Dimensional Reduction of Invariant Fields and Differential Operators. II. Reduction of Invariant Differential Operators

Transcription

1 Dimensional Reduction of Invariant Fields and Differential Operators. II. Reduction of Invariant Differential Operators Petko A. Nikolov and Nikola P. Petrov Abstract. In the present paper, which is a sequel of [1], we consider the dimensional reduction of differential operators DOs) that are invariant with respect to the action of a connected Lie group G. The action of G on vector bundles induces naturally actions of G on their sections and on the DOs between them. In [1] we constructed explicitly the reduced bundle ξ G, such that the set of all its sections, C ξ G ), is in a bijective correspondence with the set C ξ) of all G-invariant sections of the original vector bundle ξ. The main goal of the present paper is, given a G-invariant DO D : C ξ) C η) to construct the reduced DO D G : C ξ G ) C η G ). Our construction of D G uses the geometrically natural language of jet bundles which best reveals the geometry of the DOs and reduces the manipulations with DOs to simple algebraic operations. Since ξ G was constructed in [1] by restricting a certain bundle to a submanifold of its base, an essential ingredient of the dimensional reduction of a DO is the restriction of the DO to a submanifold of the base. To perform such a restriction, one has to find splittings of certain short exact sequences of jet bundles, which in practice can be achieved by choosing an appropriate auxiliary DO a highly nontrivial procedure involving arbitrary choices. However, in the case of a G-invariant DO D, this splitting is provided automatically by the G- invariance of D this uses the Lie derivative of the action of G). Certain properties of the DOs in particular, their formal integrability turn out to be crucial in our construction. We discuss this in detail and give an explicit example showing what can go wrong if one uses an auxiliary DO that is not formally integrable. Finally, we discuss the structure of the set of all G-invariant DOs that lead to the same reduced DO. Mathematics Subject Classification 2010). Primary 53C80; Secondary 58Z05, 37J15, 58D19, 70S10. Keywords. Group action and symmetry; Invariant fields; Non-compact dimensional reduction; Invariant differential operators.

2 2 1. Introduction In this paper we continue the presentation of the method for dimensional reduction initiated in [1]. In the present paper we consider the reduction of differential operators invariant with respect to the actions of a connected but not necessarily compact) Lie group on vector bundles. The natural setting for considering differential operators on vector bundles is the language of jet bundles. It interprets a differential operator as a purely algebraic object the total symbol of the differential operator which is a fiber-preserving mapping between jet bundles. This is convenient for treating the symmetry properties of differential operators and their dimensional reduction. We briefly recall the set-up described in [1]. Let ξ and η be vector bundles over the same base B, and G be a a Lie group acting on ξ and η through vector bundle morphisms, with the same projected action on the common base B. This naturally induces an action of G on the sections C ξ) and C η). In [1] we construct the reduced bundles ξ G and η G, both over B/G, such that the set C ξ G ) of all sections of ξ G is in a bijective correspondence with the set C ξ) G of all G-invariant sections of ξ and similarly for η). Let D : C ξ) C η) be an order-k differential operator. The actions of G on C ξ) and C η) induce naturally an action of G on the set Diff k ξ, η) of all order-k differential operators from ξ to η. Each G-invariant differential operator D maps C ξ) G to C η) G and, hence, determines a reduced differential operator D G : C ξ G ) C η G ) between the sections of the reduced bundles. Since the reduced bundles are constructed explicitly in [1] by restriction of the base B to a submanifold, the construction of the reduced operator D G is naturally related to the process of restricting a differential operator to a submanifold of B. In general, this restriction is not a well-defined process in the sense that it requires additional information. This missing information can be supplied by considering an appropriately chosen auxiliary differential operator. For the restriction procedure to be self-consistent, the auxiliary operator must satisfy a condition called formal integrability. If we consider the reduction of a G-invariant differential operator D, the auxiliary operator comes naturally from the Lie derivative of the action of G the infinitesimal symmetries). We give an explicit algorithm for reduction which involves only computing derivatives and solving a linear system of algebraic equations. We prove the formal integrability of the Lie derivative, which provides a theoretical justification of our algorithm. The algorithm for computing the reduced differential operator D G elucidates the geometric structures arising naturally in the process of reduction. These issues are important, for example, in Kaluza-Klein theories and in model building with a desired symmetry in elementary particle physics. An example of this approach is our construction [2] of differential operators on Minkowski space that are invariant with respect to the nonlinear) action

3 Dimensional Reduction: II 3 of the conformal group, starting from the linear) action of the orthogonal group on a bigger space. Invariant linear differential operators have invariant total symbols. These total symbols can be considered as invariant sections in appropriate vector bundles. According to our general methodology, an invariant section corresponds to a section in the reduced bundle. Therefore, the problem of description of all invariant linear differential operators can be restated as a problem of description of certain reduced bundles. We utilize this approach to study the relation between the set of all invariant linear order-1 differential operators from ξ to η and all linear order-1 differential operators between the reduced bundles ξ G and η G. The plan of the paper is the following. In Sect. 2 we introduce the jet bundle language for description of differential operators, paying special attention to the concept of formal integrability Sections 2.3 and 2.4). Sect. 3 is devoted to the problem of restriction of a differential operator to a submanifold. In Sect. 4 we develop an algorithm for reduction of an invariant differential operator, and resolve all related theoretical issues. In Sect. 5 we discuss the relation between the set of invariant differential operators and the differential operators in the reduced bundles. Although this paper is a sequel to [1], the reader does not need to have read all of [1]. In the present paper, we use mostly the material of Sections of [1], and briefly recall the basic notions of [1] as needed. 2. Differential operators on vector bundles Here we introduce the coordinate-free definition of differential operators, following [3, Chapter IV] Jet bundles Let ξ = E, π, B) and η be finite-dimensional K-vector bundles over the real finite-dimensional manifold B; the standard fiber ξ b := π 1 b) of ξ over b B is a vector space over the field K = R or C. As usual, C B) stands for the ring of all smooth K-valued functions on B, and C ξ) denotes the set of all smooth sections of ξ which is a module over C B)). Throughout the paper it is always assumed that the manifolds, vector bundles, functions and sections of bundles are smooth C ). Let I b B) be the ideal of the ring C B) consisting of all functions f C B) that vanish at the point b B: I b B) := { f C B) : fb) = 0 } C B). Let Ib kb) stand for the ideal of the ring C B) consisting of all functions on B that can be represented as a product of k functions from I b B). Let Zb kb) stand for the subspace of C ξ) consisting of all sections of ξ that can be represented as products of a function from I k+1 b B) and a

