On Poincaré lemma or Volterra theorem about differential forms and cohomology groups
aa r X i v : . [ m a t h . G M ] M a y On Poincaré lemma or Volterra theorem aboutdifferential forms and cohomology groups
A. Lesfari
Department of MathematicsFaculty of SciencesUniversity of Chouaïb DoukkaliB.P. 20, El Jadida, Morocco . [email protected], [email protected] Abstract.
The Poincaré lemma (or Volterra theorem) is of utmost im-portance both in theory and in practice. It tells us every differential formwhich is closed, is locally exact. In other words, on a contractible manifoldall closed forms are exact. The aim of this paper is to present some directproofs of this lemma and explore some of its numerous consequences. Someconnections with Cech-De Rham-Dolbeault cohomologies, ∂ -Poincaré lemmaor Dolbeault-Grothendieck lemma are given. Mathematics Subject Classification (2010).
Keywords. one-parameter group of diffeomorphisms, Lie derivative, interiorproduct, differential forms, analytic sheaves and cohomology groups.
Let ω be a k -differential form ( k > ) on an open subset U ⊂ R n .We assume that ω is C . Recall that ω is closed (or is a cocycle) if dω = 0 .Similarly, ω is exact (or cohomological at 0) if there is a C , ( k − -differentialform λ on U such that : ω = dλ . Notice that there are no exact -formsbecause there are no − -forms. It is well known that every exact differentialform is closed and that the reciprocal is false in general. Poincaré lemmaensures that it is true on a star-shaped open subset of R n whose definitionis as follows : let [ a, b ] = { λa + (1 − λ ) b, λ ∈ [0 , } , ( a, b ) ∈ R n × R n . We saythat an open subset U ⊂ R n is star-shaped if there exists an a in U , suchthat for all b in U , [ a, b ] ⊂ U . In other words, if the line segment from b to a is in U .We will study Poincaré lemma which is considered fundamental both intheory and in practice. Already at the university level and as evidenced byseveral scientific books, this is a key result for the study of many problems inmathematics, physics and the theorem itself has applications in areas ranging1rom electrodynamics to differential and integral calculus on varieties. Ata higher level it intervenes for example when studying the cohomology ofDe Rham varieties [7] to name only this striking example. Let’s note forinformation that the so currently called Poincaré lemma is due to Volterra.Indeed, the Poincaré lemma is really Volterra theorem ; the work of Volterrais contained in several notes published in the Rendiconti of the Accademia deiLincei [10] (see also [6]). In this work we use as everyone the name Poincarélemma instead of Volterra theorem and we leave the question to be clarifiedby historians.The lemma to be demonstrated is a typical example of a local result, soit suffices to prove it in local coordinates, for example in an arbitrarily smallopen of R n , but it must be admitted that the technical details are muchmore complicated to demonstrate when we move from the case of -forms to k -differential forms. All known proofs of this lemma, using these coordinatesare often too computerized and rarely illuminating.We offer some proofs of the Poincaré lemma in this paper because theproofs represent widely different views of the subject. The first proof is givenin a very classical setting and represents the classical point of view ; it isboth elementary and constructive but the problem is that it is a bit long andtechnical. Then, our problem is to give a quick proof of this lemma althoughrequiring certain knowledges pushed in differential geometry ; the proofs usesvery modern machinery and represent a more modern point of view. Bothof these points of view have merit and so we demonstrate them both. Thepaper is divided in some sections and subsections, each of them devoted tovarious and complementary aspects of the problem concerning, in particular,connections with Cech-De Rham-Dolbeault cohomologies, ∂ -Poincaré lemmaor Dolbeault-Grothendieck lemma. We give a little information (which will be needed in the second proof)about one-parameter group of diffeomorphisms, differential operators, Liederivative, inner product and Cartan’s formula. This discussion is brief, butshould be enough to define notation. Let M be a differentiable manifoldof dimension m . Let T M be the tangent bundle to M , i.e., the union ofspaces tangent to M at all points x , T M = S x ∈ M T x M . This bundle has anatural structure of differentiable variety of dimension m and it allows us toconvey immutably to the manifolds the whole theory of