Dhaka Univ. J. Sci. 60(2): 247-252, 2012 (July)
Exterior Algebra with Differential Forms on Manifolds Md. Showkat Ali, K. M. Ahmed, M. R Khan and Md. Mirazul Islam Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh. E-mail: and [email protected] and [email protected] Received on 12. 10. 2011. Accepted for Publication on 25. 03. 2012
Abstract The concept of an exterior algebra was originally introduced by H. Grassman for the purpose of studying linear spaces. Subsequently Elie Cartan developed the theory of exterior differentiation and successfully applied it to the study of differential geometry [8], [9] or differential equations. More recently, exterior algebra has become powerful and irreplaceable tools in the study of differential manifolds with differential forms and we develop theorems on exterior algebra with examples.
I. Introduction , , ∈ ( ) , ∈ ( ). Then we have Due to Elie Cartan’s systematic development of the (i) Distributive Law method of exterior differentiation, the alternating tensors ( ) have played an important role in the study of manifolds + = + ( ) [2]. An alternating contravariant tensor of order is also + = + (ii) Anti-commutative Law called an exterior vector of degree or an exterior - vector. The space ( ) is called the exterior space of = (1) (iii) Associative Law of degree . For convenience, we have the following ( ) = ( ) . conventions: ( ) = , ( ) = . More importantly, there exists an operation, the exterior (wedge) product, for Remark 1. Suppose , ∈ = ( ). Then the anti- exterior vectors such that the product of two exterior commutative law implies vectors is another exterior vector. Differential forms are an important component of the apparatus of differential = − , = = 0. geometry [10], they are also systematically employed in Generally, if there are repeated exterior 1-vectors in a topology, in the theory of differential equations, in polynomial wedge product, then the product is zero. mechanics, in the theory of complex manifolds and in the theory of functions of several complex variables. Currents Definition 2. Denote the formal sum ∑ ( ) by ( ). are a generalization of differential forms, similar to Then ( ) is a 2 - dimensional vector space. Let generalized functions. The algebraic analogue of the theory of differential forms makes it possible to define = , = differential forms on algebraic varieties and analytic spaces. Differential forms arise in some important physical contexts. For example, in Maxwell’s theory of where ∈ ( ), ∈ ( ). Define the exterior (wedge) electromagnetism, the Faraday 2 form, or electromagnetic product of and by 1 a b field strength, is F f dx dx , where f are = . 2 ab ab , formed from the electromagnetic fields. In this paper we have studied theorems on exterior algebra with Then ( ) becomes an algebra with respect to the exterior differential forms. product and is called the exterior algebra or Grassman algebra of . II. Exterior Algebra The set {1, (1 ≤ ≤ ), (1 ≤ ≤ ≤ Definition 1. Suppose is an exterior -vector and ), … , … } is a basis of the vector space ( ). an exterior -vector. Let Similarly, we have an exterior algebra for the dual space ∗, = ( ⊗ ) ( ∗) = ( ∗). where is the alternating mapping that defined in [4]. Then is an exterior ( + )-vector, called the An element of ( ∗) is called an exterior form of degree exterior (wedge) product of and . or exterior -form on ; it is an alternating -valued -linear Theorem 2.1. [4] The exterior product satisfies the function on . following rules. Suppose , , ∈ ( ),
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∗ The vector spaces ( ) and ( ) are dual to each Conversely, if , … , are linearly independent, then they other by a certain pairing. Suppose … ∈ ( ) can be extended to a basis { … , , … , } of . and ∗ … ∗ ∈ ( ∗) . Then Then
∗ ∗ < … , … >= det (< , >) … … ≠ 0. Therefore … ≠ 0. This completes the proof of the Thus { … , 1 ≤ < ⋯ < ≤ } and ∗ ∗ theorem. □ { … , 1 ≤ < ⋯ < ≤ }, the basis of ( ) and ( ∗) respectively satisfy the following Theorem 2.4 (Cartan’s Lemma) [4]. Suppose { , … , } relationship: and { , … , } are two sets of vectors in such that < … , ∗ … ∗ >= det (< , ∗ >) ∑ = 0. (1)
