Exterior Algebra with Differential Forms on Manifolds

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1 Dhaa Unv. J. Sc. 60(2): , 202 (July) Exteror Algebra wth Dfferental Forms on Manfolds Md. Showat Al, K. M. Ahmed, M. R Khan and Md. Mrazul Islam Department of Mathematcs, Unversty of Dhaa, Dhaa 000, Bangladesh. E-mal: and msa37@yahoo.com and meznan9@yahoo.co.u Receved on Accepted for Publcaton on Abstract The concept of an exteror algebra was orgnally ntroduced by H. Grassman for the purpose of studyng lnear spaces. Subsequently Ele Cartan developed the theory of exteror dfferentaton and successfully appled t to the study of dfferental geometry [8], [9] or dfferental equatons. More recently, exteror algebra has become powerful and rreplaceable tools n the study of dfferental manfolds wth dfferental forms and we develop theorems on exteror algebra wth examples. I. Introducton Due to Ele Cartan s systematc development of the method of exteror dfferentaton, the alternatng tensors have played an mportant role n the study of manfolds [2]. An alternatng contravarant tensor of order r s also called an exteror vector of degree r or an exteror rvector. The space Λ (V) s called the exteror space of V of degree r. For convenence, we have the followng conventons: Λ (V) = V, Λ (V) = F. More mportantly, there exsts an operaton, the exteror (wedge) product, for exteror vectors such that the product of two exteror vectors s another exteror vector. Dfferental forms are an mportant component of the apparatus of dfferental geometry [0], they are also systematcally employed n topology, n the theory of dfferental equatons, n mechancs, n the theory of complex manfolds and n the theory of functons of several complex varables. Currents are a generalzaton of dfferental forms, smlar to generalzed functons. The algebrac analogue of the theory of dfferental forms maes t possble to defne dfferental forms on algebrac varetes and analytc spaces. Dfferental forms arse n some mportant physcal contexts. For example, n Maxwell s theory of electromagnetsm, the Faraday 2 form, or electromagnetc feld strength, s F 2 f ab dx a dx b, where f ab are formed from the electromagnetc felds. In ths paper we have studed theorems on exteror algebra wth dfferental forms. II. Exteror Algebra Defnton. Suppose ξ s an exteror -vector and η an exteror l-vector. Let ξ Λ η = A (ξ η) where A s the alternatng mappng that defned n [4]. Then ξ Λ η s an exteror ( + l)-vector, called the exteror (wedge) product of ξ and η. Theorem 2.. [4] The exteror product satsfes the followng rules. Suppose ξ, ξ, ξ Λ (V), η, η, η Λ (V), ζ Λ (V). Then we have () () () Dstrbutve Law ( ξ + ξ ) Λ η = ξ Λ η + ξ Λ η ξ Λ ( η + η ) = ξ Λ η + ξ Λ η Ant-commutatve Law ξ Λ η = () η Λ ξ Assocatve Law ( ξ Λ η ) Λ ζ = ξ Λ ( η Λ ζ ). Remar. Suppose ξ, η V = Λ (V). Then the antcommutatve law mples ξ Λ η = η Λ ξ, ξ Λ ξ = η Λ η = 0. Generally, f there are repeated exteror -vectors n a polynomal wedge product, then the product s zero. Defnton 2. Denote the formal sum Λ (V) by Λ(V). Then Λ(V) s a 2 - dmensonal vector space. Let ξ = ξ, η = η where ξ Λ (V), η Λ (V). Defne the exteror (wedge) product of ξ and η by ξ Λ η = ξ Λ η., Then Λ(V) becomes an algebra wth respect to the exteror product and s called the exteror algebra or Grassman algebra of V. The set {, e ( n), e Λ e ( n),, e Λ Λ e } s a bass of the vector space Λ(V). Smlarly, we have an exteror algebra for the dual space V, Λ(V ) = Λ (V ). An element of Λ (V ) s called an exteror form of degree r or exteror r-form on V ; t s an alternatng F-valued r-lnear functon on V.