ordinary differentialequations. A vector field (we also say section of the tangent bundle) on M isan application, denoted X , which at every point x ∈ M associates a tangentvector X x ∈ T x M . In other words, X : M −→ T M , is an application suchthat if π : T M −→ M , is the natural projection, we have π ◦ X = id M . Let ( x , ..., x m ) be a local coordinate system in a neighborhood U ⊂ M . In this2ystem the vector field X is written in the form X = m X k =1 f k ( x ) ∂∂x k , x ∈ U, where the functions f , . . . , f m : U −→ R , are the components of X withrespect to ( x , ..., x m ) . A vector field X is differentiable if its components f k ( x ) are differentiable functions. Given a point x ∈ M , we write g Xt ( x ) (orsimply g t ( x ) ) the position of x after a displacement of a duration t ∈ R .There is thus an application g Xt : M −→ M , t ∈ R , which is a diffeomor-phism (a one-to-one differentiable mapping with a differentiable inverse), byvirtue of the theory of differential equations. The vector field X generatesa one-parameter group of diffeomorphisms g Xt on M , i.e., a differentiableapplication ( C ∞ ) : M × R −→ M , satisfying a group law : i ) ∀ t ∈ R , g Xt : M −→ M is a diffeomorphism. ii ) ∀ t, s ∈ R , g Xt + s = g Xt ◦ g Xs .The condition ii ) means that the mapping t g Xt , is a homomorphism ofthe additive group R into the group of diffeomorphisms of M . It implies that g X − t = (cid:0) g Xt (cid:1) − , because g X = id M is the identical transformation that leaves every pointinvariant. The one-parameter group of diffeomorphisms g Xt on M , which wehave just described is called a flow and it admits the vector field X forvelocity fields ddt g Xt ( x ) = X (cid:0) g Xt ( x ) (cid:1) , with the initial condition : g X ( x ) = x . Obviously ddt g Xt ( x ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = X ( x ) . Hence by these formulas g Xt ( x ) is the curve on the manifold that passesthrough x and such that the tangent at each point is the vector X (cid:0) g Xt ( x ) (cid:1) .The vector field X generates a unique group of diffeomorphisms of M . Withevery vector field X we associate the first-order differential operator L X .This is the differentiation of functions in the direction of the vector field X .We have L X : C ∞ ( M ) −→ C ∞ ( M ) , F L X F, where L X F ( x ) = ddt F (cid:0) g Xt ( x ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) t =0 , x ∈ M. C ∞ ( M ) being the set of functions F : M −→ R , of class C ∞ . The operator L X is linear : L X ( α F + α F ) = α L X F + α L X F , where α , α ∈ R ,and satisfies the Leibniz formula : L X ( F F ) = F L X F + F L X F . L X F ( x ) only depends on the values of F in the neighborhood of x ,we can apply the operator L X to the functions defined only in the neighbo-rhood of a point, without the need to extend them to the full variety M . Let ( x , ..., x m ) be a local coordinate system on M . In this system the vector field X has components f , . . . , f m and the flow g Xt is defined by a system of diffe-rential equations. Therefore, the derivative of the function F = F ( x , ..., x m ) in the direction of X is written L X F = f ∂F∂x + · · · + f m ∂F∂x m . In other words, in the coordinates ( x , ..., x m ) the operator L X is written L X = f ∂∂x + · · · + f m ∂∂x m , this is the general form of the first order lineardifferential operator.Let M and N be two differentiable manifolds of dimension m and n respectively, U ⊂ M , V ⊂ N two open subsets. For any differentiable appli-cation g : U −→ V , and any k -differential form in V , ω = X ≤ i ,...,i k ≤ n f i ,...,i k dx i ∧ ... ∧ dx i k , we define a k -differential form in U (called the pull-back by g or inverseimage or the transpose of ω by g ) by setting g ∗ ω = X ≤ i ,...,i k ≤ n ( f i ,...,i k ◦ g ) dg i ∧ ... ∧ dg i k , where dg i l = m X j =1 ∂g i l ∂y j dy j , are -forms in U . Note that g ∗ is a linear operator from the space of k -formson N to the space of k -forms on U (the asterisk indicates that g ∗ operatesin the opposite direction of g ). Let X be a vector field on a differentiablemanifold M . We recalled above that the vector field X generates a uniquegroup of diffeomorphisms g Xt (that we also note g t ) on M , solution of thedifferential equation ddt g Xt ( p ) = X ( g Xt ( p )) , p ∈ M, with the initial condition g X ( p ) = p . Let ω be a k -differential form. The Liederivative of ω with respect to X is the k -form differential defined by L X ω = ddt g ∗ t ω (cid:12)(cid:12)(cid:12)(cid:12) t =0 = lim t → g ∗ t ( ω ( g t ( p ))) − ω ( p ) t .
