The Geometry of Tangent Bundles: Canonical Vector Fields

Hindawi Publishing Corporation Geometry Volume 2013, Article ID 364301, 10 pages http://dx.doi.org/10.1155/2013/364301

Research Article The Geometry of Tangent Bundles: Canonical Vector Fields Tongzhu Li1 and Demeter Krupka1,2,3 1

Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China Department of Mathematics, Faculty of Science, The University of Ostrava, 30. dubna 22, 70103 Ostrava, Czech Republic 3 Department of Mathematics, La Trobe University, Melbourne, Bundoora, VIC 3086, Australia 2

Correspondence should be addressed to Demeter Krupka; [email protected] Received 14 December 2012; Accepted 13 March 2013 Academic Editor: Anna Fino Copyright © 2013 T. Li and D. Krupka. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A canonical vector field on the tangent bundle is a vector field defined by an invariant coordinate construction. In this paper, a complete classification of canonical vector fields on tangent bundles, depending on vector fields defined on their bases, is obtained. It is shown that every canonical vector field is a linear combination with constant coefficients of three vector fields: the variational vector field (canonical lift), the Liouville vector field, and the vertical lift of a vector field on the base of the tangent bundle.

1. Introduction Vector fields on tangent bundles belong to basic concepts of pure and applied differential geometry, global analysis, and mathematical physics. Recent research in geometry extends the well-known correspondence of semisprays, sprays, and geodesic sprays to the classical theory of geodesics and connections (see, e.g., [1, 2]). Vector fields on tangent bundles can be considered as an underlying geometric structure for the theory of second-order differential equations [3– 7]. The semispray theory has been used in the calculus of variations on manifolds to characterize extremal curves of a variational functional as integral curves of the Hamilton or Euler-Lagrange vector fields [2, 4, 8, 9]. Sprays and semisprays also provide a natural framework for extension of classical results of analytical mechanics to contemporary mechanical problems and stimulate a broad research in the global theory of nonconservative systems, symmetries, and the constraint theory (see, e.g., [6, 10–12]). This paper is devoted to the structure theory of vector fields on the tangent bundle 𝑇𝑋 of a manifold 𝑋; our aim will be to classify all canonical vector fields on 𝑇𝑋, independent of any geometric structure or on the topology of 𝑋. Our main theorem says that every canonical vector field is a linear combination with constant coefficients of three independent vector fields: (a) a variational vector field (the natural lift of a vector field, defined on 𝑋), (b) the Liouville vector field, and

(c) the vertical lift of a vector field on 𝑋; this completes the results obtained in [13]. Another result is the method how the main theorem has been formulated and proved; the concepts we use allow generalizations and applications to analogous problems of discovering and describing canonical geometric objects. Our approach to the problem is based on the theory of jets and differential invariants, and on an observation that the coordinate transformations on 𝑋 naturally define a Lie group 𝐿𝑟𝑛 , where 𝑛 = dim 𝑋, the differential group, and its left action on the type fibre of any natural bundle over 𝑋 [13– 17]. The general theory gives us equations that determine a canonical vector field on 𝑇𝑋 as a differential invariant of the second differential group 𝐿𝑟𝑛 . The proof of our result is not straightforward; it relies on the semidirect product structure of 𝐿𝑟𝑛 and on the orbit reduction method that has already been applied to the classification problem of differential invariants of a linear connection [18]. The method can also be used in the canonical constructions depending on any geometric objects. Throughout this work, 𝑋 is a smooth real 𝑛-dimensional manifold, 𝑇𝑋 is the tangent bundle of 𝑋, and 𝜏𝑋 : 𝑇𝑋 → 𝑋 is the tangent bundle projection. The second tangent bundle of 𝑋 is the tangent bundle 𝑇𝑇𝑋 = 𝑇2 𝑋 over 𝑇𝑋 with the tangent bundle projection 𝜏𝑇𝑋 : 𝑇𝑇𝑋 → 𝑇𝑋; its elements are second-order tangent vectors on 𝑋. The mapping 𝑇𝜏𝑋 is the differential 𝑇𝜏𝑋 : 𝑇𝑇𝑋 → 𝑇𝑥 of 𝜏𝑋 , satisfying

2

Geometry

𝜏𝑋 ∘𝑇𝜏𝑥 = 𝜏𝑋 ∘𝜏𝑇𝑋. The 1-jet prolongation of 𝑇𝑋 is denoted by 𝐽1 𝑇𝑋; elements of the set 𝐽1 𝑇𝑋 are the 1-jets 𝐽𝑥1 𝜉 with source 𝑥 ∈ 𝑋 and target 𝜉(𝑥) ∈ 𝑇𝑥 𝑋 = 𝜏𝑥−1 (𝑥); the source and target jet projections are 𝜏𝑋1 : 𝐽1 𝑇𝑋 → 𝑋 and 𝜏𝑋1,0 : 𝐽1 𝑇𝑋 → 𝑇𝑋.

𝜌𝑋 = 𝑝𝑟1 .

2. Second-Order Vectors and Jets of a Vector Field In this section, we fix basic notation, used throughout this paper. If (𝑈, 𝜑), 𝜑 = (𝑥𝑖 ), is a chart on 𝑋, we denote by (𝜏𝑋−1 (𝑈), Φ), Φ = (𝑥𝑖 , 𝑥̇ 𝑖 ), the associated chart on the tangent bundle 𝑇𝑋; the associated chart on 𝑇2 𝑋 is denoted −1 −1 ̃̇ 𝑖 ). The tangent bundle ̃𝑖, 𝑥 (𝜏𝑋 (𝑈)), Ψ), Ψ = (𝑥𝑖 , 𝑥̇ 𝑖 , 𝑥 by (𝜏𝑇𝑋 projection 𝜏𝑋 has the chart expression (𝑥𝑖 , 𝑥̇ 𝑖 ) → (𝑥𝑖 ), and ̃̇ 𝑖 ) → (𝑥𝑖 , 𝑥̇ 𝑖 ). The tangent ̃𝑖, 𝑥 𝜏𝑇𝑋 is expressed by (𝑥𝑖 , 𝑥̇ 𝑖 , 𝑥 ̃̇ 𝑖 ) → ̃𝑖, 𝑥 mapping 𝑇𝜏𝑋 is expressed in coordinates as (𝑥𝑖 , 𝑥̇ 𝑖 , 𝑥 ̃ 𝑖 ). (𝑥𝑖 , 𝑥 In the following two lemmas, we recall standard transformation formulas, needed in proofs. Lemma 1. Let a second-order tangent vector Ξ ∈ 𝑇𝜁 𝑇𝑋 be expressed in two charts (𝑈, 𝜑), 𝜑 = (𝑥𝑖 ), and (𝑉, 𝜓), 𝜓 = (𝑦𝑖 ), as Ξ = Ξ𝑘 (

