BASIC DIFFERENTIABLE MANIFOLDS AND APPLICATIONS MAPS WITH TRANSLATORS TANGENT AND COTANGENT SPACES

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1 IJMMS, Vol., No. -2, (January-December 205): Serals Publcatons ISSN: BASIC DIFFERENTIABLE MANIFOLDS AND APPLICATIONS MAPS WITH TRANSLATORS TANGENT AND COTANGENT SPACES Mohamed M. Osman* Abstract In ths paper of Remannan geometry to pervous of dfferentable manfolds ( M) p whch are used n an essental way n basc concepts of Remannan geometry we study the defectons, examples of the problem of dfferentally proecton mappng parameterzaton (U ) - system by struttng rank k on surfaces n k dmensonal s sub manfolds space of R n, we prove that tree depends only on the doublng charts mappng manfolds, sub manfolds ts quanttatve extrnsc dmenson... INTRODUCTION A manfolds s a generalzaton of curves and surfaces to hgher dmenson, t s Eucldean n {E R } n that every pont has a neghbored, called a chart homeomorphc to an open subset of R n, the coordnates on a chart allow one to carry out computatons as though n a Eucldean space, so that many concepts from R n, such as dfferentablty, pont dervatons, tangents, cotangents spaces, and dfferental forms carry over to a manfold. In ths paper we gven the basc defntons and propertes of a smooth manfold and smooth maps between manfolds, ntally the only way we have to verfy that a space, we descrbe a set of suffcent condtons under whch a quotent topologcal space becomes a manfold s exhbt a collecton of C compatble charts coverng the space becomes a manfold, gvng us a second way to construct manfolds, a topologcal manfolds C analytc manfolds, statng wth topologcal manfolds, whch are Hausdorff second countable s locally Eucldean space, we ntroduce the concept of maxmal C atlas, whch makes a topologcal manfold nto a smooth manfold, a topologcal manfold s a Hausdorff, second countable s local Eucldean of dmenson n, f every pont p n M has a neghborhood * Department of mathematcs faculty of scence Unversty of Al-Baha K.S.A, E-mal: mm.eltngary@hotmal.com

2 26 Mohamed M. Osman U such that there s a homeomorphsm from U onto a open subset of R n, we call the par a coordnate map or coordnate system on U, we sad chart (U, ) s centered at p U, (p) = 0 and we defne the smooth maps f : M N where M, N are dfferental manfolds we wll say that f s smooth f there are atlases (U, h ) on M and (V, g ) on N. 2. TOPOLOGICAL MANIFOLDS In ths secton, the bascally an m-dmensonal topologcal manfold s a topologcal space M whch s locally homeomorphc to R m, defnton s a topologcal space M s called an m-dmensonal (topologcal manfold) f the followng condtons hold () M s a hausdorff space () for any p M there exsts a neghborhood U of P whch s homeomorphc to an open subset V R m () M has a countable bass of open sets, Fgurer () coordnate charts (U, ) Axom () s equvalent to sayng that p M has a open neghborhood U P homeomorphc to open dsc D m n R m, axom () says that M can covered by countable many of such neghborhoods, the coordnate chart (U, ) where U are coordnate neghborhoods or charts and are coordnate homeomorphsms, transtons between dfferent choces of coordnates are called transtons maps =, whch are agan homeomorphsms by defnton, we usually wrte p = (x), : U V R n as coordnates for U, see Fgure (2), and p = (x), : V U M as coordnates for U, the coordnate charts (U, ) are coordnate neghborhoods, or charts, and are coordnate homeomorphsms, transtons between dfferent choces of coordnates are called transtons maps = whch are agan homeomorphsms by defnton, we usually x = (p), : U V R n as a parameterzaton U. A collecton A = {(, U } I of coordnate chart wth M = U s called atlas for M. The transton maps Fgurer (2) a topologcal space M s called ( hausdorff) f for any par p, q M, there exst open neghborhoods p U and q U such that Fgurer : The coordnate charts (U, ) Fgurer 2: : the transton maps