If ,… , are linearly independent, then can be 1, { … } = { } = … = … expressed as linear combination of : … { } 0, … ≠ { … } = ∑ ;1 ≤ ≤ with = . Thus these two bases are dual to each other. Theorem 2.5. Suppose , … , are linearly independent Theorem 2.2. Suppose : → is a linear map. Then vectors in and ∈ ( ). A necessary and sufficient ∗ commutes with the exterior product, that is, for any condition for to be expressible in the form ∈ ( ∗) and ∈ ( ∗), = + ⋯ + , (2) ∗( ) = ∗ ∗ . where , … , ∈ ( ), is that Proof. Choose any , … , ∈ . Then … = 0. (3) ∗ ( )( , … , ) = ( ( ), … , ( )) Proof. When + > (2) and (3) are trivially true. In the following we assume that + ≤ . Necessity is obvious, so 1 = . , … , . we need only show sufficiency. Extend , … , to a basis ( + )! ( ) ( ) ∈ Ѕ ( ) { , … , , , … , } of . Then can be expressed as
= + ⋯ + + ( ( ) , … , ( ) ) ⋯ 1 ∗ … … , = . , … , . ( + )! ( ) ( ) ∈ Ѕ ( ) where , … , ∈ ( ). Plugging into (3) we get ∗ ∗ ∗ ( ( ), … , ( )) = ( , … , ). ∗ ∗ ∗ Therefore ( ) = . This completes the ⋯ proof of the theorem. □ … … … = 0 (4) Theorem 2.3. A necessary and sufficient condition for the Inside the summation, the terms vectors , … , ∈ to be linearly dependent is … … ( + 1 ≤ < ⋯ < ≤ ) … = 0. are all basis vectors of ( ). Therefore (4) gives
… Proof. If , … , are linearly dependent, then we may = 0 ; + 1 ≤ < ⋯ < ≤ , assume without loss of generality that can be expressed i.e., = + ⋯ + . This completes the proof as a linear combination of ,… , : of the theorem. □ = + ⋯ + . Theorem 2.6. Suppose , ; , ; 1 ≤ ≤ are two Then … = … ( + ⋯ + sets of vectors in . If { , , 1 ≤ ≤ } is linearly independent and ) = 0. Exterior Algebra with Differential Forms on Manifolds 249
Suppose , ∈ ( ). For any ∈ , let = , (5) ( ) = ( ) ( ), where the right hand side is a wedge product of two exterior then , are linear combinations of , … , and forms. It is obvious that ∈ ( ). The space ( ) , … , are also linearly independent. then becomes an algebra with respect to addition, scalar multiplication and the wedge product. Moreover, it is a graded Proof: Wedge-multiply (5) by itself times to get algebra. This means that ( ) is a direct sum (8) of a ! ( … ) sequence of vector space and the wedge product defines a = ! ( … ). (6) map ∶ ( ) × ( ) → ( ), Since { , ; 1 ≤ ≤ } is linearly independent, the left hand side of (6) is not equal to zero, that is, where ( ) is zero when + > . { , ; 1 ≤ ≤ } is also linearly independent Remark 3. The tensor algebras ( ) and ( ∗), with (Theorem 2.3).We can also obtain from (6) that respect to the tensor product ⊗ and the exterior algebra ( ), … = 0, with respect to the exterior product are all graded algebras. which means { , , … , , , } is linearly Theorem 3.1. [7] Suppose is an -dimensional smooth dependent. Therefore can be expressed as a linear manifold. Then there exists a unique map : ( ) → ( ) combination of , … , , , … , .The above such that ( ( )) ⊂ ( ) and such that satisfies the conclusion is also true for . This completes the proof following: of the theorem. □ (i) For any , ∈ ( ), Remark 2. For a geometrical application of theorem 2.6 ( + ) = + . refer to Chern [5]. (ii) Suppose is an exterior differential - III. Exterior Differentiation form. Then Suppose is an -dimensional smooth manifold. The bundle of exterior -forms on ( ) = + (−1) .
∗ ∗ (iii) If is a smooth function on , ( ) = ( ) ∈ i.e., ∈ ( ) then, is precisely the is a vector bundle on . Use ( ) to denote the space differential of . of the smooth sections of the exterior bundle ( ∗): ( ) = ( ( ∗)). (iv) If ∈ ( ), then ( ) = 0. ( ) is a ( )-module. The elements of ( ) are The map defined above is called the exterior derivative. called exterior differential -forms on . Therefore, an Theorem 3.2. (Poincare’s Lemma). [4] = 0, i.e., for any exterior differential -form on is a smooth skew- exterior differential form , symmetric covariant tensor field of order on . ( ) = 0 . Similarly, the exterior form bundle ( ∗) = ∗ ⋃ ∈ ( ) is also a vector bundle on . The elements Theorem 3.3. [10] Let M be a C manifold. Then the set of the space of its sections ( ) are called exterior ( ) of all k-forms on M can be naturally identified with differential forms on . Obviously ( ) can be that of all multi-linear and alternating maps, as C (M ) expressed as the direct sum modules, from k-fold direct product of ( )to . C (M ) ( ) = ∑ ( ) , (7) Now, we shall characterize the exterior differentiation without i.e., every differential form can be written as = using the local expression namely, we have the following + + ⋯ + , where is an exterior differential theorem: -form. The wedge product of exterior forms can be extended to the space of exterior differential form ( ). 250 Md. Showkat Ali et al.