𝜕 𝜕 𝜕 𝜕 𝑘 𝑙 ) + Ξ̇ ( 𝑘 ) = Θ𝑙 ( 𝑙 ) + Θ̇ ( 𝑙 ) . 𝜕𝑥𝑘 𝜁 𝜕𝑦 𝜁 𝜕𝑥̇ 𝜁 𝜕𝑦̇ 𝜁 (1)

Then, Θ𝑙 = (

𝜕𝑦𝑙 ) Ξ𝑘 , 𝜕𝑥𝑘 𝜑(𝑥)

𝜕 2 𝑦𝑙 𝜕𝑦𝑙 𝑘 Θ̇ = ( 𝑝 𝑘 ) 𝑥̇ 𝑝 (𝜁) Ξ𝑘 + ( 𝑘 ) Ξ̇ . 𝜕𝑥 𝜕𝑥 𝜑(𝑥) 𝜕𝑥 𝜑(𝑥)

(2)

𝑙

The chart on 𝐽1 𝑇𝑋, associated with the chart (𝑈, 𝜑), 𝜑 = (𝑥 ), on 𝑋 is denoted by ((𝜏𝑥1 )−1 (𝑈), Φ1 ), Φ1 = (𝑥𝑖 , 𝑥̇ 𝑖 , 𝑥̇ 𝑖𝑗 ). The coordinates of a 1-jet 𝐽𝑥1 𝜉 are 𝑥𝑖 (𝐽𝑥1 𝜉) = 𝑥𝑖 (𝑥), 𝑥̇ 𝑖 (𝐽𝑥1 𝜉) = 𝑥̇ 𝑖 (𝜉(𝑥)), and 𝑥̇ 𝑖𝑗 (𝐽𝑥1 𝜉) = 𝐷𝑗 (𝑥̇ 𝑖 𝜉𝜑−1 )(𝜑(𝑥)). 𝑖

Lemma 2. For any two charts (𝑈, 𝜑), 𝜑 = (𝑥𝑖 ), and (𝑉, 𝜓), 𝜓 = (𝑦𝑖 ), on 𝑋, such that 𝑈 ∩ 𝑉 ≠ 0, the transformation equations on (𝜏𝑋1 )−1 (𝑈) ∩ (𝜏𝑋1 )−1 (𝑈) are 𝑦̇ 𝑝 =

𝜕𝑦𝑝 𝑚 𝑥̇ , 𝜕𝑥𝑚

𝑦̇ 𝑝𝑞 = (

𝜕2 𝑦𝑝 𝑚 𝜕𝑦𝑝 𝑚 𝜕𝑥𝑠 𝑥̇ + 𝑚 𝑥̇ 𝑠 ) 𝑞 . (3) 𝜕𝑥𝑠 𝜕𝑥𝑚 𝜕𝑥 𝜕𝑦

We need the pullback fibration 𝜏𝑋∗ 𝐽1 𝑇𝑋 = {(𝜁, 𝐽𝑥1 𝜉) ∈ 𝑇𝑋 × 𝐽1 𝑇𝑋|𝜏𝑋 (𝜁) = 𝑥} over 𝑇𝑋. We have the commutative diagram ∗ 1 𝜏𝑋 𝐽 𝑇𝑋

𝐽1 𝑇𝑋 1 𝜏𝑋

𝑇𝑋

𝜏𝑋

𝑋

in which the left vertical arrow is the restriction of the first Cartesian projection 𝑝𝑟1 : 𝑇𝑋 × 𝐽1 𝑇𝑋 → 𝑇𝑋 and the upper horizontal arrow is the restriction of the second Cartesian projection 𝑝𝑟2 : 𝑇𝑋 × 𝐽1 𝑇𝑋 → 𝐽1 𝑇𝑋. We denote

(4)

(5)

Any chart (𝑈, 𝜑), 𝜑 = (𝑥𝑖 ), on 𝑋 induces a chart on the pullback manifold 𝜏𝑋∗ 𝐽1 𝑇𝑋. Denoting by 𝑥𝑖 (𝜁) = 𝑥𝑖 (𝑥), 𝑥̇ 𝑖 (𝜁) = 𝜁𝑖 the coordinates of a vector 𝜁 ∈ 𝜏𝑋−1 (𝑈) at 𝑥 ∈ 𝑈 and by 𝑥𝑖 (𝐽𝑥1 𝜉) = 𝑥𝑖 (𝑥), 𝑥̇ 𝑖 (𝐽𝑥1 𝜉) = 𝜉𝑖 , 𝑥̇ 𝑖𝑗 (𝐽𝑥1 𝜉) = 𝜉𝑗𝑖 the coordinates of a 1-jet 𝐽𝑥1 𝜉 ∈ (𝜏𝑋1 )−1 (𝑈), then the induced chart on 𝜏𝑋∗ 𝐽1 𝑇𝑋, ̃ 1 ), Φ ̃ 1 = (𝑥𝑖 , 𝑥̇ 𝑖 , 𝑧̇ 𝑖 , 𝑧̇ 𝑖 ), is defined by (𝜌𝑥−1 (𝜏𝑋−1 (𝑈)), Φ 𝑗 𝑥𝑖 (𝜁, 𝐽𝑥1 𝜉) = 𝑥𝑖 (𝑥) , 𝑧̇ 𝑖 (𝜁, 𝐽𝑥1 𝜉) = 𝜉𝑖 ,

𝑥̇ 𝑖 (𝜁, 𝐽𝑥1 𝜉) = 𝜁𝑖 , 𝑧̇ 𝑖𝑗 (𝜁, 𝐽𝑥1 𝜉) = 𝜉𝑗𝑖 .

(6)

In these coordinates, 𝜌𝑋 is the mapping (𝑥𝑖 , 𝑥̇ 𝑖 , 𝑧̇ 𝑖 , 𝑧̇ 𝑖𝑗 ) → (𝑥𝑖 , 𝑥̇ 𝑖 ) and the second Cartesian projection is (𝑥𝑖 , 𝑥̇ 𝑖 , 𝑧̇ 𝑖 , 𝑧̇ 𝑖𝑗 ) → (𝑥𝑖 , 𝑥̇ 𝑖 , 𝑧̇ 𝑖𝑗 ).

