Symmetry, Integrability and Geometry: Methods and Applications SIGMA * 200*), ***, 17 pages On Tanaa s prolongation procedure for filtered structures of constant type Igor Zeleno Department of Mathematics, Texas A&M University, College Station, TX 77843-3368 E-mail: zeleno@math.tamu.edu Abstract. We present the Tanaa prolongation procedure for filtered structures on manifolds discovered in [6] in a spirit of Singer-Sternberg s description of the prolongation of usual G-structures [4], [5]. It gives a transparent point of view on the Tanaa constructions avoiding many technicalities of the original Tanaa paper. Key words: G-structures; filtered structures; generalized Spencer operator; prolongations. 2000 Mathematics Subject Classification: 58A30, 58A17 1 Introduction This note is based on series of lectures given by the author on the Woring Geometry Seminar at Department of Mathematics of Texas A&M University in Spring 2009 and devoted to prolongation procedure for filtered structures on manifolds discovered by Noboru Tanaa in the paper [6] published in 1970. The Tanaa prolongation procedure is an ingenious refinement of the Cartan method of equivalence. It provides an effective algorithm for the construction of canonical frames for filtered structures and for the calculation of the sharp upper bound for the dimension of their algebras of infinitesimal symmetries. This note is by no means a complete survey of the Tanaa theory. For such survey we refer the reader to [2]. Our goal here is to describe geometric aspects of the Tanaa prolongation procedure using the language similar to one used by Singer and Sternberg in [4] and [5] for description of the prolongation of the usual G-structures. We found that it gives a quite natural and transparent point of view on the Tanaa constructions avoiding many formal definitions and technicalities of the original Tanaa paper. We believe this point of view will be useful to anyone who is interested in studying both the main ideas and the details of this fundamental Tanaa construction. We hope that the material of sections 3 and 4 will be of interest for experts as well. Our language also allows to generalize the Tanaa procedure in several directions, namely for filtered structures with non-constant and non-fundamental symbols. These generalizations with applications to the local geometry of distributions will be given in a separate paper. 1.1 Statement of the problem Let D be a ran l distribution on a manifold M, i.e. a ran l subbundle of the tangent bundle T M. Two vector distributions D 1 and D 2 are called equivalent, if there exists a diffeomorphism F : M M such that F D 1 x) = D 2 F x)) for any x M. Two germs of vector distributions D 1 and D 2 at the point x 0 M are called equivalent, if there exist neighborhoods U and Ũ of x 0 and a diffeomorphism F : U Ũ such that F D 1 = D 2, F x 0 ) = x 0. The general question is: When two germs of distributions are equivalent?
2 I. Zeleno 1.2 Wea derived flags and symbols of distributions Taing Lie bracets of vector fields tangent to a distribution D i.e. sections of D) one can defined a filtration D 1 D 2...... of the tangent bundle, called a wea derived flag or a small flag of D). More precisely, set D = D 1 and define recursively D j = D j+1 + [D, D j+1 ], j > 1. If X 1,... X l are l vector fields constituting a local basis of a distribution D, i.e. D = span{x 1,..., X l } in some open set in M, then D j x) is the linear span of all iterated Lie bracets of this vector fields of length not greater than j evaluated at a point x. A distribution D is called bracet-generating or completely nonholonomic) if for any x there exists µx) N such that D µx) x) = T x M. The number µx) is called the degree of nonholonomy of D at a point x. A distribution D is called regular if for all j < 0, the dimensions of subspaces D j x) are independent of the point x. From now on we assume that D is regular bracet-generating distribution with degree of nonholonomy µ. Let g 1 x) def = D 1 x) and g j x) def = D j x)/d j+1 x) for j < 1. Consider the space mx) = 1 j= µ g j x) 1.1) which is the graded space corresponding to the filtration Dx) = D 1 x) D 2 x)... D µ+1 x) D µ x) = T x M. This space is endowed naturally with the structure of a graded nilpotent Lie algebra, generated by g 1 x). Indeed, let p j : D j x) g j x) be the canonical projection to a factor space.ae Y 1 g i x) and Y 2 g j x). To define the Lie bracet [Y 1, Y 2 ] tae a local section Ỹ1 of the distribution D i and a local section Ỹ2 of the distribution D j such that p i Ỹ1 x) ) = Y 1 and p j Ỹ2 x) ) = Y 2. It is clear that [Y 1, Y 2 ] g i+j x). Put [Y 1, Y 2 ] def = p i+j [ Ỹ 1, Ỹ2]x) ). 1.2) It is easy to see that the right-hand side of 1.2) does not depend on the choice of sections Ỹ1 and Ỹ2. Besides, g 1 x) generates the whole algebra mx). A graded Lie algebra satisfying the last property is called fundamental. The graded nilpotent Lie algebra mx) is called the symbol of the distribution D at the point x. Fix a fundamental graded nilpotent Lie algebra m = 1 i= µ g i. A distribution D is said to be of constant symbol m or of constant type m if for any x the symbol mx) is isomorphic to m as a nilpotent graded Lie algebra). In general this assumption is quite restrictive. For example, in the case of ran two distributions on manifolds with dim M 9, symbol algebras depend on continuous parameters, which implies, in particular, that generic ran 2 distributions in these dimensions do not have a constant symbol. For ran 3 distributions with dim D 2 = 6 the same situation happens for dim M = 7 as was shown in [1]. Following Tanaa, and for simplicity of presentation, we consider here distributions of constant type m only. One can construct the flat distribution D m of constant type m. For this let Mm) be the simply connected Lie group with the Lie algebra m and let e be its identity. Then D m is the left invariant distribution on Mm) such that D m e) = g 1. 1.3 The bundle P 0 m) and its reductions To a distribution of type m one can assign a principal bundle in the following way. Let G 0 m) be the group of automorphisms of the graded Lie algebra m, i.e., a group of all automorphisms A
