arXiv:math-ph/0102016v1 15 Feb 2001 rank. oueoe es unital dense a over module over bundle vector smooth some of sections global off eodcutbe n once.Vco ude r fin 1. are Theorem bundles Vector connected. and second-countable, dorff, seTerm65i e.[]i h otnoscase). continuous the 2. in Theorem [2] Ref. in 6.5 Theorem (see in 4,btte twsetne otesot ae(e Proposit [5]). (see Ref. case in smooth 3.2 the Theorem to 3.1, extended Proposition was and it [3] then but [4], tions discus is manifolds non-compact to extension promi Its a plays geometry. and manifolds, compact over bundles vector and etrbundle. vector etrbundle vector opc manifold compact a Xi e.[]gnrlzsti seto onncmatmnflsas manifolds non-compact to assertion this generalizes [1] Ref. in IX etrbundle vector eateto hoeia hsc,Mso tt Universit State Moscow Physics, Theoretical of Department i)Ayprojective Any (ii) nnncmuaiegoer,oeteeoetik fafiiepr finite a of thinks therefore one geometry, non-commutative In ealta mohmnflsaeasmdt era,finite-dimensio real, be to assumed are manifolds smooth that Recall oeta tm()o hoe seuvln otefloigwell-known following the to equivalent is 1 Theorem of (i) item that Note h er–wntermhdbe salse o otnosfun continuous for established been had theorem Serre–Swan The projectiv between link the provides theorem Serre–Swan The h e on ftepofo tm()o hoe sta n vector any that is 1 Theorem of (i) item of proof the of point key The eako h er–wntermfrnon-compact for theorem Serre–Swan the on Remark Let E SreSa) i The (i) (Serre–Swan). E ′ → → E X X X eavco udeoe opc manifold compact a over bundle vector a be C disabnl ta vrafiiecvrn of covering finite a over atlas bundle a admits uhta h hte sum Whitney the that such vracmatmanifold compact a over ∞ ( ∗ -al [email protected] E-mail: X sbler fa of -subalgebra ) mdl ffiiern sioopi otemdl of module the to isomorphic is rank finite of -module G.Sardanashvily manifolds Abstract C ∞ ( 1 X ) C mdl fgoa etoso smooth a of sections global of -module ∗ agbaa en non-commutative a being as -algebra X E X sapoetv oueo finite of module projective a is ⊕ . E sed. etrl nnon-commutative in role nent ′ ,173 ocw Russia Moscow, 117234 y, satiilvco bundle. vector trivial a is oue ffiierank finite of modules e ite-dimensional. X hr xssa exists There . X tosadsec- and ctions o . nRef. in 4.1 ion follows. Proposition . udeover bundle ojective a,Haus- nal, theorem ∗ - Proposition 3. A smooth fibre bundle Y → X over an arbitrary manifold X admits a bundle atlas over a finite covering of X. Its proof is based on the fact that, for any covering of a manifold X, there exists a refinement {Uij}, where j and i run through a countable set and a finite set, respectively, such that Uij ∩ Uik = ∅, j =6 k. Let {Uξ, ψξ} be a bundle atlas of a fibre bundle Y → X over a covering {Uξ} of X. Let {Uij} be the above mentioned refinement of this covering and {(Uij, ψij)} the corresponding bundle atlas of Y → X. Then Y → X has the finite bundle atlas

def def Ui = ∪ Uij, ψi(x) = ψij(x), x ∈ Uij ⊂ Ui. (1) j

It is readily observed that, if Y → X is a vector bundle, the atlas (1) is an atlas of a vector bundle. It follows that every smooth vector bundle admits a finite atlas, and the proof in Refs. [3, 5] of item (i) of Theorem 1 can be generalized straightforwardly to non-compact manifolds. Similarly, the above mentioned Theorem 6.5 in [2] can be extended to non-compact finite-dimensional topological manifolds. The proof of item (ii) in Theorem 1 does not imply the compactness of X.

References

[1] W.Greub, S.Halperin and R.Vanstone, Connections, Curvature, and Coho- mology, Vol. 1, (Academic Press, N.Y., 1972).

[2] M.Karoubi, K-Theory. An Introduction (Springer-Verlag, Berlin, 1978).

[3] G.Landi, An introduction to non-commutative spaces and their geometry, E-print arXiv: hep-th/9701078.

[4] R.Swan, Vector bundles and projective modules, Trans. Amer. Math. Soc. 105 (1962) 264.

[5] J.V´arilly and J.Grasia-Bondia, Connes’ noncommutative differential geome- try and the standard model, J. Geom. Phys. 12 (1993) 223.

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