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Advances in Pure Mathematics, 2011, 1, 245-249 doi:10.4236/apm.2011.15044 Published Online September 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Interior and Exterior Differential Systems for Lie Algebroids Constantin M. Arcuş Secondary School “CORNELIUS RADU”, Rădineşti Village, Gorj County, România E-mail: c_arcus@yahoo.com, c_arcus@radinesti.ro Received March 26, 2011; revised August 9, 2011; accepted August 20, 2011 Abstract A theorem of Maurer-Cartan type for Lie algebroids is presented. Suppose that any vector subbundle of a Lie algebroid is called interior differential system (IDS) for that Lie algebroid. A theorem of Frobenius type is obtained. Extending the classical notion of exterior diffential system (EDS) to Lie algebroids, a theorem of Cartan type is obtained. 2000 Mathematics Subject Classification: 00A69, 58A15, 58B34. Keywords: Vector Bundle, Lie Algebroid, Interior Differential System, Exterior Differential Calculus, Exterior Differential System 1. Introduction Using the exterior differential calculus for Lie algebroids (See [1,2]) the structure equations of Maurer-Cartan type are established. Using the Cartan’s moving frame method, there exists the following Theorem (E. Cartan) If n NMan is a Riemannian manifold and i i XX x , 1, n is n ortonormal moving frame, then there exists a collection of 1-forms , ,1,n uniquely defined by the requirements and ,1, F d n where ,1,n is the coframe. (see [3], p. 151) We know that an -dimensional distribution on a manifold is a mapping defined on , which assignees to each point r NDN x of an -dimensional linear subspace N r x D of x. A vector field TN X be- longs to if we have D x x X D for each x N . When this happens we write X D . The distribution on a manifold is said to be differentiable if for any DN x N there exists differen- tiable linearly independent vector fields r 1,, r X XD in a neighborhood of x . The distri- bution is said to be involutive if for all vector fields D , X YD we have , X YD . In the classical theory we have the following Theorem (Frobenius) The distribution is involut- ive if and only if for each D x N r there exists a neigh- borhood and Un linearly independent 1-forms 1,, rn on U which vanish on and satisfy the condition D 1,rn ,1, F drn for suitable 1-forms , ,1,n .(see [4], p. 58) Extending the notion of distribution we obtain the definition of an IDS of a Lie algebroid. A characteriza- tion of the ivolutivity of an IDS in a result of Frobenius type is presented in Theorem 4.7. This paper studies the intersection between the ge- ometry of Lie algebroids and some aspects of EDS. In the classical sense, an EDS is a pair , M I consisting of a smooth manifold M and a homogeneous, differen- tially closed ideal I in the algebra of smooth differen- tial forms on M . (see [5,6]) Using the notion of EDS of an arbitrary Lie algebroid ,,, N FN Id ,, , F we obtained a new result of Cartan type in the Theorem 5.1. In the particular case of standard Lie algebroid ,Id,, ,, ,Id MTMM TM there are obtained similar results those for distributions. TM M We know that a submanifold of is said to be S N C. M. ARCUŞ 246 integral manifold for the distribution if for every point D x N, x D coincides with x TS. The distribution is said to be integrable if for each point D x N there exists an integral manifold of containing D x . As a distribution is involutive if and only if