3. Differential Groups and Differential Invariants Recall that for any positive integer 𝑟, the 𝑟th differential group 𝐿𝑟𝑛 is the group of invertible 𝑟-jets with source and target at the origin 0, endowed with its natural Lie group structure; the group multiplication in 𝐿𝑟𝑛 is the jet composition 𝐿𝑟𝑛 × 𝐿𝑟𝑛 ∋ 𝐽0𝑟 𝛼 ∘ 𝐽0𝑟 𝛽 = 𝐽0𝑟 (𝛼 ∘ 𝛽) ∈ 𝐿𝑟𝑛 . The first differential group 𝐿1𝑛 is just the group 𝐺𝐿 𝑛 (R). For all 𝑟 ≥ 𝑠, we denote by 𝜋𝑟,𝑠 : 𝐿𝑟𝑛 → 𝐿𝑠𝑛 the canonical jet projection and by 𝜄1,𝑟 : 𝐿1𝑛 → 𝐿𝑟𝑛 the canonical injective Lie group morphism. The normal subgroup 𝐾𝑛𝑟 = 𝐾𝑒𝑟𝜋𝑟,1 of 𝐿𝑟𝑛 is nilpotent, and 𝐿𝑟𝑛 is the interior semidirect product of 𝜄1,𝑟 (𝐿1𝑛 ) and 𝐾𝑛𝑟 [13, 17]; we denote 𝐿𝑟𝑛 = 𝐿1𝑛 × 𝑠 𝐾𝑛𝑟 . The first canonical coordinates on 𝐿𝑟𝑛 are the functions 𝑖 𝑎𝑗1 , 𝑎𝑗𝑖 1 𝑗2 , . . . , 𝑎𝑗𝑖 1 𝑗2 ⋅⋅⋅𝑗𝑟 , where 1 ≤ 𝑗1 ≤ 𝑗2 ≤ ⋅ ⋅ ⋅ ≤ 𝑗𝑘 ≤ 𝑛, 𝑘 = 1, 2, . . . , 𝑟, defined as follows. If 𝐽0𝑟 𝛼 ∈ 𝐿𝑟𝑛 is an 𝑟jet and 𝛼 = (𝛼𝑖 ) its representative, then 𝑎𝑗𝑖 1 𝑗2 ⋅⋅⋅𝑗𝑘 (𝐽0𝑟 𝛼) = 𝐷𝑗1 𝐷𝑗2 ⋅ ⋅ ⋅ 𝐷𝑗𝑘 𝛼𝑖 (0). Similarly, the formula 𝑏𝑗𝑖1 𝑗2 ⋅⋅⋅𝑗𝑘 (𝐽0𝑟 𝛼) = 𝑎𝑗𝑖 1 𝑗2 ⋅⋅⋅𝑗𝑘 (𝐽0𝑟 𝛼−1 ) defines the second canonical coordinates 𝑏𝑗𝑖1 , 𝑏𝑗𝑖1 𝑗2 , . . . , 𝑏𝑗𝑖1 𝑗2 ⋅⋅⋅𝑗𝑟 ; clearly, these coordinates satisfy 𝑗

𝑎𝑗𝑖 𝑏𝑘 = 𝛿𝑘𝑖 .

(7)

Equations of the subgroup 𝐾𝑛𝑟 are 𝑎𝑗𝑖 = 𝛿𝑗𝑖 , and equations of the subgroup 𝜄1,𝑟 (𝐿1𝑛 ) are 𝑎𝑗𝑖 1 𝑗2 = 0,. . ., 𝑎𝑗𝑖 1 𝑗2 ⋅⋅⋅𝑗𝑟 = 0. By a differential invariant, we mean an 𝐿𝑟𝑛 -equivariant mapping 𝑓 : 𝑃 → 𝑄 of left 𝐿𝑟𝑛 -manifolds [13]. Given the actions of 𝐿𝑟𝑛 on 𝑃 and 𝑄, we get the equation for the differential invariant 𝑓: 𝑓 (𝐽0𝑟 𝛼 ⋅ 𝑝) = 𝐽0𝑟 𝛼 ⋅ 𝑓 (𝑝) ,

(8)

Geometry

3

where 𝑝 ∈ 𝑃 and 𝐽0𝑟 𝛼 ∈ 𝐿𝑟𝑛 . Equation (8) splits to an equivalent system: 𝑓 (𝐽0𝑟 𝛼

⋅ 𝑝) =

𝐽0𝑟 𝛼

𝑓 (𝐽0𝑟 𝛽

⋅ 𝑓 (𝑝) ,

⋅ 𝑝) =

𝐽0𝑟 𝛽

⋅ 𝑓 (𝑝) , (9)

where 𝐽0𝑟 𝛼 ∈ 𝐿1𝑛 and 𝐽0𝑟 𝛽 ∈ 𝐾𝑛𝑟 (the orbit reduction method [18]). The problem we consider in this paper reduces to solving (9) for some specific left 𝐿2𝑛 -manifolds 𝑃 and 𝑄. In our case, the orbit reduction method simplifies (8) and allows us to obtain its complete solution.

(1) Ξ is a canonical vector field. (2) For any points 𝑥, 𝑦 ∈ 𝑋, any charts (𝑈, 𝜑), 𝜑 = (𝑥𝑖 ), at 𝑥 and (𝑉, 𝜓), 𝜓 = (𝑦𝑖 ), at 𝑦, and any diffeomorphism 𝛼 : 𝑈 → 𝑉 such that 𝛼(𝑥) = 𝑦, 𝑖 𝑘 Ξ𝑙(𝑉,𝜓) (𝑎𝑘𝑖 𝜁𝑘 , 𝑎𝑘𝑖 𝜉𝑘 , 𝑎𝑘𝑚 𝑏𝑗𝑚 𝜉𝑘 + 𝑎𝑘𝑖 𝑏𝑗𝑚 𝜉𝑚 )