4 4 Petko A. Nikolov and Nikola P. Petrov section of ξ: Zb k ξ) := { ψ C ξ) : ψ = f κ, where f I k+1 b B), κ C ξ) }. Clearly, all partial derivatives of ψ Zb k ξ) with respect to the coordinates in the base B) up to order k vanish at b. For each b B define J k ξ) b := C ξ)/zb kξ), and let J k ψ) b be the image of ψ under the canonical projection C ξ) J k ξ) b : ψ J k ψ) b. The k-jet J k ψ) of the section ψ C ξ) is defined by J k ψ)b) := J k ψ) b for any b B. The k-jet J k ψ) b is the coordinate-free concept for the section the field ) ψ and its derivatives up to order k at b. Let J k ξ) stand for the union b B J k ξ) b endowed with the natural vector bundle structure. In more detail, this means the following. Let x µ, z a ) be local coordinates in ξ, where µ = 1,..., n with n = dim B), a = 1,..., dim ξ by definition, dim ξ is the dimension of the standard fiber of ξ). They generate local coordinates in the jet bundle J k ξ), x µ, z a, z a µ 1, z a µ 1µ 2,..., z a µ 1 µ k ), { 1 µ1 µ i n, i = 1,..., k a = 1,..., dim ξ, 2.1) so that the coordinates of J k ψ)b) are ) a J k ψ)b) := µ1 µ i ψ a b), 2.2) µ 1 µ i where µ1 µ i ψ a i ψ a b) := b). 2.3) x µ1 µi x The transition functions gluing J k ξ) come from the standard formulae for transformation of partial derivatives under a change of variables. From the definition, J 0 ξ) = ξ. The number of partial derivatives of order i of each component ψ a is equal to ) n+i 1 i, which is the number of unordered selections of i objects, with repetition allowed, out of n distinct objects. The dimension of the fibers of the vector bundle J k ξ) is, therefore, k ) ) n + i 1 n + k dim J k ξ) = dim ξ = dim ξ. 2.4) i k i=0 For integers k and l satisfying k l 0, let π k,l : J k ξ) J l ξ) 2.5) stand for the natural projections cutting off all derivatives of order l + 1,..., k). Let S k τ B) be the kth symmetrized tensor product of the cotangent bundle of B. There exists a natural morphism over the identity in B i : S k τ B) ξ J k ξ). 2.6) The morphism i is defined as follows: if ω Tb B), and e ξ b, then i S k ω e ) ) 1 = J k k! f fb))k ψ b), 2.7)

5 Dimensional Reduction: II 5 where f C B) and ψ C ξ) are such that ω = df b and e = ψb). In local coordinates x µ, z a ) of ξ, the coordinates 2.2) of is k ω e) are is k ω e) ) { a 0 if j < k, µ 1 µ j = ω µ1 ω µk e a if j = k. A basic fact in theory of jet bundles [3, Sect. IV.2] is the exactness of the following sequence: 0 S k τ B) ξ i J k ξ) πk,k 1 J k 1 ξ) ) This sequence is related to the fact that all coordinate-free i.e., geometrically natural) relations between derivatives come from the sequence J k ξ) πk,k 1 J k 1 ξ) πk 1,k 2 π 2,1 J 1 ξ) π1,0 J 0 ξ). 2.9) In particular, 2.8) tells us that the highest-order derivatives of a section ψ C ξ) are distinguished in a geometrically natural way only if J k 1 ψ)b) = 0, and the subbundle ker π k,k 1 J k ξ) of the highest-order derivatives is isomorphic to S k τ B) ξ Differential operators and their symbols Notations and natural isomorphisms. We start by introducing some useful notations. Let F be a subfield of the field K where K = R or C), and B be a real finite dimensional manifold. Let γ be an F-vector bundle over B, and ζ be a K-vector bundle over B. A vector bundle morphism F from γ to ζ over the identity in B is a mapping from γ to ζ whose restriction F γb to a fiber γ b is an F-linear mapping from γ b to ζ b. Denote by Hom γ, ζ) the set of all such morphisms; Hom γ, ζ) is naturally endowed with a structure of a K-vector space. In Sections 2 3 before we consider the action of a Lie group on the vector bundles all vector bundle morphisms and, more generally, all fiber-preserving mappings) are over the identity in B. Let Lγ, ζ) be the K-vector bundle over B whose fiber Lγ, ζ) b over b B consists of all F-linear mappings from γ b to ζ b. Let L k γ, ζ) stand for the K-vector bundle over B with L k γ, ζ) b equal to the set of all k-linear with respect to F) mappings from γ b to ζ b. Denote by L k sγ, ζ) the subbundle of L k γ, ζ) with L k sγ, ζ) b defined as the set of all symmetric k-linear with respect to F) mappings from γ b to ζ b. Each fiber-preserving mapping F : γ ζ over the identity in B in particular, each vector bundle morphism F Hom γ, ζ)) induces a mapping F between the sections of γ and ζ defined by F : C γ) C ζ) : ψ F ψ := F ψ. 2.10) Let γ stand for the vector bundle dual to γ, i.e., each fiber γ ) b is the dual to the F-vector space γ b. Denote by C γ ζ) the set of all F-linear mappings from γ to ζ over the identity in B; clearly, C γ ζ) is a K-vector space. If F Hom γ, ζ), the mapping F from 2.10) can be considered as

6 6 Petko A. Nikolov and Nikola P. Petrov an element of C γ ζ), and we obtain that the three K-vector bundles introduced above are naturally isomorphic: Hom γ, ζ) = C γ ζ) = C Lγ, ζ)). 2.11) A differential operator and its total symbol. Let ξ and η be K-vector bundles over a common base B. A differential operator DO) of order k from ξ to η is a mapping D : C ξ) C η) such that, for any b B, if ψ 1 C ξ) and ψ 2 C ξ), then J k ψ 1 )b) = J k ψ 2 )b) implies Dψ 1 )b) = Dψ 2 )b). In other words, Dψb) depends only on the values of ψ and its derivatives up to order k at the point b. The set of all DOs of order k from ξ to η will be denoted by Diff k ξ, η). A DO D Diff k ξ, η) can be identified with a fiber-preserving mapping D from J k ξ) to η over the identity in B: in the notations of 2.10), D = D J k. 2.12) The mapping D is called the total symbol of the DO D. A DO D Diff k ξ, η) is said to be linear if it is a K-linear mapping. Let LDiff k ξ, η) stand for the subset of Diff k ξ, η) that consists of all linear DOs. It is clear from the definition that J k LDiff k ξ, J k ξ)). A linear DO D LDiff k ξ, η) can be identified with a vector bundle morphism D Hom J k ξ), η) through 2.12). Using 2.11), we can write the following natural isomorphisms: LDiff k ξ, η) = Hom J k ξ), η) = C J k ξ) η) 2.13) = C LJ k ξ), η)) Universal properties of J k and J k ξ). This short paragraph discusses some aspects of the algebraic interpretation of jet bundles, and may be skipped at first reading. Since each D Diff k ξ, η) can be represented as a composition D J k = D J k as in 2.12), the DO J k : C ξ) C J k ξ)) : ψ J k ψ) is said to be universal. This terminology is consistent with the terminology from category theory see, e.g., [4, Chapter IV]), which we discuss here in more detail for the case of linear DOs the same construction works for nonlinear DOs, but the terminology there is clumsier). Let VB) be the category of all vector bundles η, ζ,... over the manifold B; the morphisms in VB) are the vector bundle morphisms, i.e., F Hom η, ζ). Let ξ VB) be some fixed vector bundle over B, and consider the category LD k ξ) of linear DOs of order k from ξ to some other bundle from VB). Define the covariant functor F = F Ob, F Mor ) from VB) to LD k ξ) by F Ob η) = LDiff k ξ, η), F Mor F ) = F : LDiff k ξ, η) LDiff k ξ, ζ) : D F D F D,