3 Basc Dfferentable Manfolds and Applcatons maps wth U U# for a topologcal space M wth topology U can be wrtten as unon of sets n, a bass s called a countable bass s a countable set. Fgurer 3: Coordnate maps for boundary ponts A topologcal M space s called an m-dmensonal topologcal manfold wth boundary M M f the followng condtons () M s hausdorff space () for any pont p M there exsts a neghborhood U of p whch s homeomorphc to an open subset V H m () M has a countable bass of open sets, Fgure (3) can be rephrased as follows any pont p U s contaned n neghborhood U to D m H m the set M s a locally homeomorphc to R m or H m the boundary M M s subset of M whch conssts of ponts p. Defnton 2. Let X be a set a topology U for X s collecton of X satsfyng () and X are n U () the ntersecton of two members of U s n U () the unon of any number of members U s n U. The set X wth U s called a topologcal space the members U u are called the open sets. let X be a topologcal space a subset N X wth x N s called a neghborhood of x f there s an open set U wth x U N, for example f X a metrc space then the closed ball D (x) and the open ball D (x) are neghborhoods of x a subset C s sad to closed f X / C s open Defnton 2.3 A functon f : X Y between two topologcal spaces s sad to be contnuous f for every open set U of Y the pre-mage f (U) s open n X. Defnton 2.4 Let X and Y be topologcal spaces we say that X and Y are homeomorphc f there exst contnuous functon f : X Y, g : Y X such that f g = d y and g f = d x we wrte X Y and say that f and g are homeomorphsms between X and Y, by the defnton a functon f : X Y s a homeomorphsms f and only f () f s a bectve () f s contnuous () f s also contnuous. 3. DIFFERENTIABLE MANIFOLDS A dfferentable manfolds s necessary for extendng the methods of dfferental calculus to spaces more general R n a subset S R 3 s regular surface f for every

4 28 Mohamed M. Osman pont p S the a neghborhood V of P s R 3 and mappng x : u R 2 V S open set U R 2 such that () x s dfferentable homomorphsm () the dfferentable (dx) q : R 2 R 3, the mappng x s called a parametnzaton of S at P the mportant consequence of dfferentable of regular surface s the fact that the transton also example below f x : U S and x : U S are x (U ) x (U ) = w #, the mappngs x x : x (w) R 2 and Are dfferentable Fgure (4) x x : x (w) R (3.) Fgure 4: a dfferentable structure on M A dfferentable structure on a set M nduces a natural topology on M t suffces to A M to be an open M set n f and only f x (A x (U )) s an open set n R n for all t s easy to verfy that M and the empty set are open sets that a unon of open sets s agan set and that the fnte ntersecton of open sets remans an open set. Manfold s necessary for the methods of dfferental calculus to spaces more general than de R n, a dfferental structure on a manfolds M nduces a dfferental structure on every open subset of M, n partcular wrtng the entres of an n k matrx n successon dentfes the set of all matrces wth R n,k, an n k matrx of rank k can be vewed as a k-frame that s set of k lnearly ndependent vectors n R n, V n,k K n s called the steels manfold,the general lnear group GL (n) by the foregong V n,k s dfferental structure on the group n of orthogonal matrces, we defne the smooth maps functon f : M N where M, N are dfferental manfolds we wll say that f s smooth f there are atlases (U, h ) on M, (V B, g B ) on N, such that the maps g B f h are smooth wherever they are defned f s a homeomorphsm f s smooth and a smooth nverse. A dfferentable structures s topologcal s a manfold t an open coverng U where each set U s homeoomorphc, va some homeomorphsm h to an open subset of Eucldean space R n, let M be a topologcal

5 Basc Dfferentable Manfolds and Applcatons maps wth space, a chart n M conssts of an open subset U M and a homeomorphsm h of U onto an open subset of R m, a C r atlas on M s a collecton (U, h )of charts such that theu cover M and h B, h the dfferentable vector felds on a dfferentable manfold M, let X and Y be a dfferentable vector feld on a dfferentable manfolds M then there exsts a unque vector feld Z such that such that, for all f D, Zf = (XY YX) f f that p M and let x : U M be a parameterzaton at p and X a, y a x y f f XYf X b, YXf Y a x x Therefore Z s gven n the parameterzaton x by z. (3.2) Z f = (X Y f Y X F), b a a b, x x Are dfferentable ths a regular surface s ntersect from one to other can be made n a dfferentable manner the defect of the defnton of regular surface s ts dependence on R 3. A dfferentable manfold s locally homeomorphc to R n the fundamental theorem on exstence, unqueness and dependence on ntal condtons of ordnary dfferental equatons whch s a local theorem extends naturally to dfferentable manfolds. For famlar wth dfferental equatons can assume the statement below whch s all that we need for example X be a dfferentable on a dfferentable manfold M and p M then there exst a neghborhood p M and U p M an nter (, ), 0, and a dfferentable mappng : (, ) U M such that curve t (t, q) and (0, q) = q a curve :(, ) M whch satsfes the condtons (t) = X ((t)) and (0) = q s called a traectory of the feld X that passes through q for t = 0. A dfferentable manfold of dmenson N s a set M and a famly of nectve mappng x R n M of open sets u R n nto M such that () u x (u ) = M () for any, wth x (u ) x (u ) () the famly (u, x ) s maxmal relatve to condtons (), () the par (u, x ) or the mappng x wth p x (u ) s called a parameterzaton, or system of coordnates of M, u x (u ) = M the coordnate charts (U, ) where U are coordnate neghborhoods or charts, and are coordnate homeomorphsms transtons are between dfferent choces of coordnates are called transtons maps, : (3.3)