On Tanaa s prolongation procedure 3 of the linear space m preserving the Lie bracets i.e. A[v, w]) = [Av), Aw)] for any v, w m) and the grading Ag i ) = g i for any i < 0). Let P 0 m) be the set of all pairs x, ϕ), where x M and ϕ : m mx) is an isomorphism of the graded Lie algebras m and mx). Obviously, P 0 m) is a principal G 0 m)-bundle over M. The right action R A of an automorphism A G 0 m) is as follows: R A sends x, ϕ) P 0 m) to x, ϕ A) or shortly x, ϕ) R A = x, ϕ A). Note that since g 1 generates the whole m the group G 0 m) can be identified with a subgroup of GLg 1 ). By the same reason a point x, ϕ) P 0 m) of a fiber of P 0 m) is uniquely defined by ϕ g 1 so that one can identify P 0 m) with the set of pairs x, ψ), where x M and ψ : g 1 Dx) can be extended to an automorphism of the graded Lie algebras m and mx). Speaing informally, P 0 m) can be seen as a G 0 m) reduction of the bundle of all frames of the distribution D. Besides, the Lie algebra g 0 m) is the algebra of all derivations a of m, preserving the grading i.e. ag i g i for all i < 0). Additional structures on distributions can be encoded by reductions of the bundle P 0 m). More precisely, let G 0 be a Lie subgroup of G 0 m) and let P 0 be a principal G 0 bundle, which is a reduction of the bundle P 0 m). Since g 0 is a subalgebra of the algebra of derivations of m preserving the grading, the subspace m g 0 is endowed with the natural structure of a graded Lie algebra. For this we only need to define bracets [f, v] for f g 0 and v m, because m and g 0 are already Lie algebras. Set [f, v] def = fv). The bundle P 0 is called a structure of constant type m, g 0 ). Let, as before, D m be the left invariant distribution on Mm) such that D m e) = g 1. Denote by L x the left translation on Mm) by an element x. Finally, let P 0 m, g 0 ) be the set of all pairs x, ϕ), where x Mm) and ϕ : m mx) is an isomorphism of the graded Lie algebras m and mx) such that L x 1) ϕ G 0. The bundle P 0 m, g 0 ) is called the flat structure of constant type m, g 0 ). Let us give two examples. Example 1. G-structures. Assume that D = T M. So, m = g 1, m is abelian, G 0 m) = GLm), and P 0 m) coincides with the bundle FM) of all frames on M. In this case P 0 is nothing but a usual G 0 -structure. Example 2. Sub-Riemannian structures of constant type see also [3]) Assume that each space Dx) is endowed with an Euclidean structure Q x depending smoothly on x. In this case one says that the pair D, Q) defines a sub-riemannian structure on a manifold M. Recall that g 1 x) = Dx). This motivates the following definition: A pair 1 m, Q), where m = g j is a fundamental graded Lie algebra and Q is an Euclidean structure on g 1, is called a sub- Riemannian symbol. Two sub-riemannian symbols m, Q) and m, Q) are isomorphic if there exists a map ϕ : m m, which is an isomorphism of the graded Lie algebras m and m, preserving the Euclidean structures Q and Q i.e. such that Q ϕv 1 ), ϕv 2 ) ) = Qv 1, v 2 ) for any v 1 and v 2 in g 1 ). Fix a sub-riemannian symbol m, Q). A sub-riemannian structure D, Q) is said to be of constant type m, Q), if for every x the sub-riemannian symbol mx), Q x ) is isomorphic to m, Q). Note that it may happen that a sub-riemannian structure does not have a constant symbol even if a distribution does. Such a situation occurs already in the case of the contact distribution in R 2n+1 for n > 1: its symbol m cont,n is isomorphic to the Heisenberg algebra η 2n+1. Obviously, a sew-symmetric form Ω is well defined on g 1, up to a multiplication by a nonzero constant. If in addition a Euclidean structure Q is given on g 1, then a sew-symmetric endomorphism J of g 1 is well defined, up to a multiplication by a nonzero constant, by Ωv 1, v 2 ) = QJv 1, v 2 ). Tae 0 < β 1... β n such that {±β 1 i,..., ±β n i} is the set of the eigenvalues of such a J. Then a sub-riemannian symbol with m = m cont,n is determined uniquely up to an isomorphism) by a point [β 1 : β 2 :... : β n ] of the projective space RP n 1. Let D, Q) be a sub-riemannian structure of constant type m, Q) and G 0 m, Q) G 0 m) be the group of automorphisms of a sub-riemannian symbol m, Q). Let P 0 m, Q) be the set j= µ
4 I. Zeleno of all pairs x, ϕ), where x M and ϕ : m mx) is an isomorhism of sub-riemannian symbols m, Q ) and mx), Q x ). Obviously, the bundle P 0 m, Q) is a reduction of P 0 m) with the structure group G 0 m, Q). Another important class of geometric structures that can be encoded in this way are CRstructures see 10 of [6] for more details). 1.4 Algebraic and geometric Tanaa prolongations In [6] Tanaa gives a solution of the equivalence problem for structures of constant type m, g 0 ). Two main Tanaa s constructions there are the algebraic prolongation of the algebra m + g 0 and the geometric prolongation of structures of type m, g 0 ), imitated by the algebraic prolongation. First he defines a graded Lie algebra, which is in essence the maximal nondegenerated) graded Lie algebra, containing the graded Lie algebra g i as its non-positive part. More precisely, Tanaa constructs a graded Lie algebra gm, g 0 ) = g i m, g 0 ), satisfying the following i 0 i Z 3 conditions: 1. g i m, g 0 ) = g i for all i 0; 2. if X g i m, g 0 ) with i > 0 satisfies [X, g 1 ] = 0, then X = 0; 3. gm, g 0 ) is the maximal graded Lie algebra, satisfying properties 1 and 2. This graded Lie algebra gm, g 0 ) is called the algebraic universal prolongation of the graded Lie algebra m g 0. An explicit realization of the algebra gm, g 0 ) will be described later in section 4. It turns out [6] 6, [7] 2) that the Lie algebra of infinitesimal symmetries of the flat structure of type m, g 0 ) can be described in terms of gm, g 0 ). If dim gm, g 0 ) is finite which is equivalent to the existence of l > 0 such that g l m, g 0 ) = 0), then the algebra of infinitesimal symmetries is isomorphic to gm, g 0 ). The analogous formulation in the case when gm, g 0 ) is infinite dimensional can be found in [6] 6. Furthermore for a structure P 0 of type m, g 0 ), Tanaa constructs a sequence of bundles {P i } i N, where P i is a principal bundle over P i 1 with an abelian structure group of dimension equal to dim g i m, g 0 ). In general P i is not a frame bundle. This is the case only for m = g 1, i.e. for G-structures. But if dim gm, g 0 ) is finite or, equivalently, if there exists l 0 such that g l+1 m, g 0 ) = 0, then the bundle P l+µ is an e-structure over P l+µ 1, i.e. P l+µ 1 is endowed with a canonical frame a structure of absolute parallelism). Note that all P i with i l are identified one with each other by the canonical projections which are diffeomorphisms in that case). Hence, already P l is endowed with a canonical frame. Once a canonical frame is constructed the equivalence problem for structures of type m, g 0 ) is solved in essence. Moreover, dim gm, g 0 ) gives the sharp upper bound for the dimension of the algebra of infinitesimal symmetries of such structures. By Tanaa s geometric prolongation we mean his construction of the sequence of bundles {P i } i N. In this note we mainly concentrate on the description of this geometric prolongation using a language different from the original one. In section 2 we mae a review of the prolongation of usual G-structures in the language of Singer and Sternberg. We do it in order to prepare the reader for the next section, where the first Tanaa geometric prolongation is given in a completely analogous way. We believe that after reading section 3 one will already have an idea how to proceed with the higher order Tanaa prolongations so that technicalities of section 4 can be easily overcome.