2 248 The vector spaces Λ (V) and Λ (V ) are dual to each other by a certan parng. Suppose v Λ Λv Λ (V) and v Λ Λv Λ (V ). Then < v Λ Λv, v Λ Λv >= det (< v, v >) Thus {e Λ Λe, < < n} and {e Λ Λe, < < n}, the bass of Λ (V) and Λ (V ) respectvely satsfy the followng relatonshp: < e Λ Λe, e Λ Λe >= det (< e, e >) = δ, f { = } = { } 0, f { } { } Thus these two bases are dual to each other. Theorem 2.2. Suppose f: V W s a lnear map. Then f commutes wth the exteror product, that s, for any φ Λ (W ) and ψ Λ (W ), f ( φ Λ ψ ) = f φ Λ f ψ. Proof. Choose any v,, v V. Then f ( φ Λ ψ )(v,, v ) = φ Λ ψ (f(v ),, f(v )) = (r + s)! = (r + s)! Ѕ () Ѕ () sgnσ. φ fv (),, fv (). ψ(fv (),, fv () ) sgnσ. f φv (),, v (). f ψ(v (),, v () ) = f φ Λ f ψ(v,, v ). Therefore f ( φ Λ ψ ) = f φ Λ f ψ. Ths completes the proof of the theorem. Theorem 2.3. A necessary and suffcent condton for the vectors v,, v V to be lnearly dependent s v Λ Λ v = 0. Proof. If v,, v are lnearly dependent, then we may assume wthout loss of generalty that v can be expressed as a lnear combnaton of v,, v : v = a v + + a v. Then v Λ Λ v Λ v = v Λ Λ v Λ ( a v + + a v ) = 0. Md. Showat Al et al. Conversely, f v,, v are lnearly ndependent, then they can be extended to a bass { v v, v,, v } of V. Then v Λ Λ v Λ v Λ Λ v 0. Therefore v Λ Λ v 0. Ths completes the proof of the theorem. Theorem 2.4 (Cartan s Lemma) [4]. Suppose {v,, v } and {w,, w } are two sets of vectors n V such that v Λ w = 0. () If v,, v are lnearly ndependent, then w can be expressed as lnear combnaton of v : w = a v ; α r wth a = a. Theorem 2.5. Suppose v,, v are r lnearly ndependent vectors n V and w Λ (V). A necessary and suffcent condton for w to be expressble n the form w = v Λ ψ + + v Λ ψ, (2) where ψ,, ψ Λ (V), s that v Λ Λ v Λ w = 0. (3) Proof. When p + r > n (2) and (3) are trvally true. In the followng we assume that p + r n. Necessty s obvous, so we need only show suffcency. Extend v,, v to a bass { v,, v, v,, v } of V. Then w can be expressed as w = v Λ ψ + + v Λ ψ + ξ v Λ Λ v, where ψ,, ψ Λ (V). Pluggng nto (3) we get ξ v Λ Λ v Λ v Λ Λ v = 0 (4) Insde the summaton, the terms v Λ Λ v Λ v Λ Λ v (r + α < < α n) are all bass vectors of Λ (V). Therefore (4) gves ξ = 0 ; r + α < < α n,.e., w = v Λ ψ + + v Λ ψ. Ths completes the proof of the theorem. Theorem 2.6. Suppose v, w ; v, w ; α are two sets of vectors n V. If {v, w, α } s lnearly ndependent and

3 Exteror Algebra wth Dfferental Forms on Manfolds 249 v Λ w = v Λ w, (5) then v, w are lnear combnatons of v,, v and w,, w are also lnearly ndependent. Proof: Wedge-multply (5) by tself tmes to get! (v Λ w Λ Λ v Λ w ) =! (v Λ w Λ Λ v Λ w ). (6) Snce { v,w ; α } s lnearly ndependent, the left hand sde of (6) s not equal to zero, that s, { v, w ; α } s also lnearly ndependent (Theorem 2.3).We can also obtan from (6) that v Λ w Λ Λ v Λ w Λ v = 0, whch means { v,w,, v, w,v } s lnearly dependent. Therefore v can be expressed as a lnear combnaton of v,, v, w,, w.the above concluson s also true for w. Ths completes the proof of the theorem. Remar 2. For a geometrcal applcaton of theorem 2.6 refer to Chern [5]. III. Exteror Dfferentaton Suppose M s an m-dmensonal smooth manfold. The bundle of exteror r-forms on M Λ (M ) = Λ (T ) s a vector bundle on M. Use A (M) to denote the space of the smooth sectons of the exteror bundle Λ (M ): A (M) = Γ(Λ (M )). A (M) s a C (M)-module. The elements of A (M) are called exteror dfferental r-forms on M. Therefore, an exteror dfferental r-form on M s a smooth sewsymmetrc covarant tensor feld of order r on M. Smlarly, the exteror form bundle Λ(M ) = Λ (T ) s also a vector bundle on M. The elements of