DEMONSTRATIO MATHEMATICA

Vol. XV No 4 19IJ

Hanna Matuszczyk, Wtodzimierz Waliszewski

A NON-CLASSICAL DEFINITION OF 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 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],

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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 (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]*«®'

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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)).

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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),

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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

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(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

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.

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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

Notice that X~^[b(0;1)] is open in the topology of (I'.I,C). Therefore, by (7), -N-"1 [x~1 [B(0;1 )]] is open in the topo- logy of the differential space T(M,C). Hence it follows that p> is locally of class C', i.e. ;3 e (Ccj = C'. So, C c C'. 1

2. The cotangent bundle In the theory of differentiable manifold, the cotangent bundle of a manifold is defined by means of an atlas -of this manifold. For a differentiable manifold (M,C) we have the set, T*(H,C), of all tangent covectors to (M,C), i.e. T*(M,C) is the union of all cotangent spaces (T (M,C))* being dual vector spaces to the tangent ones, T (¡.:,C), whe- re p 6 1.1. We have the natural projection Or : T*(r;i,C) —-M defined by the conditions w is in (T^wj(M,C))* for every w e T*U,,C). For any chart x e atl(I.!,C) we set

x(w) = (x1 (JT(W) ),xm(jf(w)), x1 (w).. ,xm(w) ) , where x^w) =w(xi(^(w))) for Similarly as in the previous we check that x is a 2m-sketch on T*(kl,C), and that the set, M , of all x, where x e atl(M,C) satisfies the conditions obtained from. 1) - 4) by taking T*(M,C) instead of M. The differential structure determined by Jl on T*(MtC) is the set, C', of all functions ,8: T*(M,C)—- R such that fio x~1eC°* . Indeed, it is easy to check that = x[-ux]x-''m foi any u= (u1,..., u", um+1,..., u2J 6 x [Da] we nave

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W r um'% V"u2m) = um+i(dxl) -1, 1 mi' + x ^u f*«•}

So, x is a 2m-sketch on the set T*(M,C). The equality

T*(M,C) = LJ D xeatl(M,C) x is obvious. (Here for any a 6 Cy, any U open in (i£,C), and any p e U we have set (da)p(v) = v(a) for v of T (M,C).) Similarly, we check that for any chart y & atl(M,C) and 1 m W0 any u = (u ,... ,u ,um+1,... ,u2m) e have

1 1 1 1 m m 1 1 m (yoi- )(a| = ((y ox- )(u u )f...,(y oX- )(u ,...,u ),

i 1 1 1 m um+i(x oy" )| 1(y(x" (u ,...,u ))),...

"1 "1 "1*1 m ...,um+i(x 07"')|m(y(x- (u',...,u )))).

Henoe it follows that yo x~1 is of C00 class on x[D-nD-] for x,y e atl(M,C). Now, we have to check that for any com- pact subset K of the set P = y [x"1 [k] n D-] is a closed subset of J [Wl • ** r t Let us take any u'Q 6 cl(F) n Then there exist u^e F, n= 1,2,..., such that u^ u'Q, and we have n— u'Q = y(w^), where w^ e Da. Prom the definition of P we obtain u^ = y(wn), where wn e n D* and x(wn) e K. According to the compactness of K we may assume that 1 xfw^) n-^oo e K. So, uQ = S(w ) where wQ c X" [K],

Setting pQ =^(wn), p'Q =jr(w'Q) and pQ = Jf(wQ) we get y(pn) and x(p ) ~x(p ). Because of the n—°° n—- Hausdorff axiom for the topology of (M,C), it follows that p0 = p£. On the other hand, we have

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w (Xi(pn)) w (Xi(p0)), n-^oo

wn(y,{pn)) nn—»>o —oo and

V^^n^ = "nHx^y'^tyipJ^tpjj))

i i = 7-j(P0)(x )wn(xi(pn)) - yj(P0)(* )w0(*J|{p0)J.-w0(y3(p0J).

w = u 6 K w e So, wQ = w^. Consequently, *( q) o » o

1 u'Q = y(w'0) e y[x" [K]nDA] = P. The set F is closed in

The set of all smooth vector fields on (M,C) will be denoted by X(M,C). For any Xfc3E(M,C) and for any w 6 T*(M,C) we set

X(w) = w(X(jt{w) )).