On Tanaa s prolongation procedure 5 2 Review of prolongation of G-structures Before treating the general case we give a review of the prolongation procedure for structures with m = g 1, i.e. for usual G-structures, We follow here [4] and [5]. Let Π 0 : P 0 M be the canonical projection and V λ) T λ P 0 be the tangent space at λ to the fiber of P 0 over the point Π 0 λ). The subspace V λ) is also called the vertical subspace of T λ P 0. Actually, V λ) = erπ 0 ) λ). 2.1) Recall that the space V λ) can be identified with ) the Lie algebra g 0 of G 0. The identification I λ : g 0 V λ) sends X g 0 to d dt λ R e tx t=0, where e tx is a one-parametric subgroup, generated by X. Recall also that an Ehresmann connection on the bundle P 0 is a distribution H on P 0 such that T λ P 0 = V λ) Hλ), λ P 0. 2.2) A subspace Hλ), satisfying 2.2), is called a horizontal subspace of T λ P 0. Once an Ehresmann connection H and a basis in the space g 1 g 0 are fixed, the bundle P 0 is endowed with a frame in a canonical way. Indeed, let λ = x, ϕ) P 0. Then ϕ Homg 1, T x M). By 2.1) and 2.2) the restriction Π 0 ) Hλ) of the map Π 0 ) to the subspace Hλ) is an isomorphism between Hλ) and T Π0 λ)m. Define the map ϕ Hλ) : g 1 g 0 T λ P 0 as follows: ϕ Hλ) g 1 = ϕ Hλ) g 0 = I λ. Π 0 ) Hλ) ) 1 ϕ, 2.3) If one fixes a basis in g 1 g 0, then the images of this basis under the maps ϕ Hλ) define the frame the structure of the absolute parallelism) on P 0. The question is whether an Ehresmann connection can be chosen canonically. To answer this question, first one introduces a special g 1 -valued 1-form ω on P 0 as follows: ωy ) = ϕ 1 Π 0 ) Y ) for any λ = x, ϕ) P 0 and Y T λ P 0. This 1-form is called a soldering tautological, fundamental) form of the G 0 -structure P 0. Further, fixing again a point λ = x, ϕ) P 0, one introduces a structure function a torsion) C H of a horizontal subspace H of T λ P 0, C H Homg 1 g 1, g 1 ), as follows: ) v 1, v 2 g 1, C H v 1, v 2 ) = dω ϕ H v 1 ), ϕ H v 2 ), 2.4) where ϕ H is defined by 2.3). Equivalently, C H v 1, v 2 ) = ω [Y 1, Y 2 ]λ) ) for any vector fields Y 1 and Y 2 such that ωy i ) v i and ϕ H v i ) = Y i λ), i = 1, 2. Speaing informally, the structure function C H encodes all information about horizontal parts at λ of Lie bracets of vector fields which are horizontal at λ w.r.t. the splitting 2.2) with Hλ) replaced by H). We now tae another horizontal subspace H of T λ P 0 and compare the structure functions C H and C H. By construction, for any vector v g 1 the vector ϕ Hv) ϕ H v) belongs to V λ) g 0 ). Let def Hv) = I 1 ϕ Hv) ) ϕ H v). 2.5) f H λ
6 I. Zeleno Then f H H Homg 1, g 0 ). In the opposite direction, it is clear that for any f Homg 1, g 0 ) there exists a horizontal subspace H such that f = f H H. The map defined by : Homg 1, g 0 ) Homg 1 g 1, g 1 ), fv 1, v 2 ) = fv 1 )v 2 fv 2 )v 1 = [fv 1 ), v 2 ] + [v 1, fv 2 )] 2.6) is called the Spencer operator. By direct computations [4] p.42, [5] p. 317 or proof of more general statement in Proposition 3.1 below) one can get the following identity C H = C H + f H H. 2.7) Now fix a subspace N Homg 1 g 1, g 1 ), which is complementary to Im, Homg 1 g 1, g 1 ) = Im N. 2.8) Speaing informally, the subspace N defines the normalization conditions for the first prolongation. The first prolongation of P 0 is the following bundle P 0 ) 1) over P 0 : P 0 ) 1) = {λ, H) : λ P 0, H is horizontal subspace of T λ P 0, C H N } 2.9) or, equivalently, P 0 ) 1) = {λ, ϕ H ) : λ P 0, H is horizontal subspace of T λ P 0, C H N }. 2.10) In other words, the fiber of P 0 ) 1) over a point λ P 0 is the set of all horizontal subspaces H of T λ P 0 such that their structure functions satisfy the chosen normalization condition N. Obviously, the fibers of P 0 ) 1) are not empty and if two horizontal subspaces H, H belong to the fiber, then f H H er. The subspace g 1 of Homg 1, g 0 ) defined by g 1 def = er. 2.11) is called the first algebraic prolongation of g 0 glg 1 ). Note that it is absolutely not important that g 0 is a subalgebra of glg 1 ). The first algebraic prolongation can be defined for a subspace of glg 1 ) see the further generalization below). If g 1 = 0 then the choice of the normalization conditions N determines an Ehresmann connection on P 0 and P 0 is endowed with a canonical frame. For example, for Riemannian structures, g 0 = son), where n = dim g 1, and it is easy to show that g 1 = 0. Moreover, in this case dim Homg 1 g 1, g 1 ) = dim Homg 1, g 0 ) = n2 n 1) 2. Hence, Im = Homg 1 g 1, g 1 ) and the complement subspace N must be equal to 0. So, in this case one gets the canonical Ehresmann connection with zero structure function torsion), which is nothing but the Levi-Civita connection. If g 1 0, we continue the prolongation procedure by induction. Given a linear space W denote by Id W the identity map on W. The bundle P 0 ) 1) is a frame bundle with the abelian structure group G 1 of all maps A GLg 1 g 0 ) such that A g 1 = Id g 1 + T, A g 0 = Id g 0, 2.12)