7 Dimensional Reduction: II 7 where η, ζ VB), F Hom η, ζ), and we used the notation 2.10). Let J k ξ) VB) be the kth jet bundle of ξ, and J k Hom ξ, J k ξ)) be the operation of constructing the kth jet of ξ, defined in Sect The couple J k, J k ξ)) is a universal element for the functor F in the following sense [4, Sect. IV.3]: for any η VB) and any D F Ob η) = LDiff k ξ, η) there exists a unique morphism D Hom J k ξ), η) such that F Mor D) J k = D this is nothing but the decomposition 2.10) of the linear DO D The principal symbol of a linear DO. The definition of a principal symbol of a linear DO introduced below will be used only in Sect. 5. As in Sect , let F be a subfield of the field K, and γ and ζ be respectively an F-vector bundle and a K-vector bundle over the real finite dimensional manifold B. Let P k γ, ζ) stand for the vector bundle over B whose fiber P k ξ, η) b is the K-vector space P k γ b, ζ b ) of all homogeneous polynomials of degree k from γ b to ζ b. To be explicit, let dim γ = m, and h 1,..., h m be a basis in γ b ; similarly, let dim ζ = d, and e 1,..., e d be a basis in ζ b. Then P k γ b, ζ b ) consists of all mappings of the form m d γ b Pj a v j e a ζ b, i=1 v i h i j where the first summation in the right-hand side is over all m-tuples j = j 1,..., j m ) of non-negative integers with j j m = k, and v j := v 1 ) j1 v m ) jm. We leave it to the reader to construct a natural isomorphism P k γ, ζ) = L k sγ, ζ). Let ξ and η be K-vector bundles over B, and D LDiff k ξ, η) be a linear DO between them. The principal symbol σ k D) of D is defined as the vector bundle morphism over the identity in B) a=1 σ k D) = D i : S k τ B) ξ η, 2.14) where i is the mapping from 2.6) and 2.7). If K = R, both S k τ B) ξ and η are R-vector bundles, so that the mapping σ k D) is R-linear. If K = C, we assume that S k τ B) is complexified, so that S k τ B) ξ is a C-vector bundle, and in this case σ k D) is C-linear. In more detail, if ω Tb B) and e ξ b, then ω ω) e S k Tb B) ξ b is the map T b B) T b B) v 1 v k ωv 1 ) ωv k )e ξ b. To describe the principal symbols explicitly, one can use the natural isomorphisms recall 2.11)) Hom S k τ B) ξ, η) = C LS k τ B) ξ, η)) = C LS k τ B), Lξ, η))) = C L k sτ B), Lξ, η))) = C P k τ B), Lξ, η))). Thus, σ k D) at b B can be considered as a homogeneous polynomial of degree k defined on T b B) and taking values in Lξ, η) b. To compute

8 8 Petko A. Nikolov and Nikola P. Petrov σ k D)ω)e) for ω Tb B), e ξ b by using 2.7), take f C B) and ψ C ξ) satisfying ω = df b and e = ψb), then 1 σ k D)ω)e) = D f fb)) k ψ )) b). 2.15) k! Using 2.13) and the short exact sequence 2.8), one can show that 0 LDiff k 1 ξ, η) emb LDiff k ξ, η) σ k Hom S k τ B) ξ, η) 0 is a short exact sequence [3, Sect. IV.3]. Here emb stands for the natural embedding, and σ k is the mapping from an order-k linear DO D to its principal symbol σ k D) Prolongation of a DO and formal integrability One can differentiate simultaneously both sides of a differential equation Dψ = φ. In a coordinate-free language, the result of differentiating of a DO is called its prolongation. We will define this concept only for linear DOs which is the only case that we will use in this paper). Let D = D J k LDiff k ξ, η) be a linear DO, and D Hom J k ξ), η) be its total symbol. The l-th prolongation of D is a linear DO P l D) LDiff k+l ξ, J l η)) is such that the following diagram commutes: C J k+l ξ)) P l D) C J l η)) J k+l D P l D) J l 2.16) C ξ) C η) The total symbol, P l D) Hom J k+l ξ), J l η) ), of P l D) is the only linear fiber-preserving mapping such that P l D) = P l D) J k+l in the notations of 2.10)). For D LDiff k ξ, η), we set R k,l := ker P l D), 2.17) which in general is a family of linear subspaces of the vector bundle J k+l ξ). A linear DO D LDiff k ξ, η) is said to be formally integrable if for each l 0 the following conditions are satisfied: a) regularity, constancy of rank ) R k,l is a vector subbundle of J k+l ξ); b) existence of formal solutions ) the natural projection π k+l+1,k+l : R k,l+1 R k,l is an epimorphism i.e., a surjective linear mapping). For a formally integrable DO D LDiff k ξ, η), the subbundle R k,0 J k ξ) is called its equation. Remark 2.1. Clearly, if a DO is formally integrable, all its prolongations are also formally integrable DOs.