= 𝑎𝑘𝑙 Ξ𝑘(𝑈,𝜑) (𝜁𝑖 , 𝜉𝑖 , 𝜉𝑗𝑖 ) ,

𝑘 𝑙 𝑗 𝑘 𝑖 𝑖 = 𝑎𝑗𝑘 𝜁 Ξ(𝑈,𝜑) (𝜁𝑖 , 𝜉𝑖 , 𝜉𝑚 ) + 𝑎𝑘𝑙 Ξ̇ (𝑈,𝜑) (𝜁𝑖 , 𝜉𝑖 , 𝜉𝑚 ),

4. Canonical Vector Fields on Tangent Bundles Any diffeomorphism 𝛼 : 𝑈 → 𝑋, defined on an open set 𝑈 in 𝑋, induces the corresponding lifted diffeomorphisms 𝑇𝛼 : 𝑇𝑈 → 𝑇𝑋, 𝐽1 𝑇𝛼 : 𝐽1 𝑇𝑈 → 𝐽1 𝑇𝑋, and (𝑇𝛼, 𝐽1 𝑇𝛼) : 𝜏𝑋∗ 𝐽1 𝑇𝑈 → 𝜏𝑋∗ 𝐽1 𝑇𝑋. By a canonical vector field on 𝑇𝑋, we mean a morphism Ξ : 𝜏𝑋∗ 𝐽1 𝑇𝑋 → 𝑇2 𝑋 such that 𝜏𝑇𝑋 ∘ Ξ = 𝜌𝑋 ,

(17)

𝑙 𝑖 𝑘 Ξ̇ (𝑉,𝜓) (𝑎𝑘𝑖 𝜁𝑘 , 𝑎𝑘𝑖 𝜉𝑘 , 𝑎𝑘𝑚 𝑏𝑗𝑚 𝜉𝑘 + 𝑎𝑘𝑖 𝑏𝑗𝑚 𝜉𝑚 )

where 𝑎𝑘𝑙 = (

𝜕(𝑦𝑙 𝛼𝜑−1 ) ) , 𝜕𝑥𝑘 𝜑(𝑥) 𝑏𝑗𝑘

(10)

𝑙 𝑎𝑗𝑘 =(

𝜕2 (𝑦𝑙 𝛼𝜑−1 ) ) , 𝜕𝑥𝑗 𝜕𝑥𝑘 𝜑(𝑥)

𝜕(𝑥𝑘 𝛼−1 𝜓−1 ) =( ) . 𝜕𝑦𝑗 𝜓(𝑥)

(18)

and for all diffeomorphisms 𝛼 : 𝑈 → 𝑋 of 𝑋, Ξ ∘ (𝑇𝛼, 𝐽1 𝑇𝛼) = 𝑇 (𝑇𝛼) ∘ Ξ.

(11)

Condition (10) means that the diagram ∗ 1 𝐽 𝑇𝑋 𝜏𝑋

Ξ

𝑦̇ 𝑖 (𝑇𝑥 𝛼 ⋅ 𝜁) = 𝑎𝑘𝑖 𝜁𝑘 ,

𝑇2 𝑋

𝜌𝑋

𝜏𝑇𝑋

𝑇𝑋

Proof. Consider condition (14). With the abbreviations (18), 1 the coordinates of the vector 𝑇𝑥 𝛼 ⋅ 𝜁 and the 1-jet 𝐽𝛼(𝑥) (𝑇𝛼 ∘ 𝜉 ∘ −1 𝛼 ) are

id 𝑇𝑋

1 (𝑇𝛼 ∘ 𝜉 ∘ 𝛼−1 )) = 𝑎𝑘𝑖 𝜉𝑘 , 𝑦̇ 𝑖 (𝐽𝛼(𝑥)

(12)

𝑇𝑋

1 𝑖 𝑙 𝑘 (𝑇𝛼 ∘ 𝜉 ∘ 𝛼−1 )) = 𝑎𝑘𝑙 𝑏𝑗 𝜉 + 𝑎𝑘𝑖 𝑏𝑗𝑙 𝜉𝑙𝑘 . 𝑦̇ 𝑖𝑗 (𝐽𝛼(𝑥)

commutes or, which is the same, for all (𝜁, 𝐽𝑥1 𝜉) ∈ 𝜏𝑋∗ 𝐽1 𝑇𝑋, Ξ(𝜁, 𝐽𝑥1 𝜉) is a second-order vector on 𝑋 at the point 𝜁 ∈ 𝑇𝑋, 𝜏𝑇𝑋 (Ξ (𝜁, 𝐽𝑥1 𝜉)) = 𝜁.

The coordinates of the vector 𝑇𝜁 𝑇𝛼 ⋅ Ξ(𝜁, 𝐽𝑥1 𝜉) are determined by

(13)

𝑇𝜁 𝑇𝛼 ⋅ Ξ (𝜁, 𝐽𝑥1 𝜉)

1 Since 𝐽1 𝑇𝛼(𝐽𝑥1 𝜉) = 𝐽𝛼(𝑥) (𝑇𝛼 ∘ 𝜉 ∘ 𝛼−1 ), condition (11) can also be written as 1 (𝑇𝛼 ∘ 𝜉 ∘ 𝛼−1 )) = 𝑇𝜁 𝑇𝛼 ⋅ Ξ (𝜁, 𝐽𝑥1 𝜉) . Ξ (𝑇𝑥 𝛼 ⋅ 𝜁, 𝐽𝛼(𝑥)

= 𝑎𝑘𝑙 Ξ𝑘(𝑈,𝜑) (𝜁, 𝐽𝑥1 𝜉) (

(14)

𝜕 𝜕 𝑘 ) + Ξ̇ (𝑈,𝜑) (𝜁, 𝐽𝑥1 𝜉) ( 𝑘 ) . 𝑘 𝜕𝑥 𝜁 𝜕𝑥̇ 𝜁 (15)

Then for fixed 𝑥, the components are functions of the coordinates 𝜁𝑖 , 𝜉𝑖 , and 𝜉𝑙𝑖 ; that is, Ξ𝑘(𝑈,𝜑)

=

Ξ𝑘(𝑈,𝜑)

𝑖

(𝜁 , 𝜉

𝑖

, 𝜉𝑙𝑖 ) ,

𝜏𝑋∗ 𝐽1 𝑇𝑋

𝑘 𝑘 Ξ̇ (𝑈,𝜑) = Ξ̇ (𝑈,𝜑) (𝜁𝑖 , 𝜉𝑖 , 𝜉𝑙𝑖 ) . (16) 2

→ 𝑇 𝑋 be a morphism over the Theorem 3. Let Ξ : identity id𝑇𝑋 . The following two conditions are equivalent:

𝜕 ) 𝜕𝑦𝑙 𝑇𝑥 𝛼⋅𝜁 (20)

𝑙 𝑗 + (𝑎𝑘𝑗 𝑥̇ (𝜁) Ξ𝑘(𝑈,𝜑) (𝜁, 𝐽𝑥1 𝜉)