Theorem 3.4. Let M be a C manifold and ∈ ( ) 1 k1 d(X ,, X ) { (1)s1 f }. an arbitrary k-form on M. Then for arbitrary vector fields 1 k1 j1jsjk1 (k 1)! s1 x j , … , ∊ ( ), we have On the other hand, when we calculate the right hand side of the formula using [X , X ] 0 , we obtain the same value. 1 k 1 i j i1 d(X 1 , , X k 1 ) { (1) X i ((X 1 ,, This finishes the proof. □ k 1 i1 We can consider theorem 3.4 as a definition of the exterior ˆ i j ˆ differentiation that is independent of the local coordinates. Xi,, Xk1)) (1) ([Xi, Xj],X1,, Xi, i j Example 1. Suppose the Cartesian coordinates in ℝ are ˆ given by ( , , ). , X j ,, Xk1)}. 1) If is a smooth function on ℝ , then ˆ = + + . Here the symbol X i means X i omitted. In particular, the often-used case of k = 1 is The vector formed by its coefficients ( , , ) is the 1 d(X ,Y ) {X (Y ) Y (X ) ([X ,Y ])}. gradient of , denoted by . 2 2) Suppose = + + , where , , are Proof. If we consider the right hand side of the formula to smooth functions on ℝ . Then be proved as a map from the (k + 1)-fold direct product of = + + = − + X (M ) to C (M ) , we see that it satisfies the conditions − + − . of degree (k +1) alternating form as a map between modules over C (M ) . Since it is easy to verify this by Let be the vector ( , , ), then the vector Theorem 2.8, we see that the right hand side is a (k +1) form on M. If two differential forms coincide in some − , − , − neighborhood of an arbitrary point, they coincide on the whole. Then consider a local coordinate system formed by the coefficients of is just the of the vector field , denoted by . (U , x1 , xn ) around an arbitrary point p M . Let the local expression of with respect to the local coordinate 3) Suppose = + + . Then system be = , , f dx dx . Then we have i1ik i1 ik i1 ik = d df dx dx . (8) where means the divergence of the vector field i1ik i1 ik i1 ik = ( , , ). From the linearity of differential forms with respect to the From theorems, two fundamental formulas in a vector calculus functions on M, it is enough to consider only vector fields follow immediately. Suppose is a smooth function on ℝ and is a smooth tangent vector field on ℝ . Then X such that in a i X i (i 1,,k 1) ( ) = 0, x j ( ) = 0. neighborhood of P. Then [X , X ] 0 near P. i j Theorem 3.5. Suppose is a differential 1-form on a smooth Moreover by the alternating property of differential manifold . and are smooth tangent vector fields on . forms, we may assume that j1 jk1.Then, if we Then apply (8) to (X 1 ,, X k 1 ), we have < , > = < , > − < , > − < [ , ] , > Exterior Algebra with Differential Forms on Manifolds 251
Proof: Given a “natural” differential operator that is depending only on the differential structure of M. This operator gives information on < , > = < , > − < , > − < the topology of the manifolds. [ , ] , > (9) Remark 4. Using Theorem 3.5 the Frobenius condition for - Since both sides of (9) are linear with respect to , we dimensional distributions method can be rephrased in its dual may assume that is a monomial form. Suppose = { , … , } is a smooth -dimensional = ; where and are smooth functions on distributions on . Then for any point ∈ , ( ) is an - ⇒ = dimensional linear subspace of . Let ∗ L.H.S: < , > = < , > ( ( )) = ∈ < , > = 0 ∈ ( ) . < , > < , > ∗ = ( ( )) is certainly ( − )-dimensional subspace of , < , > < , > called the annihilator subspace of ( ) . In a neighborhood of an arbitrary point, there exist − linearly independent = = . − . differential 1-forms , … , that span the annihilator subspace ( ( )) at any point in the neighborhood. In fact, R.H.S: is spanned by linearly independent smooth tangent vector < , > − < , > − < [ , ] , > = < fields , … , in a neighborhood. Therefore there exist [ ] , > − < , > − < , , = − smooth tangent vector fields , … , such that ( ) − ( ) − [ , ] = . + { , … , , , … , } is linearly independent everywhere − . − − + = in that neighborhood. . − . Suppose { , … , , , … , } are the dual differential 1 - Therefore L.H.S = R.H.S. This completes the proof of the forms in that neighborhood. Then at every point , ( ( )) is theorem. □ spanned by , … , . Locally the distribution is IV. Differential Forms therefore equivalent to the system of equations
The most important tensors are differential forms. The = 0, + 1 ≤ ≤ , main reason for their importance in the fact under mild Often called a Pfaffian system of equations [1]. compactness assumptions, it is possible to define the integration of a form of degree k on a manifold of By (9), we have dimension k. < , > = < , > − < , > Definition 3. A differential form of degree k on a −< [ , ] , > = −< manifold M is a smooth section of the bundle ( ) and [ , ] , >. we denoted by ( ) = . Hence the distribution = { , … , } satisfies the Frobenius For a vector space the exterior product of ∈ ∗ and condition ∈ ∗ is the ( + ) antisymmetric form defined by [ , ] ∈ , 1 ≤ , ≤ Λg = Ant(f ⊗ g). ! ! if and only if < , > = 0, 1 ≤ , ≤ , + Example 2. Let ∈ ( ) be a differential 1-form on 1 ≤ ≤ . such that for any ∈ (3), ∗ = holds. Then = 0. Theorem 4.1. [3] Suppose is an -dimensional distribution satisfying the Frobenius condition on a manifold . Then Exterior forms are more interesting than tensors, for the through any point ∈ , there exists a maximal integral following reasons: we shall define on manifold ℒ( ) of such that any integral manifold of through is an open submanifold of ℒ( ) with respect to the topology . Ω The term maximal integral manifold in this theorem means that it is not proper subset of another integral manifold [6]. 252 Md. Showkat Ali et al.