On Tanaa s prolongation procedure 7 where T g 1. The right action R A of A G 1 on a fiber of P 0 ) 1) is defined by the following rule: R A ϕ) = ϕ A. Obviously, g 1 is isomorphic to the Lie algebra of G 1. Set P 1 = P 0 ) 1). The second prolongation P 2 of P 0 is by definition the first prolongation of the frame bundle P 1, P 2 def = P 1 ) 1) and so on by induction: the ith prolongation P i is the first prolongation of the frame bundle P i 1. Let us describe the structure group G i of the frame bundle P i over P i 1 in more detail. For this one can define the Spencer operator and the first algebraic prolongation also for a subspace W of Homg 1, V ), where V is a linear space, which does not necessary coincide with g 1 as before. In this case the Spencer operator is the operator from Homg 1, W ) to Homg 1 g 1, V ), defined by the same formulas, as in 2.6). The first prolongation W 1) of W is the ernel of the Spencer operator. Note that by definitions g 1 = g 0 ) 1). Then the ith prolongation g i of g 0 is defined by the following recursive formula: g i = g i 1 ) 1). Note that g i Homg 1, g i 1 ). By 2.12) and the definition of the Spencer operator the bundle P i is a frame bundle with the abelian structure group G i of all maps A GL i 1 g p) such that p= 1 A g 1 = Id g 1 + T, A i 1 p=0 gp = Id i 1 p=0 gp, 2.13) where T g i. In particular, if g l+1 = 0 for some l 0, then the bundle P l is endowed with the canonical frame and we are done. 3 First Tanaa s Prolongation Now consider the general case. As before P 0 is a structure of constant type m, g 0 ). Let Π 0 : P 0 M be the canonical projection. The filtration {D i } i<0 of T M induces a filtration {D i 0 } i 0 of T P 0 as follows: D0 0 = erπ 0 ), { D0λ) i = v T λ P 0 : Π 0 ) v D i Π 0 λ) )}, i < 0. 3.1) We also put D0 i = 0 for all i > 0. Note that D0 0 λ) is the tangent space at λ to the fiber of P 0 and therefore can be identified with g 0. Denote by I λ : g 0 D0 0 λ) the identifying isomorphism. Fix a point λ P 0 and let π0 i : Di 0 λ)/di+2 0 λ) D0 i λ)/di+1 0 λ) be the canonical projection to the factor space. Note that Π 0 induced an isomorphism between the space D0 i λ)/di+1 0 λ) and the space D i Π 0 λ))/d i+1 Π 0 λ)) for any i < 0. We denote this isomorphism by Π i 0. The fiber of the bundle P 0 over a point x M is a subset of the set of all maps ϕ i<0 Hom g i, D i x)/d i+1 x) ), which are isomorphisms of the graded Lie algebras m = g i and D i x)/d i+1 x). We are i<0 i<0 going to construct a new bundle P 1 over the bundle P 0 such that the fiber of P 1 over a point λ = x, ϕ) P 0 will be a certain subset of the set of all maps ˆϕ i 0 Hom g i, D i 0λ)/D i+2 0 λ) )
8 I. Zeleno such that ϕ g i = Π i 0 π i 0 ˆϕ g i, i < 0, ˆϕ g 0 = I λ. 3.2) For this fix again a point λ = x, ϕ) P 0. For any i < 0 choose a subspace H i D0 i λ)/di+2 0 λ), which is a complement of D0 i+1 λ)/d0 i+2 λ) to D0 i λ)/di+2 0 λ): D i 0λ)/D i+2 0 λ) = D i+1 0 λ)/d i+2 0 λ) H i. 3.3) Then the map π i 0 H i defines an isomorphism between Hi and D i Π 0 λ) ) /D i+1 Π 0 λ) ). So, once a tuple of subspaces H = {H i } i<0 is chosen, one can define a map ϕ H i 0 Hom g i, D i 0λ)/D i+2 0 λ) ) as follows ϕ H g i = { π i 0 H i) 1 ϕ g i if i < 0 I λ if i = 0, 3.4) Clearly ˆϕ = ϕ H satisfies 3.2). Tuples of subspaces H = {H i } i<0 satisfying 3.3) play here the same role as horizontal subspaces in the prolongation of the usual G-structures. Can we choose a tuple {H i } i<0 in a canonical way? For this, by analogy with the prolongation of G-structure, we introduce a partial soldering form of the bundle P 0 and the structure function of a tuple H. The soldering form of P 0 is a tuple Ω 0 = {ω0 i } i<0, where ω0 i is a gi -valued linear form on D0 i λ) defined by ω0y i ) = ϕ 1 ) ) Π 0 ) Y ), 3.5) i ) where Π 0 ) Y ) is the equivalence class of Π 0) Y ) in D i x)/d i+1 x). Obviously D0 i+1 λ) = i er ω0 i, so, the form ωi 0 induces the gi -valued form ω 0 i on Di 0 λ)/di+1 0 λ). The structure function CH 0 of the tuple H = {Hi } i<0 is an element of the space 2 A 0 = Homg 1 g i, g i ) Homg 1 g 1, g 1 ) 3.6) i= µ defined as follows: Let pr H i be the projection of D0 i λ)/di+2 0 λ) to D0 i+1 λ)/d0 i+2 λ) parallel to H i or corresponding to the splitting 3.3)). Given vectors v 1 g 1 and v 2 g i tae two vector fields Y 1 and Y 2 in a neighborhood of λ in P 0 such that Y 1 is a section of D0 1, Y 2 is a section of D0 i, and ω0 1 1) v 1, ω0y i 2 ) v 2, Y 1 λ) = ϕ H v 1 ), Y 2 λ) ϕ H v 2 ) mod D0 i+2 λ). Then set CHv 0 1, v 2 ) def = ω 0 i pr H i 1 [Y1, Y 2 ]λ) )). 3.8) In the previous formula we actually tae the equivalence class of the vector [Y 1, Y 2 ]λ) in D0 i 1 λ)/d0 i+1 λ) and then apply pr H i 1 to it. One has to show that C0 H v 1, v 2 ) does not depend on the choice of vector fields Y 1 and Y 2, satisfying 3.7). Indeed, assume that Ỹ1 and Ỹ2 3.7)