the space of ts sectons A(M) are called exteror dfferental forms on M. Obvously A(M) can be expressed as the drect sum A(M) = A (M), (7).e., every dfferental form w can be wrtten as w = w + w + + w, where w s an exteror dfferental -form. The wedge product of exteror forms can be extended to the space of exteror dfferental form A(M). Suppose w, w A(M). For any p M, let w Λ w (p) = w (p) Λ w (p), where the rght hand sde s a wedge product of two exteror forms. It s obvous that w Λ w A(M). The space A(M) then becomes an algebra wth respect to addton, scalar multplcaton and the wedge product. Moreover, t s a graded algebra. Ths means that A(M) s a drect sum (8) of a sequence of vector space and the wedge product Λ defnes a map Λ A (M) A (M) A (M), where A (M) s zero when r + s > m. Remar 3. The tensor algebras T(V) and T(V ), wth respect to the tensor product and the exteror algebra Λ(V), wth respect to the exteror product Λ are all graded algebras. Theorem 3.. [7] Suppose M s an m-dmensonal smooth manfold. Then there exsts a unque map d: A(M) A(M) such that d(a (M)) A (M) and such that d satsfes the followng: () For any ω, ω A(M), d(ω + ω ) = dω + dω. () Suppose ω s an exteror dfferental r- form. Then d(ω Λ ω ) = dω Λ ω + ( ) ω Λ dω. () If f s a smooth functon on M,.e., f A (M) then, df s precsely the dfferental of f. (v) If f A (M), then d(df) = 0. The map d defned above s called the exteror dervatve. Theorem 3.2. (Poncare s Lemma). [4] d = 0,.e., for any exteror dfferental form ω, d(dω) = 0. Theorem 3.3. [0] Let M be a C manfold. Then the set A (M) of all -forms on M can be naturally dentfed wth that of all mult-lnear and alternatng maps, as C (M ) modules, from -fold drect product of (M)to C (M ). Now, we shall characterze the exteror dfferentaton wthout usng the local expresson namely, we have the followng theorem:

4 250 Theorem 3.4. Let M be a C manfold and ω A (M) an arbtrary -form on M. Then for arbtrary vector felds,, (M), we have d(,, ) { ( ) ( (,, ˆ,, ˆ, )) ( ) ([, ],,,, ˆ,, )}. Here the symbol ˆ means often-used case of = s omtted. In partcular, the d(, Y ) { ( Y ) Y ( ) ([, Y ])}. 2 Proof. If we consder the rght hand sde of the formula to be proved as a map from the ( + )-fold drect product of (M ) to C (M ), we see that t satsfes the condtons of degree ( +) alternatng form as a map between modules over C (M ). Snce t s easy to verfy ths by Theorem 2.8, we see that the rght hand sde s a ( +) form on M. If two dfferental forms concde n some neghborhood of an arbtrary pont, they concde on the whole. Then consder a local coordnate system ( U, x, xn ) around an arbtrary pont p M. Let the local expresson of wth respect to the local coordnate system be f dx dx. Then we have d df dx dx. (8) From the lnearty of dfferental forms wth respect to the functons on M, t s enough to consder only vector felds such that (,, ) n a x neghborhood of P. Then [, ] 0 near P. Moreover by the alternatng property of dfferental forms, we may assume that. Then, f we apply (8) to,, ), we have ( d( Md. Showat Al et al. s,, ) { ( ) f }. s ( )! s x On the other hand, when we calculate the rght hand sde of the formula usng [, ] 0, we obtan the same value. Ths fnshes the proof. We can consder theorem 3.4 as a defnton of the exteror dfferentaton that s ndependent of the local coordnates. Example. Suppose the Cartesan coordnates n gven by (x, y, z). ) If f s a smooth functon on R, then df = dx + dy + dz. The vector formed by ts coeffcents gradent of f, denoted by grad f. (,, R are ) s the 2) Suppose a = Adx + Bdy + Cdz, where A, B, C are smooth functons on R. Then da = da Λ dx + db Λ dy + dc Λ dz = dz Λ dx + dx Λ dy. Let