Suppose : → is a smooth map from a smooth ∗( ) = ∗( ) = ∗ ∗ manifold to a smooth manifold . Then it induces a = ( ∗ ) ( ∗ ) = ( ∗ ). linear map between the spaces of exterior differential Now assume that (11) holds for exterior differential forms of forms: ∗: ( ) → ( ). degree < . We need to show that it also holds for exterior In fact, induces a tangent mapping ∗ ∶ ( ) → differential -forms. Suppose ( ) ( ) at every point ∈ and the definition of the = , map ∗: ( ) → ( ) for each homogeneous part of ( ) and ( ) as follows: where is a differential 1-form on and is an exterior ∗ differential ( − 1) form on . Then by the induction If ∈ ( ), ≥ 1, then ∈ ( ) such that for hypothesis we have any smooth tangent vector fields , … , on , ∗( ) ∗ ∗ ( ∗ ) ∗ ∗ = ( ) = − < … , > =< ∗ … ∗ , > ( ) , ∈ ∗ ( ∗ ) (10) = ∗( ) − ∗( ) where < , > is the pairing. If ∈ ( ), we define ∗ ∗ ∗ = ∈ ( ) the map distributes over the = ( ). exterior product, that is, for any , ∈ ( ) , This completes the proof of the theorem. □ ∗( ) = ∗ ∗ . The importance of the induced map ∗ also rests on the …………….. fact that it commutes with the exterior derivative . 1. Arnold’s V.I., 1978. Mathematical Methods of Classical Theorem 4.2. Suppose : → is a smooth map from Mechanics, Springer Verlag. a smooth manifold to a smooth manifold . Then the induced map ∗: ( ) → ( ) commutes with the 2. Brickell, F. and R. S. Clark, 1970. Differential Manifolds: An Introduction,Van Nostrand Reinhold Company, London. exterior derivative , that is, ∗ ∗ 3. Carno, M.P. do, 1992, Riemannian geometry, Birkh auser, = : ( ) → ( ). Boston. (11) 4. Chern, S.S. Chern, W.H., K.S., 2000, Law, Lectures on In other words, the following diagram commutes. Differential Geometry.
( ) ⎯⎯⎯⎯⎯ ( ) 5. Chern, S.S., 1967, Curves and Surfaces in Euclidean Space, in Global Geometry and Analysis, MAA Studies in Mathematics, Vol. 4, 16-56. ∗ ∗ 6. G. de Rham, 1984, Differential manifolds, Springer.
7. Isham, J. Chrish, 1989, Modern Differential Geometry for ( ) ⎯⎯⎯⎯⎯ ( ) Physicists, World Scientific Publishing Co. Pte. Ltd. ∗ Proof: Since both and are linear, we need only 8. Kobayashi, S. and K. Nomizu, 1996. Foundations of consider the operation of both sides of (11) on a Differential Geometry, Volume 1, John Wiley and Sons, monomial . Interscience, New York.
First suppose is a smooth function on i.e., ∈ 9. Novikov, S.P. and A. T. Fomenko, 1990. Basic Elements of ( ). Choose any smooth tangent vector field on . Differential Geometry and Topology. Then it follows from (11) that 10. Sternberg, S., 1964, Lectures on differential geometry, Prentice- ∗ Hall. < , ( ) > = < ∗ , > = ∗ ( ) = ( ) = < , ( ∗ ) >. Therefore ∗( ) = ( ∗ ). Next suppose = , where , are smooth functions on . Then
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