9 Dimensional Reduction: II 9 The formal integrability is crucially important for the dimensional reduction of invariant DOs considered in Sect. 4. Condition b) from the definition of formal integrability deserves some discussion. Let D LDiff k ξ, η) and ψ C ξ). By differentiating the equation Dψ = 0, we obtain the equation P l D)ψ = 0 of order k +l). In general, it may happen that the equation P l D)ψ = 0 contains some condition only on derivatives of order k +l 1) that was not contained in the prolongation P l 1 D)ψ = 0. If this is the case, then one can never be sure that all the conditions on the partial derivatives of some order are obtained by any finite number of prolongations. Condition b) of the definition of formal integrability guarantees that this cannot happen for a formally integrable DO. The example below clarifies this important point. Example. Here is an example of a DO that is not formally integrable, suggested in the classic work of Janet [5, pp ] and discussed in [6, Introduction] and [7, pp ]. Let ξ and η be globally trivial vector bundles over the manifold B = {x 1, x 2, x 3 ) : x 2 > 1} R 3 with standard fibers R and R 2, respectively. The condition x 2 > 1 is not essential for this example, but it will be convenient in the example in Sect , where the results of this example) are used.) Let the DO M LDiff 2 ξ, η) be defined 11 x as M := Then one can easily check that π 3,2 : R 2,1 R 2,0 22 is an epimorphism, i.e., that the equation P 1 M)ψ) = 0 does not impose any additional conditions on the first and second derivatives that were not present in the equation Mψ = 0. The second prolongation, however, contains the condition 233 ψ = 0, which was not present in R 2,1 to see this, differentiate the first component of Mψ = 0 twice with respect to x 2, then note that the second component of Mψ = 0 i.e., the equation 22 ψ = 0) implies that 1122 ψ = 0 and 2233 ψ = 0. In formal language, this means that the projection π 4,3 : R 2,2 R 2,1 is not an epimorphism. The differential equation Mψ = 0 is not difficult to solve explicitly, and its solution can be shown to be contain only 12 arbitrary constants, namely, ψx 1, x 2, x 3 ) = α 1 x 1 [ x 1 ) 2 x 2 + x 3 ) 2] x 3 + α 2 [ 3x 1 ) 2 x 2 + x 3 ) 2] x 3 + α 3 x 1 x 2 x 3 + α 4 x 1 [ x 1 ) 2 x 2 + 3x 3 ) 2] + α 5 [ x 1 ) 2 x 2 + x 3 ) 2] + α 6 x 1 x 2 + α 7 x 1 x 3 + α 8 x 2 x 3 + α 9 x 1 + α 10 x 2 + α 11 x 3 + α 12. This happens because there are infinitely many conditions that the higher prolongations of Mψ = 0 imply on the lower-order derivatives like the condition on 112 ψ given above), which results in a general solution depending only on a finite number of parameters instead of depending on arbitrary functions Involutivity and formal integrability The concept of involutivity of a collection of vector fields on a manifold is closely related to the formal integrability of the differential operator defined

10 10 Petko A. Nikolov and Nikola P. Petrov as the set of all these vector fields. To clarify this, we start with two motivating examples. Let ξ and η be globally trivial vector bundles over the manifold B = {x 1, x 2, x 3 ) : x 1 > 0} R 3 with standard fibers ) R and R 2, respectively. X Let X and Y be vector fields on B, and M := LDiff Y 1 ξ, η). Consider the following two cases: If X = 1 and Y = x , then one can easily check that the general solution of the equation Mψ = 0 i.e., of the system 1 ψ = 0, x )ψ = 0) is ψx 1, x 2, x 3 ) = gx 2 x 3 ), where g is an arbitrary differentiable function of one variable. Note that in this case [X, Y ] = X span {X, Y }. If X = 1, Y = 2 + x 1 3, then the general solution of Mψ = 0 is ψx 1, x 2, x 3 ) = const. Note that in this case [X, Y ] = 3 / span {X, Y }. The reason for the fact that in the latter case the general solution had smaller amount of arbitrariness only one arbitrary constant instead of one arbitrary function of one variable) is due to the fact that the collection of vector fields {X, Y } was not involutive. The involutivity of a set of vector fields is closely related to the formal integrability of the differential operator these vector fields define. We address this connection in the theorem below. Theorem 2.2. Let B be a real n-dimensional manifold, and x µ ) be some local coordinates. Let ξ = B K, π 1, B), η = B K d, π 1, B) be vector bundles over B, where K is some field. Let X a a = 1,..., d) be vector fields on B that are linearly independent at each point, and the DO D LDiff 1 ξ, η) be defined as Dψ)b) = X 1 b) µ µ ψb). X d b) µ µ ψb), ψ C ξ). 2.18) Then D is formally integrable if and only if the collection of vector fields {X a } d a=1 is involutive. Proof. The local coordinates x µ, z) in ξ and x µ, w a ) in η generate local jet bundle coordinates x µ, z, z µ ) in J 1 ξ), x µ, z, z µ, z µν ) in J 2 ξ), and x µ, w a, w a µ) in J 1 η) recall 2.1)). Let b = x µ ) B be an arbitrary point. The total symbols of D and its first prolongation are respectively D : J 1 ξ) η : x µ, z, z µ ) x µ, w a ) with w a = X a b) µ z µ, and P 1 D) : J 2 ξ) J 1 η) : x µ, z, z µ, z µν ) x µ, w a, w a ν) with w a = X a b) µ z µ, wν a = ν X a b) µ z µ + X a b) µ z µν. Let R k,l b be the fiber over b B of R k,l J k+l ξ) defined in 2.17)), and π k+l,k : J k+l ξ) J k ξ)

11 Dimensional Reduction: II 11 be the canonical projections 2.5). In these notations, x µ, z, z µ ) R 1,0 b X a b) µ z µ = ) { x µ, z, z µ, z µν ) R 1,1 Xa b) b µ z µ = 0 ν X a b) µ z µ + X a b) µ 2.20) z µν = 0 it is understood that the conditions in the right-hand side hold for all a = 1,..., d and ν = 1,..., n). Multiplying ν X a b) µ z µ + X a b) µ z µν = 0 by X c b) ν, and ν X c b) µ z µ + X c b) µ z µν = 0 by X a b) ν yields the system X c b) ν ν X a b) µ z µ + X c b) ν X a b) µ z µν = 0 X a b) ν ν X c b) µ z µ + X a b) ν X c b) µ z µν = 0 from which, by subtracting the first equation from the second one, we obtain [X a, X c ] b) µ z µ = 0, where [X a, X c ] b) µ is the µth component of the commutator of X a and X c at b. This implies that { x µ, z, z µ ) π 2,1 R 1,1 Xa b) b ) = µ z µ = 0 [X a, X c ] b) µ 2.21) z µ = 0 c = 1,..., d. Assume that the DO D 2.18) is formally integrable. Then π 2,1 : R 1,1 R 1,0 is an epimorphism, so from 2.19) and 2.21) we obtain that X a b) µ z µ = 0 implies [X a, X c ] b) µ z µ = 0 for any c = 1,..., d. Since this holds for every point b B, the formal integrability of D implies the involutivity of the collection of vector fields {X a } d a=1. Conversely, assume that {X a } d a=1 is involutive. Then, by the Frobenius Theorem see, e.g., [7, Sect. 2.4]), it is possible to choose local coordinates x µ ) in B such that { span {X 1,, X d } = span x n d+1,..., } x n In these coordinates, the equation Dψ = 0 where D is defined in 2.18) is equivalent to the system ψ x n d+1 = 0. ψ x n = 0, hence the equations determining R 1,l become R 1,0 = {z β = 0} ; R 1,1 = {z β = 0, z βµ = 0} ; R 1,2 = {z β = 0, z βµ = 0, z βµν = 0} ; R 1,3 = {z β = 0, z βµ = 0, z βµν = 0, z βµνρ = 0} ;.