it is integrable, then the study of the integral manifolds of an IDS or EDS is a new direction by research. D 2. Preliminaries In general, if is a category, then we denote CC the class of objects and for any , A BC, we denote the set of morphisms of , BCA A source and target. Let B lgeA ,Li , M od and be the category of Lie algebras, modules and vector bundles respectively. v B We know that if ,π,v EMB, and , then πEM Ma ,,πanMEu FM n ,π ,u M : M Id ,,EM M ,R , is a F M-module. We know that a Lie algebroid is a vector bundle ,, v F NB so that there exists ,,,, v N ,, N I dB FN N TN and also an operation , ,, ,,,, ,, F F F NFN FN uv uv with the following properties: 1 LA . the equality holds good N d ,,, FF uf vuvIu f v f , N for all and , ,uv F f FN , 2 LA . the 4-tuple ,, ,,,, F FN is a Lie F N LA -algebra, 3. the M od -morphism , N I d is a -morphism of lgALie ,,,, ,, F FN source and ,, NN , ,,, TN TN target. Let , ,Id ,,FN,, F N be a Lie algebroid. Locally, for any ,1,p , we set ,F tt L t . We easily obtain that ,L L for any ,, 1,p. The real local functions ,,, 1,Lp are called the structure functions. We assume that ,, F vN is a vector bundle with type fibre the real vector space and structure group a Lie subgroup of p R ,, ,p,RGL . We denote , i x z the canonical local coordinates on ,, F vN , where and 1,in1, .p , i Consider , i x zx z a change of coordi- nates on ,, F vN . Then the coordinates z change to ' z according to the rule: z If ,,ztF N is arbitrary, then ,i Ni f I dztfx zx x (2.2) for any f FN and x N . The coefficients i change to i according to the rule: i ii i x x , (2.3) where 1. The following equalities hold good: , ii ii f f fFN xx (2.4) and . kk ki j i Lj x x (2.5) 3. Interior Differential Systems Let ,,,, ,,N FN Id F Definition 3.1 Any vector subbundle of the vector bundle be a Lie algebroid. ,π,EM ,, F vN will be called interior dif- ferential system (IDS) of the Lie algebroid ,N Id,,,, ,FN F Remark 3.1 If . ,π,EN is an IDS of the Lie alge- broid ,,FNd ,, ,, FI N then we obtain a vector subbundle 00 ,N,πE of the dual vector bundle ** ,, F N so that 00 ** ,π, ,, :0,,π,. EN FNSS EN The vector subbundle 00 ,π,EN will be called the annihilator vector subbundle of the IDS . ,π,EM Proposition 3.1 If ,π,EN is an IDS of the Lie al- gebroid ,,,, ,,N F F N I d so that ,π,EN z (2.1) 1,, , r SS then it exists ** 1,, ,, rp F N linearly independent so that 00 1 ,π,,, rp EN . Definition 3.2 The IDS of the Lie alge- broid ,π,EN N Id ,,,, ,, F FN will be called involut- ive if ,,π,STE N , F for any ,,π,STE N . Proposition 3.2 If N,π,E is an IDS of the Lie al- gebroid ,, ,FN , ,, FId N and is a base of the 1,, , r SS F N-submodule then ,π,,EN ,, ,π,EN is involutive if and only if ,π,,E N,SS ab F for any ,1,ab r. Copyright © 2011 SciRes. APM C. M. ARCUŞ247 4. Exterior Differential Calculus Let ,,,, ,,N F FN Id ,, q be a Lie algebroid. We denote F N . If ,, q the set of differential forms of de- gree q 0q ,, F NF ,,zF N , the - tio Definition 4.1 For any n we obtain the exterior differential algebra ,, ,,,FN . N the applica n ,,,, , z L F NF N defined by for any , zN LfId z f , f FN and 1q for any 1 1 1 ,, ,,, ,,,,, , zq N q iq F i Lz zIdzz z zzz z ,, q F N e covariant Lie and , is e 4.1 If , 1,, ,, q zz FN tive with respect to called thderivathe section z. Theor m ,,zFN ,, q F N and ,, r F N , then . zz LL z L (4.1) Definition 4.2 If , then the a q ,,zFN pplication 1 ,, ,, ,, ,, z i q z FN FN F NiF N defined by 0 z if, for any f FN and 22 ,, ,,, z qq iz zzzz , for any , is called the interior 2,, ,, q zz FN sociated to the sectionproduct as . z Theorem 4.2 If ,zF ,N, then for any ,, q F N and ,, r F N we obtain the equality 1. q z zz ii i (4.2) Theorem 4.3 For any we ,,,zvF N obtain ,. F vz zzvz L i Li i (4. 