On Tanaa s prolongation procedure 9 are another pair of vector fields in a neighborhood of λ in P 0 such that Ỹ1 is a section of D 1 0, Ỹ 2 is a section of D i 0, and they satisfy 3.7) with Y 1, Y 2 replaced by Ỹ1, Ỹ1. Then Ỹ 1 = Y 1 + Z 1, Ỹ 2 = Y 2 + Z 2, 3.9) where Z 1 is a section of the distribution D0 0 such that Z 1λ) = 0 and Z 2 is a section of the distribution D0 i+1 such that Z 2 λ) D0 i+2 λ). It follows that [Y 1, Z 2 ]λ) D0 i+1 λ) and [Y 2, Z 1 ]λ) D0 i+1 λ). This together with the fact that [Z 1, Z 2 ] is a section of D0 i+1 imply that [Ỹ1, Ỹ2]λ) [Y 1, Y 2 ] mod D i+1 0 λ), and by definition 3.8) we get that the structure function is independent of the choice of vector fields Y 1 and Y 2. We now tae another tuple H = { H i } i<0 such that D i 0λ)/D i+2 0 λ) = D i+1 0 λ)/d i+2 0 λ) H i 3.10) and consider how the structure functions C 1 H and C1 H are related. vector v g i the vector ϕ Hv) ϕ H v) belongs to D0 i+1 λ)/d0 i+2 λ). Let ) def f H Hv) = ω 0 i+1 I 1 λ ϕ Hv) ϕ H v) ϕ Hv) ) ϕ H v) if v g i with i < 1 if v g 1. By construction, for any Then f H H i<0 Homg i, g i+1 ). In the opposite direction, it is clear that for any f Homg i, g i+1 ) there exists a tuple H = { H i } i<0, satisfying 3.10), such that f = f H H. i<0 Further, let A 0 be as in 3.6) and define a map 0 : i<0 Homg i, g i+1 ) A 0 by 0 fv 1, v 2 ) = [fv 1 ), v 2 ] + [v 1, fv 2 )] f[v 1, v 2 ]), 3.11) where the bracets [, ] are as in the Lie algebra m g 0. The map 0 coincides with the Spencer operator 2.6) in the case of G-structures. Therefore it is called the generalized Spencer operator for the first prolongation. Proposition 3.1. The following identity holds C 0 H = C 0 H + 0 f H H. 3.12) Proof. Fix vectors v 1 g 1 and v 2 g i and let Y 1 and Y 2 be two vector fields in a neighborhood of λ, satisfying 3.7). Also tae two vector fields Ỹ1 and Ỹ2 in a neighborhood of λ in P 0 such that Ỹ1 is a section of D 1 0, Ỹ2 a section of D i 0, and ω 1 0 Ỹ1) v 1, ω i 0Ỹ2) v 2, Ỹ 1 λ) = ϕ Hv 1 ), Ỹ 2 λ) ϕ Hv 2 ) mod D i+2 0 λ). 3.13)
10 I. Zeleno Further, assume that vector fields Z 1 and Z 2 are defined as in 3.9). Then Z 1 is a section of D 0 0 and Z 2 is a section of D i+1 0 such that 1 λ) = I λ fh Hv 1 ) ) Z { 3.14) f H Hv 2 ) = ω 0 i+1 Z 2 λ)) if v g i, i < 1 I 1 λ Z 2λ)) if v g 1. 3.15) Hence [Z 1, Y 2 ] and [Y 1, Z 2 ] are sections of D0 i, while [Z 1, Z 2 ] is a section of D0 i+1. It implies that ω 0 i pr H i 1 [ Ỹ 1, Ỹ2]λ) )) = ω 0 i pr H i 1 [Y1, Y 2 ]λ) )) + ω 0 i [Z1, Y 2 ] ) + ω 0 i [Y1, Z 2 ] ). 3.16) Further, directly from the definitions of f H H, pr H i 1, and pr H i 1 it follows that ω 0 i pr Hi 1 w) ) ) = ω 0 i pr H i 1 i 1 w) f H ω H 0 w) ), w D0 i 1 λ)/d0 i+1 λ). 3.17) Besides, from the definition of the soldering form, the fact that ϕ is an isomorphism of the Lie algebras m and mx) = i<0 D i x)/d i+1 x), and relations 3.15) for i < 1 it follows that ω i 0 [Y1, Z 2 ] ) = [v 1, f H Hv 2 )] i < 1. 3.18) Taing into account 3.7) we get ω i 1 0 [Y 1, Y 2 ]) = [v 1, v 2 ]. 3.19) Finally, from 3.14), 3.15) for i = 1, and the definition of the action of G 0 on P 0 it follows that identity 3.18) holds also for i = 1 and that ω i 0 [Z1, Y 2 ] ) = [f H Hv 1 ), v 2 ]. 3.20) Substituting 3.17)-3.20) into 3.16) we get 3.12). Now we proceed as in the case of G-structures. Fix a subspace N 0 A 0 which is complementary to Im 0, A 0 = Im 0 N 0. 3.21) As for G-structures, the subspace N 0 defines the normalization conditions for the first prolongation. Then from the splitting 3.21) it follows trivially that there exists a tuple H = {H i } i<0 such that C 0 H N 0 3.22) and a tuple H = { H i } i<0 satisfies C 0 H N 0 if and only if f H H er 0. In particular if er 0 = 0 then the tuple H is fixed uniquely by condition 3.22). Let g 1 def = er 0. 3.23) The space g 1 is called the first algebraic prolongation of the algebra m g 0. Here we consider g 1 as an abelian Lie algebra. Note that the fact that the symbol m is fundamental, i.e. that g 1 generates the whole m, implies that { } g 1 = f i<0 Homg i, g i+1 ) : f[v 1, v 2 ]) = [fv 1 ), v 2 ] + [v 1, fv 2 )] v 1, v 2 m.