12 12 Petko A. Nikolov and Nikola P. Petrov etc.; here µ, ν, ρ,... take values from 1 to d, while β = n d + 1,..., n. This makes it obvious that all projections π l+2,l+1 : R 1,l+1 R 1,l, l = 0, 1, 2,... are epimorphisms, i.e., D is formally integrable. The interested reader can find more about the relation between involutivity and formal integrability in [7, Ch. 4] and [8, Chapters IX and X]. 3. Restriction of a DO to a submanifold Let ξ and η be vector bundles over the manifold B. Let C B be a submanifold of B and i : C B be the natural embedding. We will denote by ξ C or, equivalently, by ξ C, the bundle i ξ induced by i. In other words, ξ C ξ C = E, π, C) where E := π 1 C), π is the restriction of π to E, and the fiber ξ C ) c = π ) 1 c) of ξ C over a point c C is the same as ξ c = π 1 c). In this section we will discuss the problems that occur in attempting to restrict a DO D Diff k ξ, η) to a DO D C D C Diff k ξ C, η C ) in general, k may not be equal to k). These problems are central in the process of dimensional reduction of invariant DOs which is considered in Sect. 4, so below we consider them in detail. In what follows we adopt the following definition. We say that two linear subspaces, L 1 and L 2, of the the linear space L are transversal if: they are complementary in the sense of linear algebra, i.e., each v L can be decomposed as v = v 1 + v 2 with v 1 L 1, v 2 L 2 ; they intersect trivially: L 1 L 2 = {0}. If L 1 and L 2 are transversal, we write L = L 1 L 2. We say that two submanifolds B 1 and B 2 of the manifold B intersect transversely at b B 1 B 2 if T b B = T b B 1 T b B 2. We say that two subbundles ξ 1 and ξ 2 of the vector bundle ξ are transversal and write ξ = ξ 1 ξ 2 if ξ b = ξ 1 ) b ξ 2 ) b for each b in the base of ξ Set-up and notations In the construction of the reduced bundle from [1, Sect. 2.3], the base B/G of the reduced bundle ξ G is glued by a family of submanifolds {Ũα} α A recall [1, Eqn. 2.17)]). Each Ũα is transversal to the orbits of action t of the Lie group G on the base B, so that, for any α A and any b Ũα, if O b is the orbit of the point b under the action t, then T b B = T b Ũ α T b O b. Let ξ and η be vector bundles over B, and D Diff k ξ, η). Since the coordinate realizations of the reduced bundles ξ G and η G are glued from vector bundles over the submanifolds Ũα of B, it is natural to ask how one can restrict a DO D Diff k ξ, η) to a submanifold of B. Let Ũ be a submanifold of the common base B of ξ and η, and ξ Ũ ξ Ũ and ηũ be the corresponding restrictions. If D Diff k ξ, η), then in general the natural embedding i : Ũ B does not provide us with enough information in order to construct a restricted DO DŨ. Indeed, let dim Ũ = ñ,

13 Dimensional Reduction: II 13 and assume that the local coordinates x 1,..., x n ) in B are adapted to Ũ in the sense that Ũ = {xñ+1 = = x n = 0} ; 3.1) in these coordinates, the natural embedding i : Ũ B is given by i x 1,..., xñ) ) = x 1,..., xñ, 0,..., 0). We will call x 1,..., xñ) internal for Ũ and xñ+1,..., x n ) external for Ũ coordinates. Let ψ C ξ) and ψũ ψ Ũ := ψ i C ξũ ) 3.2) be the restriction of its domain to Ũ. After we restrict the domain of ψ to Ũ, we lose all information about the dependence of ψ on the external for Ũ coordinates, hence Dψ Ũ ) cannot be computed if D contains partial derivatives with respect to the external coordinates. Because of the bijective correspondence 2.12) between the DOs from Diff k ξ, η) and their total symbols, we will concentrate to the problem of restricting the domain of a jet of a section ψ C ξ). Taking the jet of a section does not commute with restricting the domain of the section. If we first find the k-jet J k ψ) of the section ψ C ξ), and then restrict J k ψ) to Ũ, we obtain J k ψ)ũ J k ψ) Ũ = J k ψ) i, which is a section of J k ξ)ũ. For any b Ũ, J k ψ)ũ b) which is the same as J k ψ)b)) contains derivatives with respect to all coordinates x 1,..., x n, namely J k ψ)ũ b) ) { a 1 µ1 µ µ 1 µ i = µ1 µ i ψ a i n, i = 1,..., k b), a = 1,..., dim ξ in the notations introduced in 2.2) and 2.3)). On the other hand, if we first restrict ψ to Ũ and after that compute the kth jet of the restriction ψũ 3.2), the result, J k ψ Ũ Ũ ), is a section of J k ξ Ũ Ũ ). Here we introduced the notations J k ξ Ũ Ũ ) and J k LDiff Ũ kξũ, J k ξũ )), which simply mean that we work with the sections of the restricted bundle ξũ as in 3.2)). For any b Ũ, J k ψ Ũ Ũ )b) contains only derivatives with respect to the internal for Ũ coordinates x1,..., xñ: J ψũ )b) kũ ) { a 1 µ1 µ µ 1 µ i = µ1 µ i ψ a i ñ, i = 1,..., k b), a = 1,..., dim ξ. 3.3) The dimensions of the fibers of) J k ξ)ũ and J k ξ Ũ Ũ ) are ) ñ ) n + k + k dim J k ξ)ũ = dim ξ, dim J k k ξ Ũ Ũ ) = dim ξ. 3.4) k Denote by j k : J k ξ)ũ J k Ũ ξ Ũ ) 3.5)

14 14 Petko A. Nikolov and Nikola P. Petrov the natural projection given by cutting off all non-internal for Ũ derivatives, i.e., derivatives containing at least one external for Ũ partial derivative. In a coordinate-free language, for any b Ũ, the map jk is given by As usual, let j k ) J k ψ)ũ b) = J k ψ Ũ Ũ )b). 3.6) j k j k ) : C J k ξ)ũ ) C J k Ũ ξ Ũ )) : J k ψ)ũ J k Ũ ψ Ũ ) 3.7) be the map between the sections that is induced by j k as in 2.10)) Internal for Ũ DOs and their restriction to Ũ There is a situation in which the restriction of a DO D Diff k ξ, η) to Ũ is naturally and uniquely determined by the embedding i : Ũ B. This happens when, for an arbitrary ψ C ξ), the section Dψ C η) evaluated at the points of Ũ which, in formal notation, is Dψ) Ũ Dψ) i) contains only differentiations with respect to internal for Ũ coordinates. In this case the DO D is said to be internal for Ũ). Let D = D J k Diff k ξ, η) be internal for Ũ, and b Ũ. Then Dψ)b) does not contain any non-internal for Ũ derivatives, i.e., derivatives J k ψ)ũ b) ) a µ 1 µ i = µ1 µ i ψ a b) for which at least one of the indices µ 1,..., µ i exceeds ñ. Clearly, in this case Dψ)b) can be expressed only in terms of the coordinates J k ψ Ũ Ũ )b)) a µ 1 µ i of the k-jet J k ψ Ũ Ũ )b) of the restricted section ψũ see 3.3)). Therefore, for an internal for Ũ DO D, we can define the restricted to Ũ DO D Ũ Diff kξũ, ηũ ) as follows: given a section ρ C ξũ ), we can think of it as a restriction ψũ = ψ i of a section ψ C ξ) to Ũ, and then set DŨ ρ)b) := Dψ)b), b Ũ. 3.8) Since D is internal for Ũ, Dψ)b) does not contain non-internal for Ũ derivatives, so that the arbitrariness in the choice of ψ C ξ) such that ψũ = ρ is immaterial. Since differentiation is a local operation, the section ψ does not need to be defined on B, but only on an open subset of B that contains Ũ. Clearly, if D is internal for Ũ, then the order of the restricted to Ũ DO D Ũ is the same as the order of D. In terms of the map j k from 3.5) and 3.6), for an internal for Ũ DO D we can naturally define the total symbol DŨ of the restricted DO DŨ by D Ũ = DŨ j k, 3.9)