4n general, for t = 0 , we have ddt g ∗ t ω = dds g ∗ t + s ω (cid:12)(cid:12)(cid:12)(cid:12) s =0 = g ∗ t dds g ∗ s ω (cid:12)(cid:12)(cid:12)(cid:12) s =0 = g ∗ t ( L X ω ) . (1)It is easily verified that for the k -differential form ω ( g t ( p )) at the point g t ( p ) ,the expression g ∗ t ω ( g t ( p )) is indeed a k -differential form in p . For all t ∈ R ,the application g t : R −→ R being a diffeomorphism then dg t and dg − t areapplications, dg t : T p M −→ T g t ( p ) M,dg − t : T g t ( p ) M −→ T p M. The Lie derivative of a vector field Y in the direction X is defined by L X Y = ddt g − t Y (cid:12)(cid:12)(cid:12)(cid:12) t =0 = lim t → g − t ( Y ( g t ( p ))) − Y ( p ) t . In general, for t = 0 , we have ddt g − t Y = dds g − t − s Y (cid:12)(cid:12)(cid:12)(cid:12) s =0 = g − t dds g − s Y (cid:12)(cid:12)(cid:12)(cid:12) s =0 = g − t ( L Y ) . An interesting operation on differential forms is the inner product thatis defined as follows : the inner product of a k -differential form ω by a vectorfield X on a differentiable manifold M is a ( k − -differential form, denoted i X ω , defined by ( i X ω )( X , ..., X k − ) = ω ( X, X , ..., X k − ) , where X , ..., X k − are vector fields. If X = m X j =1 X j ( x ) ∂∂x j , is the local expression of the vector field on the variety M of dimension m and ω = X i j , it is obviously enough to swap the indices i and j . Then, ∂h∂x i ( x ) = Z t n X j =1 ∂f i ∂x j ( a + t ( x − a ))( x j − a j ) + f i ( a + t ( x − a )) dt, = Z (cid:18) t ddt f i ( a + t ( x − a )) + f i ( a + t ( x − a )) (cid:19) dt, = Z ddt ( tf i ( a + t ( x − a ))) dt, = f i ( x ) , where ≤ i ≤ n and x ∈ U . The -differential forms in U , i.e., continuousapplications U −→ R and since ω = n X i =1 f i dx i , the applications f i are C andwe deduce from the relations above that h is C on U .b) (For a k -differential form, k ≥ ). Without restricting the generality,we suppose that U is a star-shaped open subset with respect to . Let ψ be the application from the set of l -differential forms on U to the set of ( l − -differential forms on U defined by ψ ( λ ) = X ≤ i ,...,i k ≤ n l X j =1 ( − j − (cid:18)Z t l − g i ,...,i l ( t. ) dt (cid:19) π i j dx i ∧ ... ∧ d dx i j ∧ ... ∧ dx i l , where λ = X ≤ i ,...,i l ≤ n g i ,...,i l dx i ∧ ... ∧ dx i l ,g i ,...,i l ( t. ) : U −→ R , x g i ,...,i l ( tx ) , t ∈ [0 , , the application π i j is the projection on the i th j coordinate and d dx i j denotesthe term omitted. Let ω = X ≤ i ,...,i k ≤ n f i ,...,i k dx i ∧ ... ∧ dx i k ,
