Meta tags:

Headings (most frequently used words):

submersion, mathematics, contents, definition, theorem, examples, local, normal, form, topological, manifold, submersions, see, also, notes, references, further, reading, maps, between, spheres, families, of, algebraic, varieties,

Text of the page (most frequently used words):
the (61), displaystyle (52), #submersion (20), and (20), mathbb (20), edit (15), manifolds (14), differential (14), map (14), for (14), smooth (13), are (13), point (13), this (10), dimension (10), manifold (9), differentiable (9), isbn (9), theorem (9), submersions (8), mathematics (7), wikipedia (7), theory (7), vector (7), form (7), local (7), that (7), regular (7), between (7), colon (7), lie (6), complex (6), surjective (6), 978 (6), varieties (6), algebraic (6), partial (6), page (5), maps (5), riemannian (5), topological (5), geometry (5), cambridge (5), also (5), such (5), equal (5), dim (5), projection (5), given (5), whose (5), contents (4), search (4), with (4), from (4), space (4), affine (4), almost (4), structure (4), main (4), new (4), york (4), 1994 (4), see (4), all (4), there (4), these (4), points (4), fibers (4), more (4), matrix (4), examples (4), rank (4), bmatrix (4), critical (4), definition (4), hide (4), move (4), sidebar (4), toggle (3), view (3), contact (3), history (3), related (3), tensor (3), tangent (3), normal (3), fibration (3), field (3), curvature (3), derivative (3), bundle (3), pushforward (3), 1999 (3), lang (3), 521 (3), university (3), press (3), arnold (3), singularities (3), general (3), where (3), value (3), ldots (3), then (3), coordinates (3), because (3), left (3), right (3), families (3), get (3), example (3), called (3), begin (3), end (3), inverse (3), spheres (3), maximal (3), frac (3), jacobian (3), each (3), tools (3), languages (2), table (2), about (2), privacy (2), policy (2), under (2), terms (2), may (2), non (2), use (2), was (2), categories (2), short (2), description (2), wikidata (2), https (2), index (2), singularity (2), principal (2), generalized (2), ehresmann (2), stable (2), algebra (2), fiber (2), dual (2), cotangent (2), product (2), volume (2), flow (2), ricci (2), pullback (2), one (2), rham (2), closed (2), curve (2), immersion (2), sard (2), list (2), submanifold (2), atlas (2), basic (2), what (2), necessary (2), further (2), reading (2), 2012 (2), mineola (2), dover (2), publications (2), 486 (2), physics (2), sternberg (2), springer (2), kosinski (2), 2007 (2), gallot (2), lafontaine (2), 2004 (2), hulin (2), frankel (2), 1997 (2), 8176 (2), carmo (2), crampin (2), pirani (2), curves (2), 1985 (2), varchenko (2), gusein (2), zade (2), references (2), notes (2), some (2), continuous (2), charts (2), preimage (2), empty (2), content (2), open (2), neighborhood (2), line (2), plane (2), since (2), equivalently (2), mathcal (2), large (2), class (2), consider (2), family (2), used (2), special (2), framed (2), groups (2), bordism (2), downarrow (2), rightarrow (2), gamma (2), take (2), have (2), when (2), two (2), has (2), function (2), dimensions (2), around (2), psi (2), circ (2), phi (2), than (2), above (2), its (2), df_ (2), let (2), everywhere (2), here (2), variety (2), appearance (2), upload (2), file (2), changes (2), links (2), read (2), article (2), log (2), create (2), account (2), donate (2), menu (2), add, topic, mobile, cookie, statement, statistics, developers, code, conduct, legal, safety, contacts, disclaimers, text, available, additional, apply, using, site, you, agree, registered, trademark, profit, organization, wikimedia, foundation, inc, creative, commons, attribution, sharealike, license, rendered, parsoid, last, edited, october, 2025, utc, hidden, different, articles, functions, retrieved, org, php, title, submersion_, oldid, 1319126227, stratified, supermanifold, stratifold, sheaf, over, commutative, algebras, secondary, calculus, orbifold, hausdorff, hilbert, fréchet, diffiety, diffeology, banach, generalizations, moving, frame, morse, gauge, classification, parallel, transport, levi, civita, koszul, cartan, connections, subbundle, spinor, jet, associated, adjoint, bundles, wedge, density, riemann, interior, exterior, valued, cohomology, covariant, exact, covectors, torsion, bracket, distribution, vectors, tensors, tame, symplectic, stein, sasakian, rizza, sub, pseudo, hypercomplex, quaternionic, prime, poisson, parallelizable, oriented, nilmanifold, boundary, group, kenmotsu, kähler, hyper, hyperbolic, hermitian, hadamard, flat, finsler, fibered, einstein, complete, collapsing, calabi, yau, types, section, integral, foliation, exponential, geodesic, diffeomorphism, whitney, embedding, noether, hopf, rinow, stokes, frobenius, darboux, atiyah, singer, theorems, concepts, category, glossary, mathoverflow, net, questions, 