DEMONSTRATIO MATHEMATICA Vol. XV No 4 19IJ Hanna Matuszczyk, Wtodzimierz Waliszewski A NON-CLASSICAL DEFINITION OF TANGENT BUNDLE AND COTANGENT BUNDLE In this paper there is given a proof of the concordsnoe of the definitions of the tangent bundle and the cotangent bundle of a differentiable manifold with corresponding con- cepts in category of differential spaces, when the manifold as a differential space is regarded. 0. Preliminaries In the paper [2] the concept of the tangent bundle and the cotangent bundle of a Sikorski'o differential space [.'>] is given. The differentiable manifold is meant as a diffe- rential space locally tiiffeomorphic with an Euclidean space. R.Sikorski in [4] characterized a manifold by means cf an atlas of sketchs. Namely, he considers an m-sketch on LI as one-one mapping of subset of the set M into m-dimensional Suclidean space and* proves the following theorem. A.Every atlas Ji of an m-dimensional differentiable ma- nifold (M ,C) has the following properties: 1) if xeM, then x is an m-sketch on LI, 2) 1,1 = M D , (I» is the domain oi xeM x x m 3) if x,yeM , then i^nDx y. j is open in JR , -1 «0 nc y o x is a C -mapping of DXDD] onto y[DxnD], - 913 - 2 H.Matuszczyk, W.Waliszewski 4) if x,y eM and K is a compact subset of x[^x]> than y[x~1 [k] fi is closed subset of y [Dy] (in the usual topology of 3?m). Conversely, if M is any non-empty set, and Ji is a set of mappings satisfying the conditions 1) -4), then there exists exactly one differential structure C on I»I such that (M,C) is an m-dimensional differentiable manifold and M Is an atlas of this manifold. If 1} - 4) are satisfied, then C is the set of all a : M — ]R such that a ox"^ is of class C°° for any xel. The above theorem allows us to introduce the structure of differentiable manifold if we have got an atlas of sketchs. In particular case, if we have a differentiable manifold (K,C) we may consider the full atlas, atl(MfC)y of all charts of this manifold, and we have got the tangent bundle T(i;i,C) (and, similarly, the cotangent bundle T*(M,C}) defined as follows. Denote the set of all tangent vectors to (M,C) by T(LutC), and a projection of the tangent bundle we regard as a mapping jrt T(ti,C) —- M such that for any v £ T(M,C) we have v in the vector space (see [2]}. For any chart x c atl(K>C) we set (1) x(v) c (x1(*(v)) x°(jr(v))tv(x1),...iv(xm)) for v«*~1[dJ. Here x(p) = (x1(p) xm(p)) for p«Dx. It \s easy to state that x is a 2n-sketch of I(U,C) for x c atl(l£,C). Je will check that the set Jt of all x, where x t ail(:J,C), satisfies the conditions obtained from 1) - 4;, where we set T(;qtC) instead of tt. Indeed, let »e take any xt atl(L:,C). 'Jc have, of cour- se , D- - [dJ end - {(u1,...,u2ci)j (u\...,ua) « *[l>x] and (a*1 a2a|l*,j - »[»x]*«®' - 914 - Definition of tangent bundle 3 Let {a1,... ,u2m) e We will find v e D- such that (2) x(v) = (u1,...,u2m). Pfom (1) and (2) we get (3) X1(JT(V)) = u1, vU1) = um+i, i = 1 IB. Hence it follows that (4) X(JT(V) ) = (U1 UM). Taking the vectors, being the base vectors corres- ponding to the chart x at the point p e Dx, i.e. the vec- tors defined by the formulas ^(pH«) = (a<> i(x(p)) for oteC(p), i=1,...,n, where G(p) = U|clT; pe'uercJ (cf. [l] ) we have V = v(xX j »Xj ( p) , p = IT (v). ^ denotes the partial derivative of tne function JS with respect to i-th variable. This, (3) and (4) yield v = am+i x±(x"1(u1,...,um)J. Thus, (5) x-1(u1 u2m) = um+i x^x-UJ,...,**)). Let us take cny y e atl(I.!,C). We have then (6) y(v) = (yV(v)) yV(v)), v(y1) v(ym)). - 915 - 4 J.Matuazczyk, W.Waliszewski For