7e a C , k -differential form in U . The idea of the proof is to show that ω = ψ ( dω ) + d ( ψ ( ω )) , because in this case since by hypothesis ω is closed, then ω = d ( ψ ( ω )) (byconstruction, ψ (0) = 0 ) and therefore, the form ω is exact (note that ψ ( ω ) is C if ω is). We have dω = X ≤ i ,...,i k ≤ n ∂f i ,...,i k ∂x i ∧ dx i ∧ dx i ∧ ... ∧ dx i k , and ψ ( dω )= n X i =1 X ≤ i ,...,i k ≤ n (cid:20)(cid:18)Z t k ∂f i ,...,i k ∂x i ( t. ) dt (cid:19) p i dx i ∧ ... ∧ dx i k + k X j =1 ( − j (cid:18)Z t k ∂f i ,...,i k ∂x i ( t. ) dt (cid:19) p i j dx i ∧ dx i ∧ ... ∧ d dx i j ∧ ... ∧ dx i k (cid:21) , or ψ ( dω )= X ≤ i ,...,i k ≤ n " n X i =1 (cid:18)Z t k ∂f i ,...,i k ∂x i ( t. ) dt (cid:19) p i dx i ∧ ... ∧ dx i k − X ≤ i ,...,i k ≤ n n X i =1 k X j =1 ( − j − (cid:18)Z t k ∂f i ,...,i k ∂x i ( t. ) dt (cid:19) p i j dx i ∧ dx i ∧ ... ∧ d dx i j ∧ ... ∧ dx i k i . Similarly, we have d ( ψ ( ω ))= X ≤ i ,...,i k ≤ n k X j =1 ( − j − d (cid:20)(cid:18)Z t k − f i ,...,i k ( t. ) dt (cid:19) p i j (cid:21) ∧ dx i ∧ ... ∧ d dx i j ∧ ... ∧ dx i k , or d ( ψ ( ω ))= X ≤ i ,...,i k ≤ n k X j =1 ( − j − n X i =1 (cid:20)(cid:18)Z t k − ∂f i ,...,i k ∂x i ( t. ) tdt (cid:19) p i j + (cid:18)Z t k − f i ,...,i k ( t. ) dt (cid:19) δ i j ,i (cid:21) dx i ∧ dx i ∧ ... ∧ d dx i j ∧ ... ∧ dx i k , d ( ψ ( ω ))= X ≤ i ,...,i k ≤ n n X i =1 k X j =1 ( − j − (cid:18)Z t k ∂f i ,...,i k ∂x i ( t. ) dt (cid:19) p i j dx i ∧ dx i ∧ ... ∧ d dx i j ∧ ... ∧ dx i k + X ≤ i ,...,i k ≤ n k X j =1 ( − j − (cid:18)Z t k − f i ,...,i k ( t. ) dt (cid:19) dx i j ∧ dx i ∧ ... ∧ d dx i j ∧ ... ∧ dx i k , and finally, d ( ψ ( ω ))= X ≤ i ,...,i k ≤ n n X i =1 k X j =1 ( − j − (cid:18)Z t k ∂f i ,...,i k ∂x i ( t. ) dt (cid:19) p i j dx i ∧ dx i ∧ ... ∧ d dx i j ∧ ... ∧ dx i k + k X ≤ i ,...,i k ≤ n (cid:18)Z t k − f i ,...,i k ( t. ) dt (cid:19) dx i ∧ ... ∧ dx i k . Hence, ψ ( dω ) + d ( ψ ( ω ))= X ≤ i ,...,i k ≤ n n X i =1 (cid:18)Z t k ∂f i ,...,i k ∂x i ( t. ) dt (cid:19) p i dx i ∧ ... ∧ dx i k + X ≤ i ,...,i k ≤ n (cid:18)Z kt k − f i ,...,i k ( t. ) dt (cid:19) dx i ∧ ... ∧ dx i k , = X ≤ i ,...,i k ≤ n (cid:18)Z ddt (cid:16) t k f i ,...,i k ( t. ) (cid:17) dt (cid:19) dx i ∧ ... ∧ dx i k , = X ≤ i ,...,i k ≤ n f i ,...,i k dx i ∧ ... ∧ dx i k , = ω, and proof 1 ends. Proof 2 : Using the notions and properties mentioned at the beginning ofthis section, we give a quick proof of the lemma in question. Indeed, considerthe differential equation ˙ x = X ( x ) = xt , as well as its solution g t ( x ) = x t. x , depending on how C ∞ of the initial condition and is a parameter group of diffeomorphisms. Wehave, g ( x ) = 0 , g ( x ) = x , g ∗ ω = 0 , g ∗ ω = ω. Hence, ω = g ∗ ω − g ∗ ω, = Z ddt g ∗ t ωdt, = Z g ∗ t ( L X ω ) dt ( by (1) ) , = Z g ∗ t ( di X ω ) dt, ( according to (2) and the fact that dω = 0)= Z dg ∗ t i X ωdt, ( because df ∗ ω = f ∗ dω ) . We can therefore find a differential form λ such that : ω = dλ , where λ = Z g ∗ t i X ωdt, which completes the proof 2 and ends the two proofs of the lemma. (cid:3) Remark 1
