surface-to-Air_missile
Surface-to-air_missile
Surface-to-air_missiles
Surface-to-air_missiles
Surface-to-surface_missile
Surface-to-surface_missiles
Surface-to-surface_piercing
Surface-to-surface_piercing
Surface-water_hydrology
Surface-water_hydrology
Surface-wave-sustained_mode
Surfaces
Surfaces
SurfaceWarrior
Surface_(disambiguation)
Surface_(magazine)
Surface_(Myst)
Surface_(physics)
Surface_Acoustic_Wave
Surface_acoustic_wave
Surface_air_temperature
Surface_air_temperature
Surface_area
Surface_area_of_earth
Surface_brightness
Surface_bundle
Surface_bundles_over_the_circle
Surface_bundle_over_the_circle
Surface_caching
Surface_caching
Surface_chemistry
Surface_combatant
Surface_core_level_shift
Surface_current_collection
Surface_effect_ship
Surface_energy
Surface_energy
Surface_epithelial-stromal_tumour
Surface_feature_nomenclature_of_solar_system_bodies
Surface_forces_apparatus
Surface_functionalization
Surface_gravity
Surface_hardening
Surface_integral
Surface_layer
Surface_Marker_Buoy
Surface_Marker_Buoy
Surface_marker_buoy
Surface_Marshal
Surface_micromachining
Surface_Mining
Surface_mining
Surface_mount
Surface_mount_technology
Surface_normal
Surface_normal
Surface_normals
Surface_of_revolution
Surface_Patterns
Surface_Patterns
Surface_physics
Surface_piercing
Surface_piercing
Surface_piercings
Surface_plasmon
Surface_Plasmon_Resonance
Surface_plasmon_resonance
Surface_plate
Surface_pressure
Surface_run_off
Surface_selection_rule
Surface_ship
Surface_structure
Surface_supplied_diver
Surface_supplied_diving
Surface_target
Surface_temperature
Surface_Temperature_Inversion
Surface_temperature_inversion
Surface_temperature_record
Surface_temperature_record
Surface_Tension
Surface_tension
Surface_tension
Surface_texel
Surface_to_air_missile
Surface_Transportation_Act
Surface_Transportation_Board
Surface_Transportation_Board
Surface_warfare
Surface_Warfare_Badge
Surface_Warfare_insignia
Surface_Warfare_Officer_Pin
Surface_Warfare_Pin
Surface_Water
Surface_water
Surface_wave
Surface_waves
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