On Tanaa s prolongation procedure 11 The first geometric) prolongation of the bundle P 0 is the bundle P 1 over P 0 defined by P 1 = {λ, H) : λ P 0, C 0 H N 0 } 3.24) or, equivalently, by P 1 = {λ, ϕ H ) : λ P 0, C 0 H N 0 }. 3.25) It is a principal bundle with the abelian structure group G 1 of all maps A i<1 Homg i, g i g i+1 ) such that A g i = Id g i + T i, i < 0 A g 0 = Id g 0, 3.26) where T i Homg i, g i+1 ) and T µ,..., T 1 ) g 1. The right action R 1 A of A G1 on a fiber of P 1 is defined by the following rule: R 1 A ϕh ) = ϕ H A. Obviously, G 1 is an abelian group of dimension equal to dim g 1. 4 Higher order Tanaa s prolongations More generally, define the th algebraic prolongation g of the algebra m g 0 by induction for any N. Assume that spaces g l i<0 Homg i, g i+l ) are defined for all 0 < l <. Put Then let [f, v] = [v, f] = fv) f g l, 0 l <, and v m. 4.1) g def = { f i<0 Homg i, g i+ ) : f[v 1, v 2 ]) = [fv 1 ), v 2 ] + [v 1, fv 2 )] v 1, v 2 m }. 4.2) Directly from this definition and the fact that m is fundamental i.e. it is generated by g 1 ) it follows that if f g satisfies f g 1 = 0, then f = 0. The space i Z gi can be naturally endowed with the structure of a graded Lie algebra. Bracets of two elements from m are as in m. Bracets of an element with non-negative weight and an element from m are already defined by 4.1). It only remains to define bracets [f 1, f 2 ] for f 1 g, f 2 g l with, l 0. It is defined inductively with respect to and l: if = l = 0 then the bracet [f 1, f 2 ] is as in g 0. Assume that [f 1, f 2 ] is defined for all f 1 g, f 2 g l such that a pair, l) belongs to the set {, l) : 0, 0 l l}\{, l)}. Then define [f 1, f 2 ] for f 1 g, f 2 g l to be an element of i<0 Homg i, g i+ + l) such that [f 1, f 2 ]v def = [f 1 v), f 2 ] + [f 1, f 2 v)] v m. It is easy to see that [f 1, f 2 ] g +l and that i Z gi with bracet product defined as above is a graded Lie algebra. As a matter of fact [6] 5) this graded Lie algebra satisfies properties 1-3 from subsection 1.4, i.e. it is a realization of the algebraic universal prolongation gm, g 0 ) of the algebra m + g 0. Now we are ready to construct the higher order geometric prolongations of the bundle P 0 by induction. Assume that all lth order prolongations P l are constructed for 0 l. We also set P 1 = M. We will not specify what the bundles P l are exactly. As in the case of the first prolongation P 1, their construction depends on the choice of normalization conditions on each step. But we will point out those properties of these bundles that we need in order to construct the + 1)th order prolongation P +1. Here are these properties:
12 I. Zeleno 1. P l is a principal bundle over P l 1 with an abelian structure group G l of dimension equal to dim g l and with the canonical projection Π l. 2. The tangent bundle T P l is endowed with the filtration {Dl i } as follows: For l = 1 it coincides with the initial filtration {D i } i<0 and for l 0 we get by induction Dl l = erπ l) { Dl i λ l) = v T λ P l : Π l ) v Dl 1 i Πl λ l ) )}, i < l. 4.3) The subspaces D l l λ l), as the tangent spaces to the fibers of P l, are canonically identified with g l. Denote by I λl : g l D l l λ l) the identifying isomorphism. 3. The fiber of P l, 0 l, over a point λ l 1 P l 1 will be a certain subset of the set of all maps from Hom g i, Dl 1 i λ l 1)/D i+l+1 l 1 λ l 1 ) ). i<l If l > 0 and λ l = λ l 1, ϕ l ) P l, then ϕ l g l 1 coincides with the identification of g l 1 with D l 1 l 1 λ l 1) and the restrictions ϕ l g i with i 0 are the same for all λ l from the same fiber. 4. Assume that 0 < l, λ l 1 = λ l 2, ϕ l 1 ) P l 1 and λ l = λ l 1, ϕ l ) P l. The maps ϕ l 1 and ϕ l are related as follows: if πl i : Di l λ l)/d i+l+2 l λ l ) Dl i λ l)/d i+l+1 l λ l ) 4.4) are the canonical projections to a factor space and Π i l : Di l λ l)/d i+l+1 l λ l ) Dl 1 i Πl λ l ) ) /D i+l+1 l 1 Π l λ l )) 4.5) are the canonical maps induced by Π l ), then i < l ϕ l 1 g i = Π i l 1 πi l 1 ϕ l g i. 4.6) Note that the maps Π i l are isomorphisms for i < 0 and the maps πi l we set Dl i = 0 for i > l). are identities for i 0 Now we are ready to construct the + 1)st order Tanaa geometric prolongation. Fix a point λ P and assume that λ = λ 1, ϕ ), where ϕ i< Hom g i, D i 1 λ 1)/D i++1 1 λ 1 ) ). Let H = {H i } i< be the tuple of spaces such that H i = ϕ g i ). Tae a tuple H +1 = {H i +1 } i< of linear spaces such that 1. for i < 0 the space H+1 i D i λ )/D i++2 λ ), is a complement of Di++1 λ )/D i++2 λ ) in Π i πi ) 1 H i) Π i πi ) 1 H i ) = Di++1 λ )/D i++2 λ ) H+1 i ; 4.7) 2. for 0 i < the space H i +1 is a complement of D λ ) in Π i ) 1 H i ), Π i ) 1 H i ) = D λ ) H i +1. 4.8)