Any exact differential form is closed. It is well known that theconverse is false in general and depends on the open U on which the dif-ferential form is C . For example, if U = R \{ (0 , } then the differentialform ω = − x x + x dx + x x + x dx , is closed but is not exact. Indeed, this form is obviously closed. To show thatit is not exact, we use the fact that in general if ω is an exact -differentialform on an open subset and γ a closed path in this C piecewise open subset,then Z γ ω = 0 . In the present example, U = R \{ (0 , } and let γ be the unitcircle of parametric equations : x ( t ) = cos t , x ( t ) = sin t , t ∈ [0 , π ] . Wehave Z γ ω = Z π (cid:18) − x x + x x ′ ( t ) + x x + x x ′ ( t ) (cid:19) = Z π (sin t +cos t ) dt = 2 π. Since Z γ ω = 0 , then ω is not exact This example shows that in Poincarélemma, the hypothesis that the open is starred is essential (here, the open U = R \{ (0 , } is not a star-shaped subset). roposition 1 The Poincaré lemma assures the existence of λ but not itsuniqueness.Proof : Indeed, let ω be a closed k -differential form on a star-shaped opensubset of R n . By Poincaré lemma, there exist a ( k − -differential form λ such that ω = dλ . If µ is any ( k − -differential form, then λ + dµ satisfiesthe same equation : d ( λ + dµ ) = dλ + d ( dµ ) = dλ = ω, (because the exterior derivative obeys the rule : d ( dµ ) = 0 , see for example[7]). Conversely, if λ and λ are any two ( k − -differential forms suchthat : ω = dλ = dλ , then d ( λ − λ ) = 0 . By Poincaré lemma, there exista ( k − -differential form θ such that : λ − λ = dθ , i.e., λ = λ + dθ .From this we deduce that the general solution can be expressed as the sumof a particular solution and the derivative of an arbitrary ( k − -differentialform. The proof is completed. (cid:3) On manifods the Poincaré lemma can be stated as follows :
Proposition 2
Any closed k -differential form ω is exact in the neighborhoodof an n -manifold M (or, in R n any closed differential form is exact).Proof : We have seen that for a star-shaped open subset of R n , the form ω is exact. Since M is locally diffeomorphic to an open subset of R n , then forevery point p ∈ M there exists also a neighborhood U of p and a ( k − -differential form λ such that ω = dλ on U . (cid:3) We will give in the subsections below several formulations of Poincarélemma. Similarly, some examples and questions closely related to the Poin-caré lemma will be discussed.