376129, sufficient, conditions, 47855, shlomo, zvi, graduate, texts, 387, 98593, fundamentals, serge, antoni, albert, 1993, 46244, sylvestre, jacques, 3rd, berlin, 540, 20493, verlag, dominique, theodore, 1481707, 38753, 3490, manfredo, perdigao, michael, felix, edward, england, 23190, applicable, bruce, james, giblin, peter, 1984, 0774048, 42999, birkhäuser, 3187, alexander, sabir, vladimir, 378, 181, 185, 243, well, defined, surjection, follows, full, either, possibly, known, particular, conclusion, holds, disconnected, exists, standard, considering, spaces, note, should, actually, remove, double, root, times, another, underlying, weierstrass, widely, studied, includes, many, technical, complexities, demonstrate, perverse, sheaves, intersection, homology, elliptic, mathfrak, kind, fact, cobordism, intimately, homotopy, omega, images, elements, path, higher, surjectivity, condition, existence, trivialization, diffeomorphisms, any, subset, hence, only, subsets, dimensional, setminus, every, except, consequence, giving, exist, restricts, which, expressed, becomes, ordinary, application, corresponding, denoted, can, equipped, equals, difference, orthogonal, authors, term, describe, not, indeed, useful, notion, greater, notions, coincide, however, less, according, cannot, but, still, commonly, formulation, case, otherwise, constant, linear, them, concept, topology, redirects, singular, free, encyclopedia, item, other, projects, printable, version, download, pdf, print, export, switch, legacy, parser, shortened, url, cite, information, permanent, link, actions, english, talk, tiếng, việt, українська, português, nederlands, 한국어, 日本語, italiano, français, فارسی, deutsch, català, subsection, top, personal, pages, recent, community, portal, learn, help, contribute, random, current, events, navigation, jump,

Text of the page (random words):
submersion mathematics wikipedia jump to content main menu main menu move to sidebar hide navigation main page contents current events random article about wikipedia contact us contribute help learn to edit community portal recent changes upload file special pages search search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 definition 2 submersion theorem 3 examples toggle examples subsection 3 1 maps between spheres 3 2 families of algebraic varieties 4 local normal form 5 topological manifold submersions 6 see also 7 notes 8 references 9 further reading toggle the table of contents submersion mathematics 12 languages català deutsch فارسی français italiano 日本語 한국어 nederlands português українська tiếng việt 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia differential map between manifolds whose differential is everywhere surjective regular point redirects here for regular point of an algebraic variety see singular point of an algebraic variety in mathematics a submersion is a differentiable map between differentiable manifolds whose differential pushforward is everywhere surjective it is a basic concept in differential topology dual to that of an immersion definition edit let m and n be differentiable manifolds and let f m n displaystyle f colon m to n be a differentiable map between them the map f is a submersion at a point p m displaystyle p in m if its differential d f p t p m t f p n displaystyle df_ p colon t_ p m to t_ f p n is a surjective linear map 1 in this case p is called a regular point of the map f otherwise p is a critical point a point q n displaystyle q in n is a regular value of f if all points p in the preimage f 1 q displaystyle f 1 q are regular points a differentiable map f that is a submersion at each point p m displaystyle p in m is called a submersion equivalently f is a submersion if its differential d f p displaystyle df_ p has constant rank equal to the dimension of n some authors use the term critical point to describe a point where the rank of the jacobian matrix of f at p is not maximal 2 indeed this is the more useful notion in singularity theory if the dimension of m is greater than or equal to the dimension of n then these two notions of critical point coincide however if the dimension of m is less than the dimension of n all points are critical according to the definition above the differential cannot be surjective but the rank of the jacobian may still be maximal if it is equal to dim m the definition given above is the more commonly used one e g in the formulation of sard s theorem submersion theorem edit given a submersion f m n displaystyle f