6 30 Mohamed M. Osman Whch are anse homeomorphsms by defnton, we usually wrte x = (p), : U V R n collecton U and p = (x), : V U M for coordnate charts wth s M = U called an atlas for M of topologcal manfolds. A topologcal manfold M for whch the transton maps, = ( )for all pars, n the atlas are homeomorphsms s called a dfferentable, or smooth manfold, the transton maps are mappng between open subset of R m, homeomorphsms between open subsets of R m are C maps whose nverses are also C maps, for two charts U and U the transtons mappng, = : (U U ) (U U ) (3.4) And as such are homeomophsms between these open of R m, for example the dfferentablty ( ) s acheved the mappng ( ) and ( ) whch are homeomorphsms snce (A A) by assumpton ths establshes the equvalence (A A), for example let N and M be smooth manfolds n and m respecpectvely, and let f : N M be smooth mappng n local coordnates f = ( f ): (U) (V) Fgurer (5), wth respects charts (U, ) and (V, ), the rank of f at p N s defned as the rank of fat (p)(.e) rk(f) p = rk (J f) (p) s the Jacobean of f at p ths defnton s ndependent of the chosen chart, the commutatve dagram n that f ( ) f ( ) (3.5) Fgurer 5: coordnate maps the cone C Snce ( ) and ( ) are homeomorphsms t easly follows that whch show that our noton of rank s well defned (J f) x = J( ) y J f( ), f a map has constant rank for all p N we smply wrte rk (f), these are called constant rank mappng. The product two manfolds M and be two C k -manfolds of dmenson n and n 2 respectvely the topologcal space M are arbtral unons of sets of the form U V where U s open n M and V s open n, can be gven the structure C k manfolds of dmenson n, n 2 by defnng charts as follows for any charts M on (V, )on we declare that (U V, ) (s chart on M where :

7 Basc Dfferentable Manfolds and Applcatons maps wth... 3 U V R (n +n 2 ) s defned so that (p, q) = ( (p), (q)) for all (p, q) U V. A gven a C k n-atlas, A on M for any other chart (U, ) we say that (U, ) s compatble wth the atlas A f every map ( ) and ( ) s C k whenever U U # 0 the two atlases A and A s compatble f every chart of one s compatble wth other atlas see Fgure (6) Fgurer 6: coordnate dfeomorphsms = and = A sub manfolds of others of R n for nstance S 2 s sub manfolds of R 3 t can be obtaned as the mage of map nto R 3 or as the level set of functon wth doman R 3 we shall examne both methods below frst to develop the basc concepts of the theory of Remannan sub manfolds and then to use these concepts to derve a equanttve nterpretaton of curvature tensor, some basc defntons and termnology concernng sub manfolds, we defne a tensor feld called the second fundamental form whch measures the way a sub manfold curves wth the ambent manfold, for example X be a sub manfold of Y of : E X and g : E Y be two vector brndled and assume that E s compressble, let f : E Y and g : E Y be two tubular neghborhoods of X n Y then there exsts a C p. 4. DIFFERENTIABLE MANIFOLDS TANGENT SPACE In ths secton s defned tangent space to level surface be a curve s n R n, :t ( (t), 2 (t),..., n (t)) a curve can be descrbed as vector valued functon converse a vector valued functon gven cur ve, the tangent lne at the pont dt dt dt n d d d ( t) t0..., t0 we many k bout smooth curves that s curves wth all contnuous hgher dervatves cons the level surface f(x, x 2,..., x n ) = c of a dfferentable functon f where x to th coordnate the gradent vector of f at