In the theory of differential spaces there has been introduced the concept of the cotangent bundle [2] in the following ways C' = the smallest differential strudture containing the set |oco£; aecjujx; XeX(M,C)}, T*(M,C) = (T*(M,C), C'). Theorem' 2. If (M,C) is a differentiable mani- fold, then C' = C'. Proof. Let >5=

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(jjorn») =x(rn(u)) = x-1{u)(x(x-1(u1,...,um))) =

= o^itdx1)' - i (X(x^1(u1,...,um))) = x (u ,..., u )

- dm+iX(x-1(u1,...,um))(xi) = am+i(azxi)(x-1(u1,...,am)}.

So,y0 ox"1eC°° . Hence it follows that C'c C' (cf. [5]). A To prove the inverse inclusion, take any Ji e C' and any wQ 6 T*(M.C). There exist a set U open in _ff?2m and a function coeC -(ZR2111) such that

^ox"1|u = co | U, u0 6 U C U = U^x u2, Ur U2 being open in J2m, u0 = x(w0), cl(U) c x[dJ,

(9) U2 = J^ ... x Jm,

J^ is an open interval around u0j_> where _ / 1 m > u0 ~ ^ uo'* *''ao'uo,m+1»•••»ao,2m''

x1!*"1^] B 0,1 *~1[u1]t where ot1 e C, and

xi|x"1[ui] = ^jac"1 [u J . where X± 6 3(M,C)t i = 1,...,m. Let us set

w = r1[u].

Then W = [x"1 [ui,]] n x[u2], where

(10) x(w) = (x1(w),...,xm(w)), x^w) = w(xi(^(w)) for w e L XJ - 922 - Definition of tangent bundle 11

Take w e W. Then jt (w) e x"1[u.|]. So, we have ^(¿riw)) =

=Xi(i(w)),

(11) x^w) =w(Xi(^(w))) = X±(w).

(9), (10) and (11) yield X^w) 6 J±, i = 1,...,m. Henoe it follows that w e So we have

W C 3r"1 [x~1 [u.,]] D [jJ. i=1 Conversely, take w 6 jt"1 [x"1 [u.,] fi Then, Jr[w) e and X^(w) e J^. So, consequently,

x^fw)) = Xi(^(w)),

x^w) = w(Xi(ir(w))) = Xj_(w) e J^ for i = 1,...,m. Henca it follows that x(w) e J^x ,,, * J • 1 1 1 = Ug. In other words, w e jt"" [x~ [uj] O 3T [u2]. Therefore, we have the equality

w.i-V'w]" n i=1

Thus, W is open in the topology of the differential space (T*(M,C), C'). Now, for any w € W we have y3(w) = (/Jox-1)(x(w)) =co(x(w)) = 1 m = w (x (¿r(w)) ,... ,x (ir(w)) ,w(Xl (£(w))),... ,w(xm(£(w)))) = 1 m = co (oc (JT(W) ),... ,« (£(w)) ,w(X1 (£( w))),... .wU^Jrfw)))) = 1 m = co(ot C£w)),..., a (^(w)),X1(w),...,Xm(w)).

Hence, fie(c') cj = C' So, C'CC', This ends the proof. - 923 - 12 H.Matuszczyk, W.Waliszewski

REFERENCES

[l]H. Matuszczyk: On the tangent spaces to a differential slpace, Demonstratio Math. 14 (1981) 937-942. H. Matuszczy.k-j On tahgent bundle and cotan- gent bundle of differential spaoe (to appear). [3]r. Sikorski: Abstract covariant derivative, Cplloq. Math. 18 (1967) 251-279. [ 4] Sikorski: Wst§p do geometrii rdfcniczkowej (Introduction to ) (in Polish), • tfarszawa 1972. [5] W. Waliszewski: Regular and ooregular map- pings of differential spaces, Ann. Polon. Math. 30 (1975) 263-281.

INSTITUTE OP MATHEMATICS, TECHNICAL UNIVERSITY OP WROCLAW; INSTITUTE OP MATHEMATICS, POLISH ACADEMY OP SCIENCES, BRANCH OP LÖVi Received December 19, 1980.

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