We express conditions (13) and (14) in coordinates. We write in a chart (𝑈, 𝜑), 𝜑 = (𝑥𝑖 ), Ξ (𝜁, 𝐽𝑥1 𝜉) = Ξ𝑘(𝑈,𝜑) (𝜁, 𝐽𝑥1 𝜉) (

(19)

𝑘

+𝑎𝑘𝑙 Ξ̇ (𝑈,𝜑) (𝜁, 𝐽𝑥1 𝜉)) (

𝜕 ) . 𝜕𝑦̇ 𝑙 𝑇𝑥 𝛼⋅𝜁

Writing 1 Ξ (𝑇𝑥 𝛼 ⋅ 𝜁, 𝐽𝛼(𝑥) (𝑇𝛼 ∘ 𝜉 ∘ 𝛼−1 )) 1 = Ξ𝑙(𝑉,𝜓) (𝑇𝑥 𝛼 ⋅ 𝜁, 𝐽𝛼(𝑥) (𝑇𝛼 ∘ 𝜉 ∘ 𝛼−1 )) (

𝑙

𝜕 ) 𝜕𝑦𝑙 𝑇𝑥 𝛼⋅𝜁

1 + Ξ̇ (𝑉,𝜓) (𝑇𝑥 𝛼 ⋅ 𝜁, 𝐽𝛼(𝑥) (𝑇𝛼 ∘ 𝜉 ∘ 𝛼−1 )) (

𝜕 ) , 𝜕𝑦̇ 𝑙 𝑇𝑥 𝛼⋅𝜁 (21)

4

Geometry

we can express condition (14), with help of (19) and (20), as 𝑖 𝑏𝑗𝑚 𝑥̇ 𝑘 (𝐽𝑥1 𝜉)+𝑎𝑘𝑖 𝑏𝑗𝑚 𝑥̇ 𝑘𝑚 (𝐽𝑥1 𝜉)) Ξ𝑙(𝑉,𝜓) (𝑎𝑘𝑖 𝑥̇ 𝑘 (𝜁) , 𝑎𝑘𝑖 𝑥̇ 𝑘 (𝐽𝑥1 𝜉) , 𝑎𝑘𝑚

= 𝑎𝑘𝑙 Ξ𝑘(𝑈,𝜑) (𝜁, 𝐽𝑥1 𝜉) , 𝑙 𝑖 Ξ̇ (𝑉,𝜓) (𝑎𝑘𝑖 𝑥̇ 𝑘 (𝜁) , 𝑎𝑘𝑖 𝑥̇ 𝑘 (𝐽𝑥1 𝜉) , 𝑎𝑘𝑚 𝑏𝑗𝑚 𝑥̇ 𝑘 (𝐽𝑥1 𝜉)

𝑎𝑘𝑙 =

+𝑎𝑘𝑖 𝑏𝑗𝑚 𝑥̇ 𝑘𝑚 (𝐽𝑥1 𝜉) ) =

𝑙 𝑗 𝑎𝑗𝑘 𝑥̇

(𝜁) Ξ𝑘(𝑈,𝜑)

(𝜁, 𝐽𝑥1 𝜉)

+

𝑘 𝑎𝑘𝑙 Ξ̇ (𝑈,𝜑)

(𝜁, 𝐽𝑥1 𝜉) .

𝑏𝑟𝑝 (22)

In the well-known sense, the canonical vector fields are completely determined by certain differential invariants, that 2 is, equivariant mappings from the type fibre R𝑛 × R𝑛 × R𝑛 of 𝜏𝑋∗ 𝐽1 𝑇𝑋 into the type fibre R𝑛 × R𝑛 × R𝑛 of 𝑇2 𝑋 over 𝑋 with respect to the canonical actions of the differential group 𝐿2𝑛 , induced by diffeomorphisms of 𝑋. We can characterize these actions explicitly in terms of the first and second canonical 𝑖 and 𝑏𝑟𝑝 , 𝑏𝑟𝑠𝑝 on the differential group 𝐿2𝑛 . coordinates 𝑎𝑗𝑖 , 𝑎𝑗𝑘 Note that 𝑎𝑝𝑙 𝑏𝑟𝑝 If 𝐴 =

𝑖 (𝑎𝑗𝑖 , 𝑎𝑗𝑘 )

𝛿𝑟𝑙 ,

=

𝑙 𝑎𝑝𝑞 𝑏𝑟𝑝

+

𝑎𝑝𝑙 𝑎𝑞𝑠 𝑏𝑟𝑠𝑝

= 0.

(23)

is an element of the differential group 𝑛

𝑛

𝐿2𝑛 , 𝑛2

denote by 𝛼𝐴 (resp., 𝛽𝐴 ) the transformation of R × R × R (resp., R𝑛 × R𝑛 × R𝑛 ), defined by 𝐴. For any points (𝜁𝑖 , 𝜉𝑖 , 𝜉𝑗𝑖 ) ∈ 2 𝑖 R𝑛 × R𝑛 × R𝑛 and (𝑍𝑖 , Ξ𝑖 , Ξ̇ ) ∈ R𝑛 × R𝑛 × R𝑛 , we denote 𝑖

𝑖

𝑖

𝑖

𝑖

𝑖

(24)

The following lemma defines the points (24) explicitly. 𝑖 ) be an element of the differential Lemma 4. Let 𝐴 = (𝑎𝑗𝑖 , 𝑎𝑗𝑘 group 𝐿2𝑛 . 2

(a) The canonical group action of 𝐿2𝑛 on R𝑛 × R𝑛 × R𝑛 is given by the equations: 𝑖

𝑖

𝜉 = 𝑎𝑘𝑖 𝜉𝑘 ,

𝑖

𝑖 𝑘 𝜉𝑗 = 𝑎𝑘𝑚 𝑏𝑗𝑚 𝜉𝑘 + 𝑎𝑘𝑖 𝑏𝑗𝑚 𝜉𝑚 .