Let F be a sheaf on a topological space M . Consider the set C k ( U , F ) of k -cochains of degree k with values in F , i.e., the set of applications thatassociates with each k -up of open cover of U a section on their intersection.In other words, we have C k ( U , F ) = Y α ,...,α k F ( U α ∩ ... ∩ U α k ) , ( k + 1) distinct elements α , ..., α k in the index set. In particular, we have C ( U , F ) = Y α F ( U α ) and C ( U , F ) = Y α,β F ( U α ∩ U β ) . We define the coboundary operator δ : C k ( U , F ) −→ C k +1 ( U , F ) , for s ∈ C k ( U , F ) , by ( δs )( U α , ..., U α k ) = k +1 X j − ( − j s ( U α , ..., d U α j , ..., U α k +1 ) (cid:12)(cid:12)(cid:12) U α ∩ ... ∩U αk +1 . We easily check that δ = 0 , so ( C ( U , F )) forms a cochain complex and wehave C ( U , F ) δ −→ C ( U , F ) δ −→ C ( U , F ) δ −→ · · · with δ : C ( U , F ) −→ C ( U , F ) ,s αβ s β | U α ∩U β − s α | U α ∩U β ,δ : C ( U , F ) −→ C ( U , F ) ,s αβγ s βγ | U α ∩U β ∩U γ − s αγ | U α ∩U β ∩U γ + s αβ | U α ∩U β ∩U γ , ... δ : C k ( U , F ) −→ C k +1 ( U , F ) ,s α ...α k +1 k +1 X j =0 ( − j s α ... c α j ...α k +1 (cid:12)(cid:12)(cid:12) U α ∩ ... ∩U αk +1 . For any sheaf F of abelian groups on M and for any open cover U of M , wecan find an application F −→ C ( U , F ) , so that the sequence −→ F −→ C ( U , F ) δ −→ C ( U , F ) δ −→ · · · be exact. The Cech cohomology of the sheaf F with respect to the cover U is the quotient, H k ( U , F ) = ker (cid:2) δ : C k ( U , F ) −→ C k +1 U , F ) (cid:3) Im [ δ : C k − U , F ) −→ C k U , F )] . These cohomology groups depend on the cover U . The Cech cohomology(or k -th sheaf cohomology) of F on M is defined to be the direct limit of H k ( U , F ) as U becomes finer and finer, H k ( M, F ) = lim −→U H k ( U , F ) . Theorem 2
If the open U α of a cover U of M are such that : H k ( A, F ) = 0 , for any k > and any finite intersection A ≡ U α ∩ ... ∩ U α k of open U α ,then the groups H k ( U , F ) et H k ( M, F ) are isomorphic for all k . Let Ω k be the sheaf of k -differential forms on a manifold M and Z k be the sheaf of the closed k -forms. Let d ∗ be the application of cohomologyassociated with d : Ω k −→ Z k +1 . The k -th De Rham cohomology H kDR ( M ) isdefined as the quotient space of the k -closed differential forms by the ( k − -differential forms. Since a k -form is a global section of the sheaf of k -forms,so it is an element of H ( M, Ω k ) because this latter group of cohomologyidentifies with global sections. Therefore, H kDR ( M ) = H ( M, Z k ) d ∗ H ( M, Ω k − ) . As a consequence of the Poincaré lemma, we have the following result :
Proposition 3
De Rham’s cohomology of a manifold is isomorphic to itsCech cohomology with coefficients in R .Proof : According to Poincaré lemma, any closed form is locally exact. Sothe sheaf sequence −→ Z k −→ Ω k d −→ Z k +1 −→ , is exact where Z ≡ R is the sheaf of locally constant functions and Ω thesheaf of fonctions in C ∞ . The following long exact sequence in cohomologyassociated with the above sequence is H q ( M, Ω k ) d ∗ −→ H q ( M, Z k +1 ) ∂ −→ H q +1 ( M, Z k ) −→ H q +1 ( M, Ω k ) , where d ∗ denotes the exterior derivative and ∂ the boundary operator of thelong exact sequence. The sheaf Ω p admits partitions of the unit, hence for q ≥ , H q ( M, Ω k ) = 0 , and so H q ( M, Z k +1 ) = H q +1 ( M, Z k ) . q = 0 , we have H ( M, Ω k ) d ∗ −→ H ( M, Z k +1 ) ∂ −→ H ( M, Z p ) −→ , and H ( M, Ω k ) = H ( M, Z k +1 ) d ∗ H ( M, Ω k ) . Therefore, H kDR ( M ) = H ( M, Z k − ) = H k − ( M, Z ) = H k ( M, Z ) = H k ( M, R ) , and the result follows. (cid:3) Let M and N two manifolds. Recall that two maps f : M −→ N and g : N −→ M are (smoothly) homotopic, if there exists a (smooth) homotopy h : M × I −→ N, h ( x,