colon m to n between smooth manifolds of dimensions m displaystyle m and n displaystyle n for each x m displaystyle x in m there exist surjective charts ϕ u r m displaystyle phi u to mathbb r m of m displaystyle m around x displaystyle x and ψ v r n displaystyle psi v to mathbb r n of n displaystyle n around f x displaystyle f x such that f displaystyle f restricts to a submersion f u v displaystyle f colon u to v which when expressed in coordinates as ψ f ϕ 1 r m r n displaystyle psi circ f circ phi 1 mathbb r m to mathbb r n becomes an ordinary orthogonal projection as an application for each p n displaystyle p in n the corresponding fiber of f displaystyle f denoted m p f 1 p displaystyle m_ p f 1 p can be equipped with the structure of a smooth submanifold of m displaystyle m whose dimension equals the difference of the dimensions of n displaystyle n and m displaystyle m this theorem is a consequence of the inverse function theorem see inverse function theorem giving a manifold structure for example consider f r 3 r displaystyle f colon mathbb r 3 to mathbb r given by f x y z x 4 y 4 z 4 displaystyle f x y z x 4 y 4 z 4 the jacobian matrix is f x f y f z 4 x 3 4 y 3 4 z 3 displaystyle begin bmatrix frac partial f partial x frac partial f partial y frac partial f partial z end bmatrix begin bmatrix 4x 3 4y 3 4z 3 end bmatrix this has maximal rank at every point except for 0 0 0 displaystyle 0 0 0 also the fibers f 1 t a b c r 3 a 4 b 4 c 4 t displaystyle f 1 t left a b c in mathbb r 3 a 4 b 4 c 4 t right are empty for t 0 displaystyle t 0 and equal to a point when t 0 displaystyle t 0 hence we only have a smooth submersion f r 3 0 0 0 r 0 displaystyle f colon mathbb r 3 setminus 0 0 0 to mathbb r _ 0 and the subsets m t a b c r 3 a 4 b 4 c 4 t displaystyle m_ t left a b c in mathbb r 3 a 4 b 4 c 4 t right are two dimensional smooth manifolds for t 0 displaystyle t 0 examples edit any projection π r m n r n r m n displaystyle pi colon mathbb r m n rightarrow mathbb r n subset mathbb r m n local diffeomorphisms riemannian submersions the projection in a smooth vector bundle or a more general smooth fibration the surjectivity of the differential is a necessary condition for the existence of a local trivialization maps between spheres edit a large class of examples of submersions are submersions between spheres of higher dimension such as f s n k s k displaystyle f s n k to s k whose fibers have dimension n displaystyle n this is because the fibers inverse images of elements p s k displaystyle p in s k are smooth manifolds of dimension n displaystyle n then if we take a path γ i s k displaystyle gamma i to s k and take the pullback m i s n k f i x γ s k displaystyle begin matrix m_ i to s n k downarrow downarrow f i x rightarrow gamma s k end matrix we get an example of a special kind of bordism called a framed bordism in fact the framed cobordism groups ω n f r displaystyle omega _ n fr are intimately related to the stable homotopy groups families of algebraic varieties edit another large class of submersions is given by families of algebraic varieties π x s displaystyle pi mathfrak x to s whose fibers are smooth algebraic varieties if we consider the underlying manifolds of these varieties we get smooth manifolds for example the weierstrass family π w a 1 displaystyle pi mathcal w to mathbb a 1 of elliptic curves is a widely studied submersion because it includes many technical complexities used to demonstrate more complex theory such as intersection homology and perverse sheaves this family is given by w t x y a 1 a 2 y 2 x x 1 x t displaystyle mathcal w left t x y in mathbb a 1 times mathbb a 2 y 2 x x 1 x t right where a 1 displaystyle mathbb a 1 is the affine line and a 2 displaystyle mathbb a 2 is the affine plane since we are considering complex varieties these are equivalently the spaces c c 2 displaystyle mathbb c mathbb c 2 of the complex line and the complex plane note that we should actually remove the points t 0 1 displaystyle t 0 1 because there are singularities since there is a double root local normal form edit if f m n is a submersion at p and f p q n then there exists an open neighborhood u of p in m an open neighborhood v of q in n and local coordinates x 1 x m at p and x 1 x n at q such that f u v and the map f in these local coordinates is the standard projection f x 1 x n x n 1 x m x 1 x n displaystyle f x_ 1 ldots x_ n x_ n 1 ldots x_ m x_ 1 ldots x_ n it follows that the full preimage f 1 q in m of a regular value q in n under a differentiable map f m n is either empty or a differentiable manifold of dimension dim m dim n possibly disconnected this is the