15 Dimensional Reduction: II 15 where D Ũ : J k ξ)ũ ηũ is the restriction of the total symbol D : J k ξ) η of D to the submanifold Ũ. In other words, DŨ is defined so that the diagram C J k ξ)ũ ) D Ũ) C ηũ ) j k C J k Ũ ξ Ũ )) DŨ) 3.10) be commutative. To show that the definitions 3.8) and 3.9) are consistent for an internal for Ũ DO D Diff kξ, η), we derive 3.8) from the definition 3.6), 3.7) of the map j k and the definition 3.9) of DŨ : if ρ C ξũ ) and ψ C ξ) is such that ρ = ψũ, then DŨ ρ = DŨ ) J k ψ Ũ Ũ ) = DŨ ) jk J k ) ψ)ũ = DŨ j k) J k ) ) ψ)ũ = D Ũ J k ) ψ)ũ = D J k ψ) ) Ũ = Dψ) Ũ. 3.11) 3.3. Restriction of a non-internal DO Natural geometric objects in the problem. To restrict to Ũ a DO that is not internal for Ũ, one needs information that does not come from the natural embedding i : Ũ B. A natural geometric object that plays a crucial role is the subbundle I k of J k ξ)ũ consisting of the k-jets of all vanishing on Ũ Ũ sections of ξ, i.e., whose fiber over an arbitrary point b Ũ is Let I k Ũ ) b := { J k ψ)b) : ψ C ξ) s.t. ψ i 0 } J k ξ)ũ ) b. 3.12) i k : I k Ũ J k ξ)ũ 3.13) stand for the natural embedding; to simplify the notations, we will write I k Ũ instead of i k I k Ũ ). Let x 1,..., xñ) and xñ+1,..., x n ) be respectively the internal and the external for Ũ local coordinates in B recall 3.1)). For the partial derivatives of a section of ξ, we recall the terminology used in Sect. 3.1 and 3.2: internal derivatives are those that contain only differentiations with respect to the internal coordinates; all other derivatives are non-internal; by definition, the zeroth derivative i.e., the section itself) is internal. In jet bundle coordinates 2.1) in J k ξ)ũ, the internal jet bundle coordinates in J k ξ)ũ are z a and those zµ a 1 µ i for which µ 1 ñ,..., µ i ñ, while the non-internal ones are zµ a 1 µ i for which at least one of the µ s is strictly greater than ñ. According to 3.4), the number of internal coordinates in J k ξ)ũ is ) ñ+k k dim ξ, while number of the non-internal ones is [ ) ñ+k )] n+k k k dim ξ. To simplify the notations, we temporarily write int for the set of all internal coordinates in J k ξ)ũ, and non-int for the set of all non-internal ones. Then I k consists of those elements of J k ξ)ũ all internal jet bundle Ũ

16 16 Petko A. Nikolov and Nikola P. Petrov coordinates of which are zero, while the non-internal ones are arbitrary; symbolically this can be written as I k = {int = 0, non-int)}. The natural projection j k 3.5) maps the element int, non-int) J k ξ)ũ to int) J k ξ Ũ Ũ ), Ũ preserving the values of all the internal jet bundle coordinates i.e., j k simply cuts off all non-internal coordinates). Obviously, j k i k = 0, so that we obtain the following short exact sequence: 0 {int = 0, non-int)} ik {int, non-int)} jk {int)} ) In coordinate-free notations, the short exact sequence 3.14) can be written as the horizontal short exact sequence in the diagram ik 0 I k J k ξ)ũ Ũ Π k j k Σ k J k Ũ ξ Ũ ) 0, D Ũ DŨ ηũ 3.15) in which we have also shown the maps from 3.9), as well as the maps Π k and Σ k which will be discussed below Geometry of the restriction of a non-internal DO. In the geometric language introduced above, the gist of the problem of restricting a DO D Diff k ξ, η) to a submanifold Ũ of the base B is that while J k ξ)ũ is isomorphic to the direct sum I k J k ξ Ũ Ũ Ũ ), the bundle J k ξ Ũ Ũ ) is not naturally embedded in J k ξ)ũ. One way to define the total symbol DŨ of the restricted DO DŨ is to choose a splitting of the short exact sequence in 3.15), i.e., a vector bundle morphism Σ k Hom J k ξ Ũ Ũ ), J k ξ)ũ ) over the identity in Ũ that satisfies j k Σ k = Id J ξũ ). 3.16) kũ Equivalently, we can choose a vector bundle morphism Π k Hom J k ξ)ũ, I k Ũ ) over the identity in Ũ such that Π k i k = Id I kũ Then the total symbol of DŨ is given by, ker Π k = Σ k J k ξ Ũ Ũ )). 3.17) DŨ = D Ũ Σ k, 3.18) as shown in the diagram 3.15). Conditions 3.16) and 3.17) imposed on Σ k and Π k guarantee that if the DO D is internal for Ũ, then D Ũ defined by 3.18) is the same as its natural restriction to Ũ discussed in Sect. 3.2). We required that the maps Σ k and Π k be vector bundle morphisms for two reasons. Firstly, this is the case that occurs in dimensional reduction of invariant DOs considered in Sect. 4. Furthermore, if they are morphisms, the restriction DŨ of a linear DO D will be linear as well. In Remark 3.2 we