8 32 Mohamed M. Osman f pont P = x (P) x 2 (P),..., x n (P) s,..., f f n x x s gven a vector u = (u,..., f f u n ) the drecton dervatve Du f f u u u x x surface f(x, x 2,..., x n ) the tangent s gven by equaton. f x f x n..., n n n ( P)( x x )( P)... ( P)( x x )( P) 0 n the pont P on level (4.) For the geometrc vews the tangent space shout consst of all tangent to smooth curves the pont P, assume that s curve through t = t 0 s the level surface f(x, x 2,..., x n ) = c, f( (t), 2 (t),..., n (t) = c by takng dervatves on both f f n ( P)( ( t 0 )... ( P) ( t)) 0 n and so the tangent lne of s really normal x x orthogonal to f, where runs over all possble curves on the level surface through the pont P. The surface M be a C manfold of dmenson n wth k the most ntutve to defne tangent vectors s to use curves, p M be any pont on M and let : ]-, [ M be a C curve passng through p that s wth (M) = p unfortunately f M s not embedded n any R N the dervatve (M) does not make sense, however for any chart (U, ) at p the map ( ) at a C curve n R n and tangent vector v = ( v) / (M) s wll defned the trouble s that dfferent curves the same v gven a smooth mappng f : N M we can defne how tangent vectors n T p N are mapped to tangent vectors n T q M wth (U, ) choose charts q = f(p) for p N and (V, ) for q M we defne the tangent map or flash-forward of f as a gven tangent vector X p = []T p N and d f * : T p M, f * ([]=[f ]). A tangent vector at a pont p n a manfold M s a dervaton at p, ust as for R n the tangent at pont p form a vector space T p (M) called the tangent space of M at p, we also wrte T p (M) a dfferental of map f : N M be a C map between two manfolds at each pont p N the map F nduce a lnear map of tangent space called ts dfferental at p, F * : T p N T F(p) N as follows t X p T p N then F * (X p ) s the tangent vector n T F(p) M defned see Fgurer (7) (F * (X p )) f = X p (f F) R, f C (M)` (4.2) The tangent vectors gven any C -manfold M of dmenson n wth k for any p M, tangent vector to M at p s any equvalence class of C -curves through p on M modulo the equvalence relaton defned n the set of all tangent vectors at p s denoted by T p M we wll show that T p M s a vector space of dmenson n of m. The tangent space T p M s defned as the vector space spanned by the tangents at p to

9 Basc Dfferentable Manfolds and Applcatons maps wth Fgurer 7: coordnate representaton for f wth f(u) V all curves passng through pont p n the manfold M, and the cotangent T * p M of a manfold at p M s defned as the dual vector space to the tangent space T p M, we take the bass vectors E x for T M and we wrte the bass vectors p T* M as p the dfferental lne elements e = dx thus the nner product s gven by / x, dx. Defnton 4. Let M and be dfferentable manfolds a mappng : M s a dfferentable f t s dfferentable, obectve and ts nverse s dffeomorphsm f t s dfferentable s sad to be a local dffeomorphsm at p M f there exst neghborhoods U of p and V of (p)such that : U V s a dffeomorphsm, the noton of dffeomorphsm s the natural dea of equvalence between dfferentable manfolds, ts an mmedate consequence of the chan rule that f : M s a dffeomorphsm then d : T p M T (p) (4.5) Is an somorphsm for all : M n partcular, the dmensons of M and are equal a local converse to ths fact s the followng d : T p M T (p) s an somorphsm then s a local dffeomorphsm at p from an mmedate applcaton of nverse functon n R n, for example be gven a manfold structure agan A mappng f : M N n ths case the manfolds N and M are sad to be homeomorphsm, usng charts (U, ) and (V, ) for N and M respectvely we can gve a coordnate expresson f : M N Defnton 4.2 Let M and M be dfferentable manfolds and let : M 2 be dfferentable mappng for every P M and for each v T p M choose a dfferentable curve : (, ) M wth (M) = p and (0) = v take = the