be the vector (A, B, C), then the vector C y B z, A z C x, B x A y dy Λ dz + formed by the coeffcents of da s ust the curl of the vector feld, denoted by curl. 3) Suppose a = Ady Λ dz + Bdz Λ dx + Cdx Λ dy. Then where dv = (A, B, C). da = A x, B y, C dx Λ dyλ dz z = dv dx Λ dyλ dz means the dvergence of the vector feld From theorems, two fundamental formulas n a vector calculus follow mmedately. Suppose f s a smooth functon on R and s a smooth tangent vector feld on R. Then curl ( grad f ) = 0, dv ( curl ) = 0. Theorem 3.5. Suppose ω s a dfferental -form on a smooth manfold M. and Y are smooth tangent vector felds on M. Then < Λ Y, dω > = < Y,ω > Y <, ω > < [,Y ], ω >

5 Exteror Algebra wth Dfferental Forms on Manfolds 25 Proof: Gven < Λ Y, dω > = < Y,ω > Y <,ω > < [,Y ], ω > (9) Snce both sdes of (9) are lnear wth respect to ω, we may assume that ω s a monomal ω = g df ; where f and g are smooth functons on M dω = dg Λ df L.H.S: < Λ Y, dω > = < Λ Y, dg Λ df > <, dg > <, df > = < Y, dg > < Y, df > g f = = g. Yf f. Yg Yg Yf R.H.S: < Y, ω > Y <,ω > < [,Y ], ω > = < Y, g df > Y <, g df > < [,Y ],g df = (g Yf ) Y( g f ) g[,y ] f = g. Yf + g Yf Yg. f g Yf g Yf + g Yf = g. Yf f. Yg Therefore L.H.S = R.H.S. Ths completes the proof of the theorem. IV. Dfferental Forms The most mportant tensors are dfferental forms. The man reason for ther mportance n the fact under mld compactness assumptons, t s possble to defne the ntegraton of a form of degree on a manfold of dmenson. Defnton 3. A dfferental form of degree on a manfold M s a smooth secton of the bundle Λ (M) and we denoted by Γ(Λ M) = Ω M. For a vector space the exteror product of f Λ E and g Λ E s the ( + l) antsymmetrc form defned by fλg = Ant(f g).!! Example 2. Let w Ω (S ) be a dfferental -form on S such that for any φ SO(3), φ w = w holds. Then w = 0. Exteror forms are more nterestng than tensors, for the followng reasons: we shall defne on Ω M a natural dfferental operator that s dependng only on the dfferental structure of M. Ths operator gves nformaton on the topology of the manfolds. Remar 4. Usng Theorem 3.5 the Frobenus condton for rdmensonal dstrbutons method can be rephrased n ts dual form. Suppose L = {,, } s a smooth r-dmensonal dstrbutons on M. Then for any pont p M, L (p) s an rdmensonal lnear subspace of T. Let (L (p)) = ω T <,ω > = 0 for any L (p). (L (p)) s certanly (m r)-dmensonal subspace of T, called the annhlator subspace of L (p). In a neghborhood of an arbtrary pont, there exst m r lnearly ndependent dfferental -forms ω,, ω that span the annhlator subspace (L (p)) at any pont p n the neghborhood. In fact, L s spanned by r lnearly ndependent smooth tangent vector felds,, n a neghborhood. Therefore there exst m r smooth tangent vector felds,, such that {,,,,, } s lnearly ndependent everywhere n that neghborhood. Suppose {ω,, ω,ω,, ω } are the dual dfferental - forms n that neghborhood. Then at every pont p, (L (p)) s spanned by ω,, ω. Locally the dstrbuton L s therefore equvalent to the system of equatons ω = 0, r + s m, Often called a Pfaffan system of equatons []. By (9), we have < Λ,dω > = <,ω > <,ω > < [, ],ω > = < [, ],ω >. Hence the dstrbuton L = {,, } satsfes the Frobenus condton [, ] L, α, β r f and only f < Λ,dω > = 0, α, β r, r + s m. Theorem 4.. [3] Suppose L s an r-dmensonal dstrbuton satsfyng the Frobenus condton on a manfold M. Then through any pont p M, there exsts a maxmal ntegral manfold L(p) of L such that any ntegral manfold of L through p s an open submanfold of L(p) wth respect to the topology O. The term maxmal ntegral manfold n ths theorem means that t s not proper subset of another ntegral manfold [6].

