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Functor and Natural Operators on Symplectic Manifolds

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Functor and Natural Operators on Symplectic Manifolds Mathematical Methods and Techniques in Engineering and Environmental Science Functor and natural operators on symplectic manifolds CONSTANTIN PĂTRĂŞCOIU Faculty of Engineering and Management of Technological Systems, Drobeta Turnu Severin University of Craiova Drobeta Turnu Severin, Str. Călugăreni, No.1 ROMANIA [email protected] http://www.imst.ro Abstract. Symplectic manifolds arise naturally in abstract formulations of classical Mechanics because the phase–spaces in Hamiltonian mechanics is the cotangent bundle of configuration manifolds equipped with a symplectic structure. The natural functors and operators described in this paper can be helpful both for a unified description of specific proprieties of symplectic manifolds and for finding links to various fields of geometric objects with applications in Hamiltonian mechanics. Key-words: Functors, Natural operators, Symplectic manifolds, Complex structure, Hamiltonian. 1. Introduction. diffeomorphism f : M → N , a vector bundle morphism The geometrics objects (like as vectors, covectors, T*f : T*M→ T*N , which covers f , where −1 tensors, metrics, e.t.c.) on a smooth manifold M, x = x :)*)(()*( x * → xf )( * NTMTfTfT are the elements of the total spaces of a vector So, the cotangent functor, )(:* → VBmManT and bundles with base M. The fields of such geometrics objects are the section in corresponding vector more general ΛkT * : Man(m) → VB , are the bundles. bundle functors from Man(m) to VB, where Man(m) By example, the vectors on a smooth manifold M (subcategory of Man) is the category of m- are the elements of total space of vector bundles dimensional smooth manifolds, the morphisms of this (TM ,π M , M); the covectors on M are the elements category are local diffeomorphisms between of total space of vector bundles (T*M , pM , M). A manifolds. Recall that an operator is a rule field of vectors on the manifold M is a section of transforming the sections of a fiber bundles the vector bundles (TM ,πM , M); a field of M MpE ),,( into sections of another fiber bundle covectors on the manifold M is a section of the M ' MpE )',,'( . Regarding to the bundle functors vector bundles (T*M , pM , M). A differential form k or an exterior form of degree k or a kform is a Λ T * , the exterior derivative d, transforms sections k k +1 section of the vector bundle ( ΛkT * M , p k ,M) of Λ T * M into sections of Λ T * M for every M manifold M and d commutes with local In fact, the vector bundle (TM ,π M , M) is the value diffeomorphisms. So, d is a natural operator from the of the functor, T: Man→VB, where Man is the functor k into functor k+1 and written: category of smooth manifolds(the morphisms of Λ T * Λ T * k k+1 this category is smooth maps between manifolds) d : Λ T * Λ T * , k ∈N and k k +1 and VB is the category of vector bundles, the dM : Λ T * M→ Λ T * M for any ∈ mManM )( morphisms of this category are morphisms of vector bundles [2]. So, the tangent functor T , 2.Natural operator from tangent to associates to each manifold M , the tangent bundle cotangent functors on symplectic (TM ,π M , M) and to each smooth map f:M→N , a vector bundle morphism Tf: TM→TN , which manifolds. covers f. Recall that the couple (M,ω ) is an almost symplectic In the case of the cotangent functor T* we can not manifold if M is a smooth manifold and ω is an use the whole category Man, we use only almost symplectic form i.e. a nondegenerate 2form on manifolds of the same dimension the manifold M. The cotangent functor )(:* → VBmManT If an almost symplectic form ω∈ Ω 2( M) is closed, ω associates to each m-dimensional manifold M , the is called symplectic form and the couple (M,ω ) is cotangent bundle (T*M ,π*M , M) and to each local called symplectic manifold. ISBN: 978-1-61804-046-6 162 Mathematical Methods and Techniques in Engineering and Environmental Science If (M,ω ) is a symplectic manifold. Then, each So, Φ : T T* , is an operator from T to T* . tangent space ( Tx M , ωx ) is symplectic vector Moreover, Φ is a regular operator because every space and the manifold M is necessarily of even smoothly parameterized family of vector fields is dimension. transformed into a smoothly family of