(25)

(b) The canonical group action of 𝐿2𝑛 on R𝑛 × R𝑛 × R𝑛 is given by the equations: 𝑖

𝑎𝑘𝑖 𝑍𝑘 ,

𝑍 = 𝑖

𝜕𝑥𝑝 = 𝑟, 𝜕𝑦

𝑙 𝑎𝑘𝑝 =

𝑏𝑟𝑠𝑝

𝜕 2 𝑦𝑙 , 𝜕𝑥𝑝 𝜕𝑥𝑘

𝜕2 𝑥𝑝 = 𝑟 𝑠. 𝜕𝑦 𝜕𝑦

(27)

Since a canonical vector field is always a morphism over the identity mapping id𝑇𝑋 , the corresponding differential 2 invariant 𝐹 : R𝑛 × R𝑛 × R𝑛 → R𝑛 × R𝑛 × R𝑛 has an expression 𝑗 (𝜁𝑖 , 𝜉𝑖 , 𝜉𝑗𝑖 ) → 𝐹(𝜁𝑖 , 𝜉𝑖 , 𝜉𝑗𝑖 ) = (𝜁𝑖 , 𝐹𝑗 (𝜁𝑖 , 𝜉𝑖 , 𝜉𝑗𝑖 ), 𝐹̇ (𝜁𝑖 , 𝜉𝑖 , 𝜉𝑗𝑖 )); we 𝑖 denote 𝐹 = (𝐹𝑖 , 𝐹̇ ). 𝑖 Theorem 5. A mapping 𝐹 = (𝐹𝑖 , 𝐹̇ ) is a differential invariant if and only if 𝑝 𝑠 𝑚 𝑝 𝑠 𝑚 𝑏𝑞 𝜉 + 𝑎𝑚 𝑏𝑞 𝜉𝑠 ) 𝐹𝑖 (𝑎𝑠𝑝 𝜁𝑠 , 𝑎𝑠𝑝 𝜉𝑠 , 𝑎𝑠𝑚

= 𝑎𝑘𝑖 𝐹𝑘 (𝜁𝑠 , 𝜉𝑠 , 𝜉𝑠𝑚 ) , 𝑖 𝑝 𝑠 𝑚 𝑝 𝑠 𝑚 𝐹̇ (𝑎𝑠𝑝 𝜁𝑠 , 𝑎𝑠𝑝 𝜉𝑠 , 𝑎𝑠𝑚 𝑏𝑞 𝜉 + 𝑎𝑚 𝑏𝑞 𝜉𝑠 )

(28)

𝑘 𝑖 𝑝 𝑘 𝜁 𝐹 (𝜁𝑠 , 𝜉𝑠 , 𝜉𝑠𝑚 ) + 𝑎𝑘𝑖 𝐹̇ (𝜁𝑠 , 𝜉𝑠 , 𝜉𝑠𝑚 ) , = 𝑎𝑘𝑝 2

(𝑍 , Ξ , Ξ̇ ) = 𝛽𝐴 (𝑍𝑖 , Ξ𝑖 , Ξ̇ ) .

𝜁 = 𝑎𝑘𝑖 𝜁𝑘 ,

𝜕𝑦𝑙 , 𝜕𝑥𝑘

𝑖 ) ∈ 𝐿2𝑛 . for all (𝜁𝑠 , 𝜉𝑠 , 𝜉𝑠𝑚 ) ∈ R𝑛 × R𝑛 × R𝑛 and (𝑎𝑗𝑖 , 𝑎𝑗𝑘

(𝜁 , 𝜉 , 𝜉𝑗 ) = 𝛼𝐴 (𝜁𝑖 , 𝜉𝑖 , 𝜉𝑗𝑖 ) , 𝑖

Proof. The left 𝐿2𝑛 -actions on the type fibres of 𝜏𝑋∗ 𝐽1 𝑇𝑋 and 𝑇2 𝑋 can be derived from the general theory of differential groups and differential invariants [13]. Note that these 𝐿2𝑛 actions can also be defined by transformation properties of components of jets and tangent vectors; compare with Lemmas 1 and 2 with the substitution

𝑖

Ξ =

𝑎𝑘𝑖 Ξ𝑘 ,

𝑘 𝑖 𝑗 𝑘 Ξ̇ = 𝑎𝑗𝑘 𝜁 Ξ + 𝑎𝑘𝑖 Ξ̇ .

Proof. Equations (28) are direct consequences of (25) and (26). Remark 6. Note that our definition of the canonical vector field differs from the lifting of a vector field from a base manifold to its tangent bundle, which is defined by means of the lifting of diffeomorphisms and 1-parameter groups of diffeomorphisms to the tangent bundle [2, 9]. Remark 7. We can specify Theorem 3 to diffeomorphisms 𝛼, preserving a given point 𝑥 ∈ 𝑋, such that 𝛼(𝑥) = 𝑥, and to charts (𝑈, 𝜑) and (𝑉, 𝜓) such that (𝑈, 𝜑) = (𝑉, 𝜓) 𝑘 and 𝜑(𝑥) = 0. Then, the components Ξ𝑙(𝑈,𝜑) and Ξ̇ (𝑈,𝜑) of a canonical vector field Ξ satisfy 𝑖 𝑘 Ξ𝑙(𝑈,𝜑) (𝑎𝑘𝑖 𝜁𝑘 , 𝑎𝑘𝑖 𝜉𝑘 , 𝑎𝑘𝑚 𝑏𝑗𝑚 𝜉𝑘 + 𝑎𝑘𝑖 𝑏𝑗𝑚 𝜉𝑚 ) 𝑘 ), = 𝑎𝑘𝑙 Ξ𝑘(𝑈,𝜑) (𝜁𝑘 , 𝜉𝑘 , 𝜉𝑚 𝑙 𝑖 𝑘 Ξ̇ (𝑈,𝜑) (𝑎𝑘𝑖 𝜁𝑘 , 𝑎𝑘𝑖 𝜉𝑘 , 𝑎𝑘𝑚 𝑏𝑗𝑚 𝜉𝑘 + 𝑎𝑘𝑖 𝑏𝑗𝑚 𝜉𝑚 )

(26)

𝑘

𝑙 𝑗 𝑘 𝑘 𝑘 = 𝑎𝑗𝑘 𝜁 Ξ(𝑈,𝜑) (𝜁𝑘 , 𝜉𝑘 , 𝜉𝑚 ) + 𝑎𝑘𝑙 Ξ̇ (𝑈,𝜑) (𝜁𝑘 , 𝜉𝑘 , 𝜉𝑚 ),

(29)