3) Theorem 4.4 The application F d q 1 ,, ,, q F F NF d N defined by , F z dfzL f for any and ,,zFN 01 01 1 01 ,,, ˆ 1, ,,,,, ˆˆ 1, ,,,,,,,, q qi Ni i q i ij ijijq F ij zzz Idzz zzz zzz zzzz F d for any 01 ,, ,,, q zzzFN , property: is unique having the . (4.4) This application is called the exterior dif op following F Ldi ,,, F zzz id zFN ferentiation erator of the exterior differential algebra of the Lie algebroid ,,,, ,,N F FN Id . Theorem 4.5 The exterior differentiation operator F d given by the previous theorem has the following perties: 1) For any pro ,, q F N and ,, r F N we obtain F d 1. q F dF d (4.5) 2) For any ,,zFN we obtain . FF z z d L Ld (4.6) 3) 0. FF dd orem 4.6 (oThe f Maurer-Cartan type) If ,,,, ,,N F FN Id is a Lie algebroid ation operator of the exte differential and F d rioris the exterior differenti F N-algebra ,, ,,,FN , then we obtain there equationsn type structu of Maurer-Carta 1,1, F dtLt tp (MC) 2 1 and ,1, Fi i dxt in . (MC 2) where ,1,tp is the coframe of the vector bundle ,, F N quatio called the structure equations . These ens will be of Maurer-Cartan type associated to the Lie algebroid ,,,, ,,N F FN Id . Proof. Let 1, p be ar. Since bitrary ,, ,1,dt L Fttp it results that . F dtLt t (1) Since LL and tt tt , for any ,1,p hat , it results t 1 Lt t ,1 ,. 2Lt tp (2) Copyright © 2011 SciRes. APM C. M. ARCUŞ 248 Using the equalities (1) and (2) it results t equation (MC). Let he structure 1 1,in be arbitrary. Since ,1, Fi i dx tp it results the s2). q.e.d. Remark 4.1 In the particular case of te standard Lie algebroid tructure equation (MC h ,,,, ,,TNNId Id NTNN TN we obtain ,1, TN ii dx dxin, (MC 2)' where dx vector bund ,1, ii n is the coframe of thele and , for all ,, N TN N . As TN dd0. TN 0 i L jk ,, 1,ijk n we obtain 11,. i k jk Lx in 0, 2 Fi j d dxdxd (MC 1)' These equations are the structure equations of Maurer-Cartan type associated to the stand gebroid ard Lie al- ,,,, ,,TNNId Id . NTNN TN Theorem 4.7 (of Frobenius type) Let ,π,EN be an IDS of the Lie algebroid ,,,, ,, F FN Id . If 1, r N p is a base of, F N submodule 00 ,π,,,EN, then the IDS ,πolut- ive if and only if it exists ,EN is 1,, inv F N , ,1,rp so that 1, ,. F rp d p ,1r Proof. Let is a base of the 1,, r SS F N Let -submodule that ,πEN ,F ,,,. ,N so 1rp ,,, ,, rr SS of the ,,SS 11p S is a baseS F N-module ,, ,,FN . ** 1r Let ,, ,, F N so that 11rr p is a base of the ,, ,,, F N-mod- ule For any ** ,, ,,FN . ,1,ab r and ,1,rp whe e have t lities: , equa 0, . aa bbb a SS SS ,0 We remark that the set of the 2-forms ,1 , ,1,ab rrp ,,,; aba b is a base of the F N have , , Fbcb bc b bc b dAB C (1) where, , bc b A B and C , ,, 1,abc r, ,, 1,rp bc cb are real local functions so that A A and CC . ula Using the form ,, ,, F bcN bc Nc b bc F dSSIdSS Id S SS S that (2) we obtain ,, bc F SS bc A (3) for any ,1,bc r and 1,rp . ,π,EN is an involutive IDS of the We admit that Lie algebroid ,, N Id ,, ,, F FN . As ,N, for a,,πEny bc F SS ,bc1,r, it results that ,0 b Sc F S -module Therefore, we 2,, ,,FN . for any ,1,bc r and 1,rp . Therefore, for any ,bc1,r and 1,rp , we obtain 0 bc A and , 1 