0) = f ( x ) , h ( x,
1) = g ( x ) , ( I is the interval, ≤ t ≤ ). We say the manifolds M and N are homotopyequivalent if there exist (smooth) maps f : M −→ N and g : N −→ M suchthat the composites gof : M −→ M and f og : N −→ N are (smoothly)homotopic to the respective identity maps M −→ M , x x and N −→ N , y y . For example, the closed unit disc in R n is homotopy equivalent toa point. It easy to verify that R n \{ } and S n − are homotopy equivalent.Using reasoning similar to proof 2, we show that : Proposition 4
If two manifolds M and N are homotopy equivalent thentheir cohomology groups are isomorphic.Proof : Indeed, consider the (smooth) homotopy F : M × I −→ N betweenthe above two maps f and g . Then the induced homomorphisms f ∗ : H k ( N ) −→ H k ( M ) , ,g ∗ : H k ( M ) −→ H k ( N ) , of the cohomology groups coincide. It suffices to consider a differential form ω ∈ H k ( gof )( M ) , which implies that it exists a pullback h ∗ ( ω ) = λ + dt ∧ θ,h ∗ ( ω ) | t = t = h ∗ ( ., t )( ω ) = λ | t = t , where λ ∈ H k ( M × I ) , θ ∈ H k − ( M × I ) , and which λ , θ do not involve thedifferential dt (in the sens that λ , θ do not contain a dt term). We have h ∗ ( d M ( ω )) = d M × I ( h ∗ ( ω )) = dt ∧ (cid:18) ∂λ∂t − d M ( θ ) (cid:19) ,
1. Let V ⊂ R m , U ⊂ R n be open subsets, g : V −→ U be a differentiable applicationand ω = X ≤ i <...
Recall that a (smooth) manifold M is contractible if the identitymap on M is homotopic to a constant map or in other words if M is homo-topy equivalent to a point. For example, a star-shaped open subset of R n iscontractible and hence has De Rham cohomology of a point. The cohomologygroups of space R n (and of the ball around any point p ) are isomorphic tothose of p . Thus H k ( R n ) is trivial for k > , while H ( R n ) ≃ R . In fact if M is contractible, which means that M is homotopy equivalent to a point andsince homotopy equivalent manifolds admit isomorphic De Rham cohomologygroups, then H k ( M ) = (cid:26) R , k = 00 , k > This fact leads immediately to the Poincaré Lemma. As we mentioned abovewhen k = 0 , notice that constant functions are closed and H ( M ) is a finitedimensional vector space equal to the number of connected components of M . Here we have only -differential forms, i.e., f ( x ) functions on M . Thereare no exact differential forms. Then, H ( M, R ) = { f : f is closed } . Since df ( x ) = 0 , then in any chart ( U, x , ..., x n ) of M , we have ∂f∂x = ∂f∂x = · · · = ∂f∂x n = 0 . herefore, f ( x ) = constant locally, i.e., f ( x ) = constant on each connec-ted component of M . So the number of connected components of M is thedimension in question. Remark 3