Geometry

5 Properties of 𝐹𝑖 are completely determined by the subgroup 𝐿1𝑛 of 𝐿2𝑛 . If 𝑎𝑠𝑝 = 𝜆𝛿𝑠𝑝 , 𝜆 ≠ 0, then by (38) 𝐹𝑖 satisfies the positive homogeneity condition

where 𝑎𝑘𝑙 = (

𝜕(𝑥𝑙 𝛼𝜑−1 ) ), 𝜕𝑥𝑘 0

𝑙 𝑎𝑗𝑘 =(

𝜕2 (𝑥𝑙 𝛼𝜑−1 ) ), 𝜕𝑥𝑗 𝜕𝑥𝑘 0

𝜕(𝑥𝑘 𝛼−1 𝜓−1 ) 𝑏𝑗𝑘 = ( ). 𝜕𝑥𝑗 0

5. Canonical Vector Fields: Classification 𝑖 We find all solutions 𝐹 = (𝐹𝑖 , 𝐹̇ ) of the equations for differential invariants, associated with canonical vector fields Ξ : 𝜏𝑋∗ 𝐽1 𝑇𝑋 → 𝑇2 𝑋 (Theorem 5). These equations can be written in coordinates as 𝑝 𝑠 𝑚 𝑝 𝑠 𝑚 𝑏𝑞 𝑧̇ + 𝑎𝑚 𝑏𝑞 𝑧̇ 𝑠 ) 𝐹𝑖 (𝑎𝑠𝑝 𝑥̇ 𝑠 , 𝑎𝑠𝑝 𝑧̇ 𝑠 , 𝑎𝑠𝑚

(31)

= 𝑎𝑘𝑖 𝐹𝑘 (𝑥̇ 𝑠 , 𝑧̇ 𝑠 , 𝑧̇ 𝑚 𝑠 ), 𝑖 𝑝 𝑠 𝑚 𝑝 𝑠 𝑚 𝐹̇ (𝑎𝑠𝑝 𝑥̇ 𝑠 , 𝑎𝑠𝑝 𝑧̇ 𝑠 , 𝑎𝑠𝑚 𝑏𝑞 𝑧̇ + 𝑎𝑚 𝑏𝑞 𝑧̇ 𝑠 )

=

𝑖 𝑎𝑘𝑝 𝑥̇ 𝑝 𝐹𝑘

𝑠

𝑠

(𝑥̇ , 𝑧̇

, 𝑧̇ 𝑚 𝑠 )

+

𝑘 𝑎𝑘𝑖 𝐹̇

𝑠

𝑠

(𝑥̇ , 𝑧̇

, 𝑧̇ 𝑚 𝑠 ).

𝐹𝑖 (𝜆𝑥̇ 𝑗 , 𝜆𝑧̇ 𝑘 ) = 𝜆𝐹𝑖 (𝑥̇ 𝑗 , 𝑧̇ 𝑘 ) .

(30)

(39)

We suppose, however, that the functions 𝐹𝑖 are defined at the origin 𝑥̇ 𝑝 = 0, 𝑧̇ 𝑝 = 0; then it is easily seen that (39) also holds for 𝜆 = 0. Indeed, in this case, we have 𝐹𝑖 (𝜆 ⋅ 0, 𝜆 ⋅ 0) = 𝐹𝑖 (0, 0) = 𝜆𝐹𝑖 (0, 0) for all 𝜆 ≠ 0, hence 𝐹𝑖 (0, 0) = 0. On the other hand, the points 𝜆𝑥̇ 𝑗 and 𝜆𝑧̇ 𝑘 are always defined for 𝜆 = 0; then 𝐹𝑖 (0 ⋅ 𝑥̇ 𝑗 , 0 ⋅ 𝑧̇ 𝑘 ) = 𝐹𝑖 (0, 0) = 0 = 0 ⋅ 𝐹𝑖 (𝑥̇ 𝑗 , 𝑧̇ 𝑘 ), which proves (39) for 𝜆 = 0. Then, we have from (28) 𝐷1,𝑞 𝐹𝑖 (𝜆𝑥̇ 𝑙 , 𝜆𝑧̇ 𝑙 )𝑥̇ 𝑞 + 𝐷2,𝑞 𝐹𝑖 (𝜆𝑥̇ 𝑙 , 𝜆𝑧̇ 𝑙 )𝑧̇ 𝑞 = 𝐹𝑖 (𝑥̇ 𝑙 , 𝑧̇ 𝑙 ) by differentiation with respect to 𝜆; we see that the expression on the left does not depend on 𝜆. For 𝜆 = 0, we get 𝐷1,𝑞 𝐹𝑖 (0, 0)𝑥̇ 𝑞 +𝐷2,𝑞 𝐹𝑖 (0, 0)𝑧̇ 𝑞 = 𝐹𝑖 (𝑥̇ 𝑙 , 𝑧̇ 𝑙 ), showing that 𝐹𝑖 is linear in 𝑥̇ 𝑞 and 𝑧̇ 𝑞 ; that is, 𝐹𝑖 (𝑥̇ 𝑗 , 𝑧̇ 𝑗 ) = 𝑆𝑞𝑖 𝑥̇ 𝑞 + 𝑇𝑞𝑖 𝑧̇ 𝑖 .

(32)

(40)

Substituting now into (38), we get

The following lemma solves (31). Lemma 8. The functions 𝐹𝑖 satisfy condition (31) if and only if

𝑆𝑞𝑖 𝑎𝑠𝑞 𝑥̇ 𝑠 + 𝑇𝑞𝑖 𝑎𝑠𝑞 𝑧̇ 𝑠 = 𝑎𝑘𝑖 (𝑆𝑞𝑘 𝑥̇ 𝑞 + 𝑇𝑞𝑘 𝑧̇ 𝑞 ) .

(41)

𝑞

(33)

That is, 𝑆𝑞𝑖 𝑎𝑠𝑞 = 𝑎𝑞𝑖 𝑆𝑠𝑞 and 𝑇𝑞𝑖 𝑎𝑠𝑞 = 𝑎𝑞𝑖 𝑇𝑠𝑞 . Thus, 𝑆𝑞𝑖 𝑎𝑙 𝛿𝑠𝑙 = 𝛿𝑙𝑖 𝑎𝑞𝑙 𝑆𝑠𝑞 ,

Proof. (1) First we consider (31) for the group elements, belonging to the subgroup 𝐾𝑛2 of 𝐿2𝑛 ; equations of 𝐾𝑛2 are

Analogously, 𝑛𝑇𝑞𝑖 = 𝛿𝑞𝑖 𝑇𝑙𝑙 . These expressions together with (40) prove formula (33). (3) If condition (33) is satisfied, then we get (31) by immediate substitution.

𝑗

𝐹𝑖 (𝑥̇ 𝑗 , 𝑧̇ 𝑗 , 𝑧̇ 𝑘 ) = 𝐴𝑥̇ 𝑖 + 𝐵𝑧̇ 𝑖 , for some constants 𝐴, 𝐵 ∈ R.