17 Dimensional Reduction: II 17 consider briefly the more general case when Σ k is a nonlinear fiber-preserving map. Conditions 3.16) and 3.17) imply that the maps Σ k and Π k are completely defined by Σ k J k ξ Ũ Ũ )), which is a subbundle of J k ξ)ũ transversal to I k in J k ξ)ũ : Ũ J k ξ)ũ = I k Ũ Σk J k ξ Ũ Ũ )). 3.19) Let us take an arbitrary point b meaning of the splitting Ũ and consider in practical terms the J k ξ) b = I k Ũ ) b Σ k J k Ũ ξ Ũ )) b of the fiber of J k ξ)ũ over b. In the notations int and non-int introduced above, I k Ũ ) b = {int = 0, non-int)} recall the short exact sequence 3.14)). According to 3.4), the dimensions of the subspaces in the splitting are [ ) ñ n + k + k dim I k ) Ũ b = k k ñ + k dim Σ k J k ξ Ũ Ũ )) b = k ) dim ξ = #{int}. )] dim ξ = #{non-int}, To define a subspace Σ k J k Ũ ξ Ũ )) b of dimension #{int} in the linear space dim J k ξ) b of dimension dim J k ξ) b = #{non-int} + #{int}, we can write a system of #{non-int} independent linear equations with #{non-int} + #{int}) unknowns. To make this explicit, we denote by Z I, for I = 1,..., #{int}, the set of all internal for Ũ coordintes in J k ξ) b, and by Z N, for N = #{int} + 1,..., #{int} + #{non-int}, the set of all non-internal coordinates in J k ξ) b. In coordinates Z I, Z N ), the maps Σ k and Π k have the form ) ) ) Σ k ZI : Z I ), Π k ZI 0 :. Mb)Z I Z N Mb)Z I + Z N 3.20) The matrix Mb) is of size #{non-int} #{int}. This makes it clear that the set of all splittings of the horizontal short exact sequence in 3.15) is an affine space with a linear group Hom J k ξ Ũ Ũ ), ) Ik Ũ.

18 18 Petko A. Nikolov and Nikola P. Petrov The process of restriction of a DO D Diff k ξ, η) to a submanifold Ũ of the base B is illustrated well by the diagram 3.21). 0 C ξ) D C η) J k C J k ξ)) C I k ) Ũ D Ũ Ũ C ξũ ) DŨ C ηũ ) J k Ũ D Ũ Ũ ) DŨ ) Π k i k C J k ξ)ũ ) Σ k j k C J k Ũ ξ Ũ )) ) The triangle in the upper left corner of 3.21) represents the relation 2.12) between the DO D and its total symbol D. The dashed arrows with label Ũ are the restrictions of the domains of the sections of ξ, J k ξ), and η to Ũ, as in 3.2). The triangle involving C ξũ ), C J k ξ Ũ Ũ )), and C ηũ ) represents the relation DŨ = DŨ ) between the desired restricted DO J k Ũ DŨ and its total symbol DŨ. Note that the vertical arrows between the five rightmost objects in the diagram 3.21) come from the horizontal short exact sequence in 3.15). The triangle involving C J k ξ)ũ ), C J k ξ Ũ Ũ )), and C ηũ ) clarifies the role of the splitting Σ k in the construction of the restricted DO DŨ namely, DŨ = DŨ ) J k = ) D Ũ Ũ Σk J k. 3.22) Ũ Defining the splitting of 3.15) through an auxiliary DO. Thanks to 3.16), 3.17), the splitting of the horizontal short exact sequence in 3.15) is defined completely by specifying a subbundle of J k ξ)ũ transversal to recall 3.19)). Such a subbundle can sometimes be defined through an I k Ũ auxiliary linear DO M LDiff k ξ, η). If M Ũ Hom J k ξ)ũ, ηũ ) is the restriction to Ũ of the total symbol of M, then it may turn out that its kernel, ker M Ũ ) J k ξ)ũ, is a subbundle transversal to I k in J k ξ)ũ. In Ũ

19 Dimensional Reduction: II 19 this case we can set Σ k J k Ũ ξ Ũ )) ker Πk := ker M Ũ ). 3.23) In the light of the discussion before 3.21), equation 3.23) means that we can express all non-internal jet bundle coordinates in terms of the internal jet bundle coordinates from the equation Mψ)b) = 0, where b is an arbitrary point in Ũ. Having expressed all non-internal derivatives of ψ through the internal derivatives of ψ, we substitute them in Dψ) Ũ, and the result is DŨ ψũ an expression that contains only internal for Ũ derivatives of the restricted section ψũ C ξũ ). From DŨ ψũ we can read off the desired restricted DO DŨ Diff k ξũ, ηũ ). To define the splitting 3.19), one can sometimes use a linear DO of order lower that k. Let m < k, and M LDiff m ξ, η). Then the k m)th prolongation P k m M) of M is a DO of order k, and one can hope that the restriction to Ũ of its total symbol P k m M) can be used to define Σ k J k ξ Ũ Ũ )), similarly to 3.23): Σ k J k ξ Ũ Ũ )) ker Πk := ker[ P k m M) ) ] Ũ. 3.24) While, in principle, it is possible to use an auxiliary linear DO M to define the splitting as in 3.23) or 3.24), finding such a DO may not be easy for several reasons. First of all, on what ground will one use certain DO M and not another one? In general, one can use some DO M if this would guarantee that certain properties are preserved. A fundamental example of this is the process of dimensional reduction of a DO invariant with respect to the action of a Lie group, in which case the needed auxiliary DO is the Lie derivative see Sect. 4. A second problem is how to choose M so that the right-hand side of 3.23) or 3.24) indeed defines a subspace Σ k J k Ũ ξ Ũ )) transversal to Ik Ũ in J k ξ)ũ as in 3.19). Taking care of this transversality requirement is highly non-trivial. For an example of finding such an auxiliary DO see Sect. 6 of our paper [2]. Last but not least, the auxiliary DO must be formally integrable otherwise, unexpected complications may occur, as illustrated in the example below. Example. This example shows the dangers of using an auxiliary DO M that is not formally integrable. We use the same notations ) as in the Example in 11 x Sect. 2.3 for B, ξ, η, and the DO M := 2 33 LDiff 2 ξ, η). ) Let D := Diff 0 3 ξ, η), Ũ = {x3 = 1} B. Assume that we want to restrict D to Ũ by using a splitting of the horizontal short exact sequence in 3.15) that comes from M as an auxiliary DO.