6 252 Suppose f: M N s a smooth map from a smooth manfold M to a smooth manfold N. Then t nduces a lnear map between the spaces of exteror dfferental forms: f : A(N) A(M). In fact, f nduces a tangent mappng f T (M) T () (N) at every pont p M and the defnton of the map f : A(N) A(M) for each homogeneous part of A(N) and A(M) as follows: If β A (N), r, then f β A (M) such that for any r smooth tangent vector felds,, on M, < Λ Λ, f β > =< f Λ Λ f, β > (), p M (0) where <, > s the parng. If β A (N), we defne f β = βof A (M) the map f dstrbutes over the exteror product, that s, for any ω, η A(N), f (ω Λ η) = f ω Λ f η. The mportance of the nduced map f also rests on the fact that t commutes wth the exteror dervatve d. Theorem 4.2. Suppose f: M N s a smooth map from a smooth manfold M to a smooth manfold N. Then the nduced map f : A(N) A(M) commutes wth the exteror dervatve d, that s, f o d = d o f : A(N) A(M). () In other words, the followng dagram commutes. f A(N) A(M) A(N) A(M) Proof: Snce both f and d are lnear, we need only consder the operaton of both sdes of () on a monomal β. Frst suppose β s a smooth functon on N.e., β A (N). Choose any smooth tangent vector feld on M. Then t follows from () that <, f (dβ) > = < f, dβ > = f (β) = (β o f) = <, d(f β) >. Therefore f (dβ) = d(f β). Next suppose β = u dv, where u, v are smooth functons on N. Then f f (dβ) = f (du Λ dv) = f du Λ f dv = d(f u) Λ d(f v) = d(f β). Md. Showat Al et al. Now assume that () holds for exteror dfferental forms of degree < r. We need to show that t also holds for exteror dfferental r-forms. Suppose β = β Λ β, where β s a dfferental -form on N and β s an exteror dfferental (r ) form on N. Then by the nducton hypothess we have d o f (β Λ β ) = d( f β Λ f β ) = d( f β ) Λ f β f β Λ d(f β ) = f (dβ Λ β ) f (β Λ dβ ) = f o d(β Λ β ). Ths completes the proof of the theorem.... Arnold s V.I., 978. Mathematcal Methods of Classcal Mechancs, Sprnger Verlag. 2. Brcell, F. and R. S. Clar, 970. Dfferental Manfolds: An Introducton,Van Nostrand Renhold Company, London. 3. Carno, M.P. do, 992, Remannan geometry, Brh auser, Boston. 4. Chern, S.S. Chern, W.H., K.S., 2000, Law, Lectures on Dfferental Geometry. 5. Chern, S.S., 967, Curves and Surfaces n Eucldean Space, n Global Geometry and Analyss, MAA Studes n Mathematcs, Vol. 4, G. de Rham, 984, Dfferental manfolds, Sprnger. 7. Isham, J. Chrsh, 989, Modern Dfferental Geometry for Physcsts, World Scentfc Publshng Co. Pte. Ltd. 8. Kobayash, S. and K. Nomzu, 996. Foundatons of Dfferental Geometry, Volume, John Wley and Sons, Interscence, New Yor. 9. Novov, S.P. and A. T. Fomeno, 990. Basc Elements of Dfferental Geometry and Topology. 0. Sternberg, S., 964, Lectures on dfferental geometry, Prentce- Hall.

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