covector fields. If 2n is the dimension of manifold M, the product Let F and G be two bundle functors over manifolds, n ω = ω ∧ ω ∧ ... ∧ ω (n-factors) never vanishes, M a smooth manifold, FMM and G the fiber bundle thus M is orientable and any symplectic corresponding to M ; Γ(FM) and Γ(GM), the set of diffeomorphism preserve the volume. smooth section of this fiber bundle. By Darboux’s Theorem such a 2n-dimensional 2 nn Recall that a natural operator A : F G is a system of manifold looks locally like ≅ CR with the regular operators n AM : Γ(FM) → Γ(GM) satisfying following standard symplectic form ω0 ∑ i ∧= dydx i , i=1 conditions: (i) For every section s∈Γ(FM) and every where 21 n 21 yyyxxx n ),...,,,,...,,( are coordinates 2 nn isomorphism f : M → N in category of manifold it in . ≅ CR . holds So symplectic manifolds, in contrast to A (Ff s f -1) = Gf A s f –1 Riemannian manifolds, have no local invariants. N o o o M o A symplectomorphism between 2n-dimensional (ii) AU( s U ) = (AMs ) U for every section s ∈ Γ(FM) symplectic manifolds M ω ),( and M ω ),( is and every open submanifold U of M. 11 22 a diffeomorphism : 1 → MMf 2 satisfying the Let be T* and, T the restriction of tangent and condition: f * = ωω 12 . We denote Simp(2n), the cotangent functors to category Symp(2n). category of 2n-dimensional symplectic manifolds , Proposition. Φ : T T* is a natural operator between the morphisms in this category are the the two bundle functors T and T* . symplectomorphisms. Proof. Φ is a system of regular operators The Simp(2n) is a subcategory of Man(2n). 1 ΦM : X (M) → Ω ()M , M ∈ Ob Symp(2n) We will consider the restriction of tangent and Let be M , N∈ Ob Symp(2n) , ω and ω the cotangent functors to category Symp(2n). M N Let (TM ,π , M) be the tangent bundle and corresponding sympletic forms. M The condition (i) from previously definition is (T*M ,π*M , M) the cotangent bundle of -1 –1 symplectic manifold M . The manifold M is ΦN( Tf o X o f ) = T*f oΦM X of endower with a symplectic structure i.e. a for every vector field X∈ X (M) and for every nondegenerate closed 2form ω∈ Ω 2( M). symplectomorphism f : M → N Then, each tangent space ( Tx M , ωx ) , x∈ M is Let be x ∈M, f(x) = y∈N, Z∈ X (N) symplectic vector space. –1 [ΦN( Tf o X o f ] (Z) y = ω N y(Txf Xx , Zy ) For each x∈ M we can define the map –1 [T*f oΦM X of ](Z)y ΦMx : Tx M* →T x M , = T*f(i ω M x)(Zy) . X x Xx → ΦMx(Xx) =i ω x =ω x(Xx , ) X x 1– = ω M x(Xx , Tf Zy) Since ωx is nondegenerate this map is an 1– = f*(ω N y) (Xx , Tyf Zy) isomorphism between the tangent space Tx M and 1– cotangent space T*x M. = ω N y(Txf Xx , Txf (Tyf )Zy ) Then, the map ΦM : T *M→T M , Φ/TxM =ΦMx , = ω N y(Txf Xx , Zy ) ∀ x∈M is an isomorphism of tangent fiber In the previous calculus we have used the equality bundles TM and cotangent fiber bundle T*M. ω M =*f ω N due the fact that f is a Let X (M) be the set of vector fields of M (the symplectomorphism. 1 sections of tangent bundle ) and Ω ( M) the set of The condition (ii) is satisfied because the geometric 1forms of M (the sections of cotangent bundle objects implied in definition of Φ do not depend on (T*M ,π*M , M)). the changes of coordinates. There is a one-toone correspondence between vector fields and 1forms of manifold M , A vector field X∈ X (M) is called symplectic if ω 1 iX given by the map ΦM :) X (M → Ω ( M) , is closed. ΦM( X) = Φ o X = iX ω. ISBN: 978-1-61804-046-6 163 Mathematical Methods and Techniques in Engineering and Environmental Science If is a vector field and the Lie and linear and X∈ X (M) L X →∈ JMx x :xx → x MTMTJ derivative along , the vector field is 2 X X∈X (M) x −= IdJ . symplectic if and only if Indeed, we L X ω =0. The almost complex structure J on the manifold M is know that Because is L X =iX o d + do iX . ω tamed by the symplectic form ω if ω(X,JX)>0, closed we have dω =0 . But, X is symplectic if ∀X∈ T(M) -{0}; if moreover ω is J-invariant, J is and only if i ω is closed i.e. if and only if said to be calibrated. We know that any symplectic X manifold have a lot a almost complex structure, the d( ω) = 0 iX space of almost complex structures on a given L X ω = (iX o d) ω+ (do iX )ω symplectic manifold (M,ω) which are tamed (resp. = iX (d ω)+d(iX ω) calibrated) by ω is nonempty and contractible(in = d(i ω) = 0 particular these spaces are connected).
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