10 34 Mohamed M. Osman mappng d p : T (p) by gven by d (v) = (M) s lne of and M M 2 be a dfferentable mappng and at p M be such d : T p M T s an somorphsm then s a local homeomorphsm. n m Proposton 4.3 Let M and M be dfferentable manfolds and let : M be a dfferentable mappng, for every p M and for each v T p M choose a dfferentable curve : (, ) M wth (0) = p, (0) = v take = the mappng d : T p M T (p) gven by d p (v) = (0) s a lnear mappng that dose not depend on the choce of see Fgurer (8) Fgurer 8: y x Theorem 4.4 Let G be le group of matrces and suppose that log defnes a coordnate chart the near the dentty element of G, dentfy the tangent space g = T G at the dentty element wth a lnear subspace of matrces, va the log and then a le algebra wth [B, B 2 ] = B B 2 B, B the space g s called the le algebra of G 2 Proof: It suffces to show that for every two matrces B, B 2 g the [B, B 2 ] s also an element of g as [B, B 2 ] s clearly ant commutatve and the (Jacobs dentty) holds for A(t) = (B t) exp (B 2 t) exp ( B t) exp ( B 2 t) exp (4.6) Defne for [t] wth suffcently small a path A(T) n G such that A(O) = I usng for each factor the local formula (Bt) exp = I + Bt +/2 B 2 t 2 + O (t 2 ) A(t) = I + [B, B 2 ]t 2 + O(t), t 0 (4.7) hence B(t) = log A(t) = [B, B 2 ]t 2 + O (t 2 ) exp B(t) = A(t) hold for any suffcently that le bracket [B, B 2 ] g on algebra s an nfntesmal verson of the commutaton (g, g ) ( g, g ) n the correspondng (le group). 2 Theorem 4.5 The tangent bundle TM has a canoncal dfferentable structure makng t nto a smooth 2N-dmensonal manfold, where N = dm. The charts dentfy

11 Basc Dfferentable Manfolds and Applcatons maps wth any U p U(T p M) (TM) for an coordnate neghborhood U M, wth U R n that s hausdorff and second countable s called (The manfold of tangent vectors) Defnton 4.6 A smooth vectors felds on manfolds M s map X : M TM such that () X(P) T p M for every G () n every chart X s expressed as a (/ x ) wth coeffcents a (x) smooth functons of the local coordnates x. Theorem 4.7 Suppose that on a smooth manfold M of dmenson n there exst n vector felds (x (), x (2),..., x (n),) for a bass of T p M at every pont p of m, then T p M s somorphc to M R n m here somorphc means that TM and M R n are homeomorphsm as smooth manfolds and for every p M, the homeomorphsm restrcts to between the tangent space T p M and vector space {P } R n. Proof: defne : a T p M TM on other hand, for any M R n for some a R now defne : a TM [( s ) : (a,..., a n ) M R n ] s t clear from the constructon and the hypotheses of theorem that and are smooth usng an arbtrary chart : U M R n and correspondng chart 5 CONCLUSION (a) (b) T: (U) TM R n R m (4.8) The paper study a manfolds s a generalzaton of curves and surfaces, locally Eucldean {E n } n every pont has a neghbored s called a chart homeomorphc, so that many concepts from R n as dfferentablty manfolds. In ths paper we gve the basc defntons, theorems and propertes of smooth topologcal be comes a manfold s to exhbt a collecton of C s compatble charts (c) The tangent, cotangent vector space manfold of dmenson k wth k the most ntutve method to defne tangent vectors to use curves, tangent space T p M and tangent space at some pont p M the cotangent defnes as dual vector space of p M. References * Tp M s [] R.C.A.M van der vorst soluton manual Dr. G. J. Rdderbos,htt//creatveommons.org/ sprng 94305(202),secnd a letter to creatve commons, 559 Nathan a bbott, way, Stanford,Calforna. [2] J.Cao, M.C. shaw and L.wang estmates for the -Neumann problem and nonexstence of c 2 lev-flat hypersurfaces n p n, Math. Z.248 (2004), [3] T.W. Lorng An ntroducton to manfolds, second edton sprng 94303(202),secnd a letter to creatve commons, 559 Nathan Abbott,Way, Stanford Calforna.

12 36 Mohamed M. Osman [4] R. Arens, Topologes for homeomorphsm groups Amer, Jour. Math. 68(946), [5] Sergelang,dfferental manfolds, Addson wesley pubshng. In (972). [6] O. Abelkader, and S. Saber, soluton to -equaton wth exact support on pseudoconvex manfolds, Int. J. Geom. Meth. phys. 4(2007), [7] Yozo matsushma, dfferentable manfolds, Translated by E.T.Kobayash, Marcel Dekker Inc. Now York and Beesl (972). [8] J. Mlnor, constructon of unversal boundless II, Ann.Math. 63(956), [9] Bertschng E, Eeserved A.R.-Introducton to tensor calculus for general Relatvtysprng (999). [0] K.A. Anton-dfferental manfolds-department of mathematcs-new Brunswk, New ersey copyrght 993-Inc. bblographcal references ISBN (992). [] H.Ngel dfferentable C 3.Ib( 202). [2] K.V. Rchard, S.M. sdore Math. Theory and Applcatons, Boston, mass QA649C293 (992).

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