content of the regular value theorem also known as the submersion theorem in particular the conclusion holds for all q in n if the map f is a submersion topological manifold submersions edit submersions are also well defined for general topological manifolds 3 a topological manifold submersion is a continuous surjection f m n such that for all p in m for some continuous charts ψ at p and φ at f p the map ψ 1 f φ is equal to the projection map from r m to r n where m dim m n dim n see also edit ehresmann s fibration theorem notes edit crampin pirani 1994 p 243 do carmo 1994 p 185 frankel 1997 p 181 gallot hulin lafontaine 2004 p 12 kosinski 2007 p 27 lang 1999 p 27 sternberg 2012 p 378 arnold gusein zade varchenko 1985 lang 1999 p 27 references edit arnold vladimir i gusein zade sabir m varchenko alexander n 1985 singularities of differentiable maps volume 1 birkhäuser isbn 0 8176 3187 9 bruce james w giblin peter j 1984 curves and singularities cambridge university press isbn 0 521 42999 4 mr 0774048 crampin michael pirani felix arnold edward 1994 applicable differential geometry cambridge england cambridge university press isbn 978 0 521 23190 9 do carmo manfredo perdigao 1994 riemannian geometry isbn 978 0 8176 3490 2 frankel theodore 1997 the geometry of physics cambridge cambridge university press isbn 0 521 38753 1 mr 1481707 gallot sylvestre hulin dominique lafontaine jacques 2004 riemannian geometry 3rd ed berlin new york springer verlag isbn 978 3 540 20493 0 kosinski antoni albert 2007 1993 differential manifolds mineola new york dover publications isbn 978 0 486 46244 8 lang serge 1999 fundamentals of differential geometry graduate texts in mathematics new york springer isbn 978 0 387 98593 0 sternberg shlomo zvi 2012 curvature in mathematics and physics mineola new york dover publications isbn 978 0 486 47855 5 further reading edit https mathoverflow net questions 376129 what are the sufficient and necessary conditions for surjective submersions to b rq 1 v t e manifolds glossary list category basic concepts topological manifold atlas differentiable smooth manifold differential structure smooth atlas submanifold riemannian manifold smooth map submersion pushforward tangent space differential form vector field main theorems list atiyah singer index darboux s de rham s frobenius generalized stokes hopf rinow noether s sard s whitney embedding maps curve diffeomorphism local geodesic exponential map in lie theory foliation immersion integral curve lie derivative section submersion types of manifolds calabi yau closed collapsing complete almost complex almost contact einstein fibered finsler almost ricci flat g structure hadamard hermitian hyperbolic hyper kähler kenmotsu lie group lie algebra manifold with boundary nilmanifold oriented parallelizable poisson prime quaternionic hypercomplex pseudo sub riemannian rizza sasakian stein almost symplectic tame tensors vectors distribution lie bracket pushforward tangent space bundle torsion vector field vector flow covectors closed exact covariant derivative cotangent space bundle de rham cohomology differential form complex vector valued one form exterior derivative interior product pullback ricci curvature flow riemann curvature tensor tensor field density volume form wedge product bundles adjoint affine associated cotangent dual fiber co fibration jet lie algebra stable normal principal spinor subbundle tangent tensor vector connections affine cartan ehresmann form generalized koszul levi civita principal vector parallel transport related classification of manifolds gauge theory history morse theory moving frame singularity theory generalizations banach diffeology diffiety fréchet hilbert k theory non hausdorff orbifold secondary calculus over commutative algebras sheaf stratifold supermanifold stratified space retrieved from https en wikipedia org w index php title submersion_ mathematics oldid 1319126227 categories maps of manifolds smooth functions hidden categories articles with short description short description is different from wikidata this page was last edited on 27 october 2025 at 23 17 utc page was rendered with parsoid text is available under the creative commons attribution sharealike 4 0 license additional terms may apply by using this site you agree to the terms of use and privacy policy wikipedia is a registered trademark of the wikimedia foundation inc a non profit organization privacy policy about wikipedia disclaimers contact wikipedia legal safety contacts code of conduct developers statistics cookie statement mobile view search search toggle the table of contents submersion mathematics 12 languages add topic