2 1 b b 2 Fb b dBC b BC As 1 1 b,, 2 b BC FN , for any ,1rp Conversely, we adm ,it results the first implication. it that it exists 1,, F N , ,1rp , so that 1, , F rp d (4) for any1, .p r (4) we obtain thaUsing the affirmations (1), (2) andt 0 bc A , for any ,1bc1,rp . , r and Using the affirmation (3), we obtain ,0 bc F SS for any ,1,bc r and 1,rp . Therehafore, we ,,π, bc F SSE N ve , for any ,1,bc r. Using thPr the sec- on 5. Exterior Differential Systems Le e oposition 3.2, we obtain d implication. q.e.d. t d efinit ,,,, ,,N F FN I oid. ion 5.1 Any ideal ,,I of the exterior dif- ferential algebra of the Lie algebroid be a Lie algebr D Copyright © 2011 SciRes. APM C. M. ARCUŞ Copyright © 2011 SciRes. APM 249 ,,,, ,,N F FN Id closed under differentiation 1,, rp be a base of operator F d, namely of the Lie algebroid F dI I , is called differential ideal ,,,, ,,N F N Id . F be aDefinition 5.2 Let ,,I differential ideal of the Lie algebroid ,,,, ,,FN I for all kN and ,, N d. If it exists F an IDS ,π,EN so that k I F N we 1,, 0 k uu , for any 1,,,π, k uu EN, the have n we will say that ,,I is an sys exterior differential (EDS) of the Lie algebroid tem ,,,, ,, F FN Theo of Cartan type) S ,πE N Id . rem 5.1 ( the Lie algebroid The ID,N of ,, N Id is by ,, ,,FN involut- f the ideal generatedthe F ive, if and only i F N-submodule 00 ,π,,,EN is an EDS of the Lie algebroid ,N Id . , ,, ,FN, F Proof. Let ,π,Ee Lie algebroid N be an involutive IDS of th ,,,, ,,N FN Id . Let 1,, rp se of the F be a ba F N-su 0 π, ;, N bmodule , ,, q 00 ,π,,,EN. We know that 0 ,IE 1rp qN F N . ,, q Let qNand 1 ,, rp F N be 7 we obtain 1 1q q bitrary. Using the Theorems 4.5 FF F dd d and 4. 1 ar- F d . it sults that As 12 1,, q Fq dF N re- 00 ,π,EN. 00 ,,π,.I EN F Therefore, , F dI E sely, let dI 00 πN Conver ,π,EN broid be an IDS of the Lie alge- ,N Id,,,, , F FN so that the F N -submodule 00 ,πIE alg ,,N , is an EDS of the Lie ebroid , ,,, N FN Id . Let the F N-submod- ule , 00 ,π,,NE . As 00 ,,π,NI Eit results that it 00 ,π F dI EN exists 1,, F N , ,1rp , so that . Using the T r 00 1, ,π, F rp dIEN heorem 4.7 thereesults that ,π, is EN an involutive IDS. q.e.d. 6. ae International Foundation t Tokai University, during ould also like to thank Pro- rbanski, “Lie Algebroids and Pois- Structures,” Reports on Mathematical Phys- . 2, 1997, pp. 196-208. doi: Acknowledgments I would like to thank Matsum for the research grant a April-September 2008. I w fessors Hideo SHIMADA and Sorin Vasile SABAU from Tokai University-Japan, for useful discussions and their suggestions. In memory of Prof. Dr. Gheorghe RADU. Dedicated to Acad. Prof. Dr. Doc. Radu MIRON at his 84th anniversary. 6. References [1] J. Grabowski and P. U son-Nijenhuis ics, Vol. 40, No ,, F 10.1016/S0034-4877(97)85916-2 [2] C. M. Marle, “Lie Algebroids and Lie Pseudoalgebras,” Mathematics & Physical Sciences, Vol. 27, No. 2, 2008, pp. 97-147. L. I. Nicolescu, “Lectures on the G[3] eometry of Mani- folds,” World Scientific, Singapore, 1996. doi:10.1142/9789814261012 [4] M. de Leon, “Methods of Differential Geometry in Analitical Mechanics,” North-Holland, Amsterdam, 1989. [5] R. L. Bryant, S. S. Chern, R. B. Gardner, H. schmidt and P. A. Griffiths, “ L. Gold- Exterior Differential Sys- tems,” Springer-Verlag, New York, 1991. [6] T. A. Ivey and J. M. Landsberg, “Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems,” American Mathematical Society, Providence, 2003. |