On Tanaa s prolongation procedure 13 Here the maps π i and Πi are defined as in 4.4) and 4.5) with l =. Since D i++1 λ )/D i++2 λ ) = er π i and Πi is an isomorphism for i < 0, the map Πi π i H+1 i defines an isomorphism between Hi +1 and Hi for i < 0. Besides, by 4.8) the map Π l ) H i defines an isomorphism between H i +1 +1 and Hi for 0 i <. So, once a tuple of subspaces H +1 = {H+1 i } i<, satisfying 4.7) and 4.8), is chosen, one can define a map ϕ H +1 i Hom g i, D i λ )/D i++2 λ ) ) as follows Π i πi H i ) 1 ϕ ϕ H +1 g i if i < 0, ) +1 1 g i = Πl ) H i ϕ +1 g i if 0 i <, I λ if i =. 4.9) Can we choose a tuple or a subset of tuples H in a canonical way? To answer this question, by analogy with sections 2 and 3, we introduce a partial soldering form of the bundle P and the structure function of a tuple H +1. The soldering form of P is a tuple Ω = {ω i } i<, where ω i is a gi -valued linear form on D i λ ) defined by ω i Y ) = ϕ 1 ) )) Π ) Y ), 4.10) i ) where Π ) Y ) is the equivalence class of Π ) Y ) in D 1 i λ 1)/D i++1 1 λ 1 ). By constructions it follows immediately that D i+1 i λ ) = er ω i. So, the form ωi induces the gi -valued form ω i on Di λ )/D i+1 λ ). The structure function CH +1 of a tuple H +1 is an element of the space 2 1 ) A = Homg 1 g i, g i+ ) Homg 1 g 1, g 1 ) Homg 1 g i, g 1 ) 4.11) i= µ defined as follows: Let π i,s l : Dl iλ l)/d i+l+2 l λ l ) Dl iλ l)/d i+l+2 s l λ l ) be the canonical projection to a factor space, where 1 l, i l. Here, as before, we assume that Dl i = 0 for i > l. Note that the previously defined πl i coincides with π l i,1. By construction, one has the following two relations D i λ )/D i++2 ) λ ) = s=0 πi+s,s H i+s +1 ) D i++1 λ )/D i++2 λ ) if i < 0 4.12) ) D i λ ) = D λ ) if 0 i < 4.13) Let pr H +1 i 1 s=i Hi +1 be the projection of D i λ )/D i++2 i=0 λ ) to D i++1 λ )/D i++2 λ ) correspond- corresponding to the ing to the splitting 4.12) if i < 0 or the projection of D i λ ) to H 1 +1 splitting 4.13) if 0 i <. Given vectors v 1 g 1 and v 2 g i tae two vector fields Y 1 and Y 2 in a neighborhood U of λ in P such that for any λ = λ 1, ϕ ) U, where ϕ Hom g i, D 1 i λ 1)/D i++1 1 λ 1 ) ), one has i< Π Y 1 λ ) = ϕ v 1 ), Π Y 2 λ ) ϕ v 2 ) mod D i++1 1 λ 1 ), Y 1 λ) = ϕ H +1 v 1 ), Y 2 λ) ϕ H +1 v 2 ) mod D i++2 λ). 4.14)
14 I. Zeleno Then set C H +1 v 1, v 2 ) def = ω i+ ω 1 pr H +1 i 1 [Y1, Y 2 ] )) if i < 0, [Y1, Y 2 ] )) if 0 i <. pr H +1 i 1 4.15) As in the case of the first prolongation, CH v 1, v 2 ) does not depend on the choice of vector fields Y 1 and Y 2, satisfying 4.14). Indeed, assume that Ỹ1 and Ỹ2 is another pair of vector fields in a neighborhood of λ in P such that Ỹ1 is a section of D 1, Ỹ2 is a section of D i, and they satisfy 4.14) with Y 1, Y 2 replaced by Ỹ1, Ỹ1. Then Ỹ 1 = Y 1 + Z 1, Ỹ 2 = Y 2 + Z 2, 4.16) where Z 1 is a section of the distribution D such that Z 1λ ) = 0 and Z 2 is a section of the distribution D min{i++1,} D min{i++1,} such that Z 2 λ ) D min{i++1,}+1 λ ). It implies that [Y 1, Z 2 ]λ ) λ ). This together with the fact that [Z 1, Z 2 ] λ ) and [Y 2, Z 1 ]λ) D min{i++1,} is a section of D min{i++1,}+1 implies that [Ỹ1, Ỹ2]λ) [Y 1, Y 2 ] mod D min{i++1,} λ), Then by definition 4.15) we get that the structure function is independent of the choice of vector fields Y 1 and Y 2. Now tae another tuple H +1 = { H i +1 } i< such that 1. for i < 0 the space H +1 i D i λ )/D i++2 λ ), is a complement of Di++1 λ )/D i++2 λ ) in Π i πi ) 1 H i) Π i πi ) 1 H i ) = Di++1 λ )/D i++2 λ ) H +1 i. 4.17) 2. for 0 i < the space H i +1 is a complement of D λ ) in Π i ) 1 H i ), Π i ) 1 H i ) = D λ ) H i +1, 4.18) How the structure functions C H +1 and C H+1 are related? By construction, for any vector v g i the vector ϕ H +1 v) ϕ H +1v) belongs to D i++1 λ )/D i++2 λ ) for i < 0 and to D λ ) for 0 i <. Let ) f H+1 v) def ω i++1 ϕ H +1 v) ϕ H +1v) if v g i with i < 1 = ) H+1 I 1 ϕ H +1 v) ϕ H +1v) if v g i with 1 i <. Then λ f H+1 1 Homg i, g i++1 ) Homg i, g ). H+1 i<0 i=0 In the opposite direction, it is clear that for any f 1 Homg i, g i++1 ) Homg i, g ) i<0 there exists a tuple H +1 = { H i +1 } i<, satisfying 4.17) and 4.18), such that f = f H+1 H+1. Further, let A be as in 4.11) and define a map : 1 Homg i, g i++1 ) Homg i, g ) A i<0 i=0 i=0