As we have seen the Poincaré lemma and De Rham theorem givea characterization of the de Rham cohomology groups. Applying Poincarélemma for k -differential forms where k = 0 , we show that the cohomology ofa ball is trivial and in particular, dim H k ( R n ) = δ ( k, . If S n is the n -sphereand if T n = ( S ) n is the n -dimensional torus, i.e., the product of n circles,then by proceeding by induction on n , we obtain dim H k ( S n ) = δ ( n, k ) , dim H k ( T n ) = n ! k !( n − k )! . ∂ -Poincaré lemma (or Dolbeault-Grothendieck lemma) In each point of a complex manifold M , we define local coordinates z j and z j as well as the tangent bundle C h ∂∂z j , ∂∂z j i that we note T M . Thisadmits the decomposition
T M = T ′ M ⊕ T ′′ M , where T ′ M (resp. T ′′ M )is the holomorphic (respectively antiholomorphic) part of T M generatedby ∂∂z j (resp. ∂∂z j ). Similarly, the cotangent bundle T ∗ M (dual of T M )admits the decomposition : T ∗ M = T ′∗ M ⊕ T ′′∗ M , into holomorphic andantiholomorphic parts. The points of this bundle are the linear forms on thefibers of T M −→ M . A p -differential form is a section of the bundle Λ p T ∗ M (the exterior powers of T ∗ M ). The points of the latter being the alternatingmultilinear p -forms on T ∗ M . If z , ..., z n are holomorphic local coordinateson a complex manifold M of dimension n , then dz , ..., dz n , dz , ..., dz n forma local base of the space of differential forms. A differential form on M oftype ( p, q ) is given locally by ω = X ≤ j <... 16s holomorphic. The differential of ω is dω = X ≤ j <... 17e have shown previously that Poincaré lemma leads to the nullity ofcohomology groups in the case of star-shaped open subsets. We have anisomorphism H p,q∂ ( M ) ≃ H q ( M, Ω pM ) , for all integers p and q , where H p,q∂ ( M ) = { ∂ -closed forms of type ( p, q ) on M }{ ∂ -exact forms of type ( p, q ) on M } , is a Dolbeault cohomology group of complex manifold M . Références [1] V. I. Arnold, Mathematical methods in classical mechanics , Springer-Verlag, Berlin-Heidelberg- New York, 1978.[2] V. I. Arnold, A.B. Givental, Symplectic geometry in : Dynamical systemsIV , (eds). V.I. Arnold and S.P. Novikov, (EMS, Volume , pp. 1-136)Springer-Verlag, 1988.[3] H. Cartan, Formes différentielles. Application élémentaires au calcul desvariations at à la théorie des courbes et surfaces , Hermann, Paris, 2007.[4] B. A. Dubrovin, S.P., Novikov, A.T., Fomenko, A.T., Modern geometry.Methods and applications , Parts I, II, Springer-Verlag, 1984, 1985.[5] P.A. Griffiths, J. Harris, Principles of algebraic geometry , Wiley-Interscience, New-York, 1978.[6] S. Hans, Differential Forms, the Early Days ; or the Stories of Deah-na’s Theorem and of Volterra’s Theorem , The American MathematicalMonthly, Vol. 108, No. 6 (2001), 522-530[7] A. Lesfari, Introduction à la géométrie algébrique complexe , ÉditionsHermann, Paris, 2015.[8] A. Lesfari, Formes différentielles et analyse vectorielle (Cours et exer-cices résolus) , éditions Ellipses, Paris, 2017.[9] J. Mawhin, Analyse. Fondements, techniques, évolution , De Boeck Uni-versité, Bruxelles, 1997.[10] V. Volterra, Delle variabili complesse negli iperspazii , Rend. Accad. deiLincei, ser. IV, vol. V1 (1889), NotaI, pp. 158-165, Nota II, pp.291-299 ; Sulle funzione conjugate , ibidem, ser. Iva, vol. VI (1889), p. 158-169 ; Opere Matematiche , Accad. Nazion. dei Lincei, Roma (1954), vol. 1 ,pp. 403-410, 411-419, 420-432.[11] A. Weinstein, Symplectic manifolds and their lagrangian submanifolds ,Advances in Maths., 6, (1971), 329-346.1812] A. Weinstein, Lectures on ymplectic manifolds , American MathematicalSociety, Cbms Regional Conference Series in Mathematics, Number 29,1977.[13] A. Weinstein,