Di®erential forms on di®erential spaces R. M»aka P. Urba¶nski 1 Division of Mathematical Methods in Physics University of Warsaw Ho_za74, 00-682 Warszawa The presented approach to di®erential forms on di®erential spaces is motivated by the use of di®erential forms in the theory of static systems ([10, 11, 3]). The concept of a covector of higher order is based on duality between forms and multivectors rather then on the idea of the Grassman algebra. 1 Di®erential spaces Let M be a set and let C be a family of real functions on M. The topology induced on M by the family C, i.e., the weakest topology such that all functions from the family C are continuous, will be denoted by ¿C. If the family C separates points of M, then (M; ¿C) is a Hausdor® space. Let sc(C) denote the set of all functions of the form ® ± (f1; : : : ; fk) where f1; : : : ; fk 2 C and ® is a smooth real-valued function de¯ned on an open neighbor- k hood of f1(M) £ ::: £ fk(M) in R . It is easy to show ( [9]) that: C ½ sc(C) sc(C) = sc(sc(C)) ¿sc(C) = ¿C De¯nition 1.1 A function f on A ½ M is a local C-function if at each point p 2 A there exist g 2 C and an open neighborhood U of p such that fjU = gjU . 1Supported by Polish scienti¯c grant RP.I.10 1 The set of all local C-functions on A will be denoted by CA. We list the following properties of CA (see [9]): CjA ½ CA (in particular C ½ CM ) (CA)A = CA C ½ sc(CM ) ½ (sc(C))M De¯nition 1.2 A di®erential space is a pair (M; C) where M is a set and C is a family of real functions on M such that C = (sc(C))M . The family C will be called the di®erential structure on M and functions in C will be called smooth functions on M. We use also C1(M) to denote the family af all smooth functions. For each family C0 of functions on M there exists the smallest di®erential structure C which includes C0. We observe that C = (sc(C0))M . De¯nition 1.3 We say that a family C0 generates a di®erential structure C if C = (sc(C0))M . De¯nition 1.4 Let (M; C) and (N; D) be di®erential spaces. A mapping ®: M ! N is a smooth mapping between di®erential spaces if for each f 2 D we have f ± ® 2 C. Proposition 1.1 Let (M; C) and (N; D) be di®erential spaces and let D0 generate D.A mapping ®: M ! N is smooth if for each f 2 D0 we have f ± ® 2 C. 2 Tangent spaces and the module of smooth 1- forms Let the interval [0; 1[2 R be endowed with the natural di®erential structure (see [9]) and let (M; C1(M)) be a di®erential space. De¯nition 2.1 A smooth half-curve on M is a smooth mapping γ: [0; 1[! M. Let §1 denote the set of all smooth half-curves on M. We de¯ne a ¯bration ¿:§1 ! M by ¿(γ) = γ(0): By ¦1 we denote a trivial ¯bration M £ C1(M) with the canonical projection ¼: ¦1 ! M:(q; f) 7! q: 2 1 1 We de¯ne a mapping <; >:§ £M ¦ ! R by the formula: d < γ; (q; f) >= f(γ)j dt t=0 where q = γ(0). We refer to this mapping as a pairing between ¯brations §1 and ¦1. 1 Let be γ1; γ2 2 § . We say that γ1 and γ2 are equivalent if ¼(γ1) = ¼(γ2) = q and 1 < γ1; (q; f) >=< γ2; (q; f) > for each f 2 C (M): An equivalence class [γ] is a tangent vector at the point q.A tangent space Tq(M) is the set of all tangent vectors at the point q. De¯nition 2.2 The tangent bundle of a di®erential space M is a triple (TM; M; ¿) where S TM = q2M TqM is the disjoint sum of tangent spaces and ¿ is the canonical projection ¿:TM ! M: vq 7! q. 1 Now, we introduce an equivalence relation in ¦ : two pairs (q1; f1) and (q2; f2) are equivalent if q1 = q2 and < γ; (q1; f1) >=< γ; (q2; f2) > for each γ 2 § such that γ(0) = q1 = q2. An equivalence class [(q; f)] is called the di®erential of the function f at the point q and will be denoted by dqf. The cotangent space at q is the set of all di®erentials of ¤ functions