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 , , ∈ (), 248 Md. Showkat Ali et al. ∗ 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