20 the identity in Ũ satisfying j k Σ k = Id J kξũ ), 3.25) 20 Petko A. Nikolov and Nikola P. Petrov The kernel of the first prolongation P 1 M) restricted to Ũ in other words, the solution of the equation P 1 M)ψ) Ũ = 0 contains the equations 11 ψ x 2 33 ψ = 0 and 112 ψ x ψ 33 ψ = 0, from which we can express the non-internal derivative in D as 233 ψ = 1 x 2 112ψ 33 ψ) = 1 x 2 112ψ 1 x 2 ) 2 11ψ. Therefore, if one uses ker P 1 M) as Σ 3 J 3 ξ Ũ Ũ )) in 3.23) to define the splitting, then the restriction of the DO D to Ũ is D Ũ = x ) x 2 ) If ψũ x 1, x 2 ) := ψx 1, x 2, 1) is the restricted to Ũ section, then it is easy to show that the general solution ψũ of the restricted equation 1 DŨ ψũ )x 1, x 2 ) = x ψũ x 1, x 2 ) 1 x 2 ) 2 11 ψũ x 1, x 2 ) ) ) 0 = 0 0 is ψũ x 1, x 2 ) = h 1 x 1 )x 2 + x 1 h 2 x 2 ) + h 3 x 2 ), where h 1, h 2, and h 3 are arbitrary smooth functions of one variable. We can, however, treat D as a 4th-order DO and use Σ 4 J 4 Ũ ξ Ũ )) = ker P 2 M) J 4 ξ)ũ to define the splitting, i.e., to express the non-internal derivatives in D from the equation P 2 M)ψ) Ũ = 0. As we showed in the example in Sect. 2.3, the second prolongation of the equation Mψ = 0 contains the condition 233 ψ = 0; clearly, this condition remains unchanged after restricting it to Ũ. Therefore the restricted DO for this choice of splitting 0 becomes DŨ =, so that the general solution of the reduced equation 0) DŨ ψũ x 1, x 2 ) = 0 consists of all smooth functions of two variables. Remark 3.1. We note that, even if the auxiliary DO M provides a splitting of the horizontal short exact sequence in the diagram 3.15) and is formally integrable, the order of the restricted DO DŨ may be greater than the order of the original DO D. For a concrete example of this phenomenon we refer the reader to Sect. 6 of our paper [2]. Remark 3.2. One can define the total symbol DŨ of the restricted DO DŨ by choosing a nonlinear fiber-preserving mapping Σ k : J k Ũ ξ Ũ ) J k ξ)ũ over and defining the total symbol of DŨ by DŨ = D Ũ Σ k 3.26) cf. 3.22)). As before, the condition 3.25) imposed on Σ k guarantees that for an internal for Ũ DO its restriction defined by 3.26) is the same as its natural restriction to Ũ. Since our main goal is the reduction of invariant

21 Dimensional Reduction: II 21 DOs Sect. 4), in which case the splitting Σ k is a linear map, we will not consider the case of nonlinear splittings in detail. 4. Dimensional reduction of invariant DOs 4.1. Group actions and Lie derivatives We start this section by defining the concept of derivative of a Lie group action on a vector bundle, which plays a key role in the process of dimensional reduction of DOs invariant with respect to the action of a Lie group. For brevity, we will call it a Lie derivative of the action. Recall the notations of Sect. 2.2 of [1]: let G be a connected Lie group acting on the vector bundle ξ = E, π, B) by vector bundle morphisms T = t, T ), where t and T are the actions of G on the base B and total space E, respectively. Assume that the action of G on ξ is such that ξ is a reducible G-vector bundle recall [1, Definition 2.8]). Let x = x µ ) stand for some local coordinates in the base B; we will often identify a point b B with its coordinates x. Assume that a basis e a x)) has been chosen in each fiber ξ x = π 1 x), and let z = z a ) be the coordinates in the fibers in this basis. If the action T preserves the fibers of ξ, then its general form is T g x µ, z a ) = tg x) µ, T g x, z) a) for g G. If, in addition, the action is through vector bundle morphisms i.e., if T is linear in the fibers), then the general form of T is T g x µ, z a ) = t g x) µ, T g x) a c z c), g G. 4.1) By assumption, we consider only actions T of the form 4.1). The action T of G on ξ defines an action of G on C ξ): for g G and ψ C ξ), the transformed section gψ) C ξ) is defined by In the local coordinates x µ, z a ), gψ) := T g ψ t 1 g. 4.2) gψ) a b) = T g t 1 g b)) a c ψ c t 1 g b)). 4.3) Let g be the Lie algebra of G with generators λ i, and e sλ with s in an open interval in R containing 0) be the local 1-parameter subgroup of G generated by λ g. Let g ξ be a vector bundle whose sections are of the form Λ ψ, where Λ g is an element of g independent on the point in the base, and ψ C ξ). In other words, for any b B, Λ ψ)b) = Λ ψb) takes an element λ g and produces Λ, λ ψb) ξ b where, is the natural pairing between g and g). To emphasize this peculiarity of g ξ, for the space of its sections we will use the notation g C ξ) instead of C g ξ). The Lie derivative of the action T of G on ξ is a linear first-order DO L LDiff 1 ξ, g ξ) defined by Lψ)λ) := d ds esλ ψ) C ξ), ψ C ξ), λ g 4.4) s=0

22 22 Petko A. Nikolov and Nikola P. Petrov where we used the notation 4.2)). In local coordinates, if b = x µ ), and λ i is a generator of g, then Lψ)λ i ) a b) = d ds T ) a expsλ i) texp sλi)b) c ψ c t ) exp sλi)b) s=0 =: X i b) µ µ ψ a b) + Z i b) a c ψ c b). Here X i = X i µ µ C τb)) with 4.5) X i b) µ := d ds t expsλ i)b) µ s=0 4.6) are the fundamental vector fields of the action t of G on B, and Z i b) a c := d ds T ) a expsλ i) texp sλi)b) c. 4.7) s=0 Remark 4.1. The stationary subbundle st ξ ξ recall [1, Definition 2.7]) is invariant under the action of G, so the Lie derivative can also be considered as a DO in LDiff 1 st ξ, g st ξ) Invariant DOs and their reduction Lie group action on a DO; reduced DO. Let ξ and η be reducible G- vector bundles over B with the same action t of G on the common base B, and let T ξ = t, T ξ ) and T η = t, T η ) be the actions of G on the corresponding bundles. The actions of g G on C ξ) and C η), define an action of g on Diff k ξ, η): g ξ : ψ g ξ ψ) := T ξ g ψ t 1 g, g η : χ g η χ) := T η g χ t 1 g, 4.8) g : Diff k ξ, η) Diff k ξ, η) : D gd) := g η D g ξ ) ) We say that a DO D Diff k ξ, η) is G-invariant if gd) = D, i.e., g η D = D g ξ, for all g G. 4.10) Let Diff k ξ, η) G stand for the set of all G-invariant order-k DOs from ξ to η. If ψ C ξ) G is a G-invariant section of ξ and D Diff k ξ, η) G, then Dψ is a G-invariant section of η indeed, 4.10) yields g η Dψ) = Dg ξ ψ)) = Dψ. Recall that in [1, Sect. 2.3] we gave an explicit construction of a reduced vector bundle ξ G. This bundle is such that the set C ξ G ) of all its sections is in a bijective correspondence with the set C ξ) G of all G-invariant sections of ξ, as in [1, Eqn. 2.16)]. Moreover, this bijection is a homomorphism from the C B/G)-module C ξ G ) to the C B) G -module C ξ) G cf. [1, Remark 2.10]). Let θ ξ : C ξ G ) C ξ) G, θ η : C η G ) C η) G 4.11) be the two bijections for ξ and η, respectively. Then each G-invariant DO D Diff k ξ, η) G maps C ξ) G to C η) G and, therefore, generates a reduced DO D G := θ η ) 1 D θ ξ : C ξ G ) C η G ) 4.12)