On Tanaa s prolongation procedure 15 by fv 1, v 2 ) = { [fv 1 ), v 2 ] + [v 1, fv 2 )] f[v 1, v 2 ]) if v 1 g 1, v 2 g i, i < 0; [v 1, fv 2 )] if v 1 g 1, v 2 g i, 0 i < 1, 4.19) where the bracets [, ] are as in the algebraic universal prolongation gm, g 0 ). Clearly for = 0 this definition coincides with the definition of the generalized Spencer operator for the first prolongation given in the previous section. The reason for introducing the operator is that the following generalization of identity 3.12) holds: C H+1 = C H +1 + f H+1 H+1. 4.20) A verification of this identity for pairs v 1, v 2 ), where v 1 g 1 and v 2 g i with i < 0, is completely analogous to the proof of Proposition 3.1, while for i 0 one has to use the inductive assumption that the restrictions ϕ l g i with i 0 are the same for all λ l from the same fiber see item 3 from the list of properties of the bundles P l in the beginning of this section) and the splitting 4.13). Now we proceed as in sections 2 and 3. Fix a subspace N A which is complementary to Im, A = Im N. 4.21) Similarly to above, the subspace N defines the normalization conditions for the first prolongation. Then from the splitting 4.21) it follows trivially that there exists a tuple H +1 = {H i +1 } i<, satisfying 4.7) and 4.8), such that C H +1 N 4.22) and C H+1 N for a tuple H +1 = {H+1 i } i<, satisfying 4.17) and 4.18), if and only if f H+1 er H+1. Note also that if f er f g i = 0 0 i 1. 4.23) In other words, er Homg i, g i++1 ). i<0 4.24) Indeed, if f er, then by 4.19) for any v 1 g 1 and v 2 g i with 0 i 1 one has [v 1, fv 2 )] = fv 2 )v 1 = 0. 4.25) In other words, fv 2 ) g 1 = 0 recall that fv 2 ) g i<0 Homg i, g i+ )). Since g 1 generates the whole symbol m we get that fv 2 ) = 0 and it holds for any v 2 g i with 0 i 1, So, 4.23) is proved. Further, comparing 4.19) and 4.24) with 4.2) and using again the fact that g 1 generates the whole symbol m we obtain that er = g +1. 4.26)
16 I. Zeleno The + 1)st geometric) prolongation of the bundle P 0 is the bundle P +1 over P defined by P +1 = {λ, H +1 ) : λ P, CH +1 N } 4.27) or, equivalently, by P +1 = {λ, ϕ H +1 ) : λ P, CH +1 N }. 4.28) It is a principal bundle with the abelian structure group G +1 of all maps A Homg i, g i g i++1 ) i such that A g i = { Id g i + T i, if i < 0, Id g i, if 0 i, 4.29) where T i Homg i, g i++1 ) and T µ,..., T 1 ) g +1. The right action R +1 A of A G +1 on a fiber of P +1 is defined by the following rule: R +1 A ϕh +1) = ϕ H +1 A. Obviously, G +1 is an abelian group of dimension equal to dim g +1. It is easy to see that the bundle P +1 is constructed such that the Properties 1-4, formulated in the beginning of the present section, hold for l = + 1 as well. Finally, assume that there exists l 0 such that g l 0 but g l+1 = 0. Since the symbol m is fundamental, it follows that g l = 0 for all l > l. Hence, for all l > l the fiber of P l over a point λ l 1 P l 1 is a single point, which belongs to l 1 i= µ Hom g i, Dl 1 i λ l 1)/D i+l+1 l 1 λ l 1 ) ), where, as before, µ is a degree of nonholonomy of the distribution D. Besides, by our assumptions Dl i = 0 if l l and i l. Therefore, if l = l + µ, then i + l + 1 > l for i µ and the fiber of P l l 1 over P l is an element of Hom g i, T λl 1 P l 1 ). In other words, P l+µ defines a canonical frame i= µ on P l+µ 1. But all bundles P l with l l are identified one with each other by the canonical projections which are diffeomorphisms in that case). As a conclusion we get an alternative proof of the main result of the Tanaa paper [6]: Theorem 4.1. If the l + 1)th algebraic prolongation of the graded Lie algebra m g 0 is equal to zero then for any structure P 0 of constant type m, g 0 ) there exists a canonical frame on the lth geometric prolongation P l of P 0. Acnowledgements I would lie to than my colleagues Joseph Landsberg, Colleen Robles, and Dennis The for encouraging me to give this series of lecture and to Boris Doubrov for stimulating discussions. References [1] O. Kuzmich, Graded nilpotent Lie algebras in low dimensions, Lobachevsii Journal of Mathematics, Vol.3, 1999, pp.147-184. [2] T. Morimoto, Lie algebras, Geometric Structures, and Differential Equations on Filtered manifolds, Advanced Studies in Pure Mathematics 33, 2002, Lie Gropus, Geometric Structures and Differential Equations One Hundreds Years after Sophus Lie, 205 252. [3] T. Morimoto, Cartan connection associated with a subriemannian structure, Differential Geometry and its Applications, Volume 26, Issue 1, February 2008, pages 75-78
On Tanaa s prolongation procedure 17 [4] I. M Singer, S. Sternberg, The infinite groups of Lie and Cartan. I. The transitive groups. J. Analyse Math. 15,1965, 1 114. [5] S. Sternberg, Lectures on Differential Geometry. Prentice Hall, Englewood Cliffs, N.J., 1964. [6] N. Tanaa, On differential systems, graded Lie algebras and pseudo-groups, J. Math. Kyoto. Univ., 10 1970), 1 82. [7] K. Yamaguchi, Differential Systems Associated with Simple Graded Lie algebras, Advanced Studies in Pure Mathematics 22, 1993, Progress in Differential Geometry, 413 494.