𝑎𝑠𝑝 = 𝛿𝑠𝑝 ,

(34)

𝑝 𝑚 𝐹𝑖 (𝑥̇ 𝑝 , 𝑧̇ 𝑝 , 𝑎𝑞𝑚 𝑧̇ + 𝑧̇ 𝑝𝑞 ) = 𝐹𝑖 (𝑥̇ 𝑠 , 𝑧̇ 𝑠 , 𝑧̇ 𝑚 𝑠 ).

(35)

and we get the system

This equation is obviously satisfied at all points where 𝑧̇ 𝑚 = 0. On the other hand, suppose that there exists at least one index 𝑗 such that 𝑧̇ 𝑗 ≠ 0; then to every point 𝑦̇ 𝑝𝑞 one can find the 𝑝 𝑝 𝑚 group parameters 𝑎𝑞𝑚 𝑧̇ + 𝑧̇ 𝑝𝑞 = 𝑦̇ 𝑝𝑞 . Indeed, if, such that 𝑎𝑞𝑚 for example, 𝑧̇ 1 ≠ 0, we set 𝑝

𝑎𝑞,1 =

𝑦̇ 𝑝𝑞 − 𝑧̇ 𝑝𝑞 𝑧̇ 1

𝑝

𝑝

𝑝 𝑎𝑞,2 , 𝑎𝑞,3 , . . . , 𝑎𝑞,𝑛 = 0.

,

(36)

hence 𝑆𝑞𝑖 𝛿𝑠𝑙 = 𝛿𝑞𝑖 𝑆𝑠𝑙 and by the trace operation, 𝑛𝑆𝑞𝑖 = 𝛿𝑞𝑖 𝑆𝑙𝑙 .

Now, we wish to solve (32). In view of Lemma 8, these equations are of the form 𝑖 𝑝 𝑠 𝑚 𝑝 𝑠 𝑚 𝑏𝑞 𝑧̇ + 𝑎𝑚 𝑏𝑞 𝑧̇ 𝑠 ) 𝐹̇ (𝑎𝑠𝑝 𝑥̇ 𝑠 , 𝑎𝑠𝑝 𝑧̇ 𝑠 , 𝑎𝑠𝑚 𝑘

𝑖 = 𝑎𝑘𝑝 𝑥̇ 𝑝 (𝐴𝑥̇ 𝑘 + 𝐵𝑧̇ 𝑘 ) + 𝑎𝑘𝑖 𝐹̇ (𝑥̇ 𝑠 , 𝑧̇ 𝑠 , 𝑧̇ 𝑚 𝑠 ).

We prove separately the following lemma. Lemma 9. The following two conditions are equivalent: 𝑖𝑞

𝑝 𝑠𝑘 = 𝑎𝑘𝑙 𝑎𝑠𝑖 𝑄𝑚 (𝑥̇ 𝑗 , 𝑧̇ 𝑗 ) . 𝑄𝑝𝑖𝑙 (𝑎𝑠𝑗 𝑥̇ 𝑠 , 𝑎𝑠𝑗 𝑧̇ 𝑠 ) 𝑎𝑚

(43)

𝑖𝑞

(37)

𝑗

which shows that 𝐹𝑖 (𝑥̇ 𝑗 , 𝑥̇ 𝑗 , 𝑥̇ 𝑘 ) is independent of 𝑥̇ 𝑝𝑞 . (2) In view of (37), we can write 𝐹𝑖 (𝑥̇ 𝑗 , 𝑥̇ 𝑗 ) instead of 𝑗 𝐹𝑖 (𝑥̇ 𝑗 , 𝑥̇ 𝑗 , 𝑥̇ 𝑘 ). Turning back to conditions (31), we have the following equations for the functions 𝐹𝑖 : 𝐹𝑖 (𝑎𝑠𝑗 𝑥̇ 𝑠 , 𝑎𝑠𝑗 𝑧̇ 𝑠 ) = 𝑎𝑘𝑖 𝐹𝑘 (𝑥̇ 𝑗 , 𝑧̇ 𝑗 ) .

𝑖𝑞

(a) The functions 𝑄𝑝 = 𝑄𝑝 (𝑥̇ 𝑗 , 𝑧̇ 𝑗 ) satisfy

This choice of the group parameters yields 𝐹𝑖 (𝑥̇ 𝑝 , 𝑧̇ 𝑝 , 𝑦̇ 𝑝𝑞 ) = 𝐹𝑖 (𝑥̇ 𝑝 , 𝑧̇ 𝑝 , 𝑧̇ 𝑚 𝑠 ),

(42)

(38)

(b) The functions 𝑄𝑝 are of the form 𝑖𝑘 (𝑥̇ 𝑗 , 𝑧̇ 𝑗 ) 𝑄𝑚 𝑖 𝑘 𝑖 𝑖 𝑘 𝑖 + 𝛿𝑚 𝛿𝑗 ) + 𝑄 (𝛿𝑗𝑘 𝛿𝑚 − 𝛿𝑚 𝛿𝑗 )) 𝑥̇ 𝑗 = (𝑃 (𝛿𝑗𝑘 𝛿𝑚 𝑖 𝑘 𝑖 𝑖 𝑘 𝑖 + (𝑅 (𝛿𝑗𝑘 𝛿𝑚 + 𝛿𝑚 𝛿𝑗 ) + 𝑆 (𝛿𝑗𝑘 𝛿𝑚 − 𝛿𝑚 𝛿𝑗 )) 𝑧̇ 𝑗 ,

where 𝑃, 𝑄, 𝑅, 𝑆 ∈ R.

(44)

6

Geometry

𝑖𝑙 Proof. (1) If 𝑎𝑗𝑖 = 𝜆𝛿𝑗𝑖 , Lemma 9, equation (43) 𝜆𝑄𝑚 (𝜆𝑥̇ 𝑗 , 𝜆𝑧̇ 𝑗 ) 2 𝑖𝑙 𝑗 𝑗 = 𝜆 𝑄𝑚 (𝑥̇ , 𝑧̇ ) yields 𝑖𝑙 𝑖𝑙 𝑗 𝑖𝑙 𝑗 (𝑥̇ 𝑗 , 𝑧̇ 𝑗 ) = 𝑆𝑚𝑗 𝑧̇ , 𝑄𝑚 𝑥̇ + 𝑇𝑚𝑗

(45)

𝑖𝑙 𝑖𝑙 , 𝑆𝑚𝑗 ∈ R (Lemma 8). Substituting back to (43), for some 𝑆𝑚𝑗

𝑖𝑙 𝑖𝑙 we get conditions for the coefficients 𝑆