at the point q and will be denoted Tq M. De¯nition 2.3 The cotangent bundle (T¤M; M; ¼) of a di®erential space M is the dis- ¤ S ¤ joint sum of cotangent spaces T M = q2M Tq M with the canonical projection ¤ ¼:T M ! M: dqf 7! q: The pairing between §1 and ¦1 de¯nes a pairing between TM and T¤M by the formula d < v; d f >= f(γ)j q dt t=0 where γ represents v. Remark. C1(M) has a natural structure of a linear space. This structure de¯nes a linear space structure in T¤M. Let be f; h 2 C1(M) and a 2 R then: dqf + dqh = dq(f + h) adqf = dq(af): In general, the dimension of a cotangent space is not the same at each point of M and a tangent space is not a linear space. 3 De¯nition 2.4 The di®erential of a function f 2 C1(M) is a mapping df:TM ! R de¯ned by the formula df(v) =< v; df >. A di®erential structure C1(TM) on a tangent bundle TM we de¯ne in the following way: 1 ¤ 1 1 C (TM) = (sc(f¿ f: f 2 C (M)g [ fdg: g 2 C (M)g))TM : It follows that that the projection ¿:TM ! M is smooth and that if C1(M) separates points of M then C1(TM) separates points of TM. We may introduce a di®erential structure on T¤M generated by functions of the form ¤ 1 ¼ f where f 2 C (M) and of the form dqf 7! hX(q); dqfi where X is a smooth vector ¯eld on M (X: M ! TM). However this di®erential structure does not, in general, separate points of T¤M and, consequently, is not very usefull. De¯nition 2.5 A smooth 1-form on M is a smooth function Á:TM ! R such that for 1 each point q 2 M there exists f 2 C (M) such that ÁjTqM = dqf. The space ¤1M of all smooth 1-forms has natural structure of a module over the ring of smooth functions on M. The structure of a linear space and of a module in ¤1M is given by: ('1 + '2)(vq) = '1(vq) + '2(vq) (f')(vq) = f(q)'(vq): It is evident that the di®erential of a smooth function is a smooth 1-form. 3 The ¯bration of 2-vectors and the module of smooth 2-forms A 2-cube is a smooth mapping »: [0; 1[£[0; 1[! M. Let §2 be the set of all 2-cubes on M. We de¯ne a ¯bration ¿:§2 ! M by §2 3 » 7! »(0; 0). Let ¼: ¦2 = M £ ¤1M ! M be the trivial ¯bration de¯ned by ¼(q; ') = q. For each t 2 [0; 1[ half-curves 2 t1 ! »t (t1) = »(t1; t) and 1 t2 ! »t (t2) = »(t; t2) 2 1 are smooth and, consequently, represent tangent vectors denoted by [»t ] and [»t ], respec- 1 2 tively. Mappings [0; 1[3 t 7! [»t ] 2 TM and [0; 1[3 t 7! [»t ] 2 TM are smooth half-curves. Thus we can de¯ne a pairing between ¯brations ¿ and ¼ by the formula: 4 d < »; (q; ') >= ('([»1]) ¡ '([»2]))j dt t t t=0 We say that two 2-cubes » and »0 are equivalent if ¿(») = ¿(»0) = q and < »; (q; ') >=< »0; (q; ') > for each ' 2 ¤1M. (2) An equivalence class [»] of 2-cubes is called a 2-vector at point q. By Tq M we denote the space of all 2-vectors at point q. De¯nition 3.1 The ¯bre bundle of 2-vectors on the di®erential space M is the triple (2) (2) S (2) (2) (T M; M; ¿2) where T M = q2M Tq M and ¿2:T M ! M:[»] 7! »(0; 0) Now, we introduce an equivalent relation in ¦2. Two pairs (q; ') and (q; '0) are equivalent if < »; (q; ') >=< »; (q; ') > for each » 2 §2 such that ¿(») = q. An equivalence class [(q; ')] is called the di®erential of 1-form ' at the point q and it will be denoted by dq'. De¯nition 3.2 The di®erential of a smooth 1-form ' is a map d':T(2)M ! R de¯ned by the formula: d'([»q]) =< »q; dq' > . We introduce a di®erential structure C1(T(2)M) on T(2)M by 1 (2) 1 ¤ 1 C (T M) = (sc(fd': ' 2 ¤ Mg [ f¿ f: f 2 C (M)g))T(2)M : (2) 1 (2) We observe that the projection ¿2:T M ! M is projection and that C (T M) sepa- rates points of T(2)M if C1(M) separates points of M. De¯nition 3.3 A smooth 2-form on M is a smooth function !:T2M ! R such that for 1 2 each point q 2 M there exists ' 2 ¤ M such that !jTqM = dq'.