any u = (u1,...,u2m) e xjD-nD-], b7 (5), we have x-1(u1,...,u2m)(y3) = am+i xi(x-1(u1,...,um))(7;5) = = um+i (yJ. *-1)|i(x(x"1(tt1,...,aaJ)) = = um+i (y30 x^^u1 um). This and (6) yield (yo 5c-1)(u1,...,u2m) = = ((y1o *-1)(u1t...taa),...t(7n0 x-1)(u1 um), um+i (y1o x"1)|i(u1,...,uni) um+i (ymox-1)U(a1,...,um)). Hence it follows that jux"1 is of class C°" on x D- o D-1. L x yj Now, let K be any compact subset of x [l)-~j. Set F ='[K]n 1^]. We will prove that F is closed in y [D-]. For this purpose take any u'Q e cl(F) fl y |D-J , where cl(F) denotes the closure of F in R 2m. Then, there exists un6f, n = 1,2,... , such that u'o< So, there are v e G—»oo £ DynDx 3uch that un = and e K' n=1>2 Similarly, u'Q = y(v^), v'Q 6 D_. Because of the compactness of K we may assume that x(v j - u , where u e K. n—*<*> So, v/e have u = x(v ), v € D-. Thus, y(vn) -y(v^) and x(vn) "»2(v ). n—» n—-<» Set Pn-*(vn)t n=1,2,... , p0=jr{v0), p'0«»(Vg), - 916 - Definition of tangent bundle 5 So, we have y(pn) -yip^) aad n—oo x(pn) ). n—oo Prom the assumption that p'Q ^ pQ it would follow that p' # D or else p £ D . Then there exist neighbourhoods o x ( o y ( U and U of the points pQ and p0 contained in Dx and D , respectively, U n uj' =0. Because of yip^) e ?[U'J and x( p ) e x[u] , we have for sufficiently large n y(pn)e y[u'] and x(pn) e x[Uj. Hence it follows that pn 6 U fi U' for some n, which is impossible.| Therefore, po' = pu e Dx O Dy and for i,j = 1,...,m we get j 1 1 vn(y ) and v^x ) -v (x ). n—o° n—oo 1 v xi an v xl 1 This yields v (x ) - '0( ) <3 n( ) -v (x ), n—oo 0 n n —— °° 1 i=1,...,m. Hence it follows that v' = vQ e x~ [k]fi D-. So, u; e P. 7 1. The tangent bundle According to Theorem A we have got a differential structu- re C' on the set T(M,C), C' • Jj9 ;y3 tT(M,C) —R and y3ox"1eC°° for all xe atl(M,C)j. The differential space (T(M,C), c') is the tangent bundle of the differentiable manifold (M,C) regarded in the category of differentiable manifolds. This differential space has been defined by means of the atlas { x; x e atl(M,C)J. On the other hand, in the theory of differential spaces we have [2] the tangent bundle T(M,C) = (T(M,C), C'), where - 917 - 6 H.Matuszczyk, 'ff.Waliszewski (7) C' = the smallest differential structure containing the set |«0 jr ; a e cj U jdaj. oC fc C J, where da(v) = v(a) for all v 6 T(M,C). Theorem 1. If (M,C) is a differentiable ma- nifold, then C' = C'. Proof. To prove that C'c C' take ae C and set ft =«o jt . For any x 6 atl(IwfC), according to (5), we have *-(x"1(u1,...,u2m}) = x"1(u1 um). Hence it follows that (j8 o x~1) (u1,... ,u2m) = (a o x"1) (u1 um). So, jJ.S"1tf" . Thus, /3eC'. How, set 0 = d«. Then (d«o x"1)(u1,...,u1,...,u2o)(a) = m+i _1 1 m 1 1 = u xi(x (u u ))(oO = u^lcox" ),^ a"). Thus, d«ox"16f" . So, fi fc C' . Therefore (cf. [5]) C'c C'. To prove the inverse inclusion take any Ji e C and any vQ e T(M,C). Set p0 =JT(V0). Then there exists, in atl(M,C), s chart x around pQ with condition: x(pQ) = 0 m and x[Dx] = B(0;2) (B(u;r) stands for the ball in ffi of the centre u and radius r). Thus, there exist a func- tion <o e €°° (R2m) and functions a1 e C, such that U|B(0J1)*= x~1| B(OjU«tf?m and a i| B(0;1) = x1 B(0;1), 1 = 1,...,m. Hence it follows that dai(v) = vi«1) = vix1) for v e V, where V = [x"1 [B(0;1 )]] = X~1 [B(0;1 )*.KM], and fi[v) = (/Jo x"1 )(x(v)) =w(x(v)) for v e V. - 918 - Definition of tangent bundle 7 Thus, for v e V we have A(v) = w(x1(*(v)} xm(*(v)), v(x1) v(xm)j = = co (a1 (JT(V) ),...,«VM), dcx1(v),...,d<xm(v)}. Notice that X~^[b(0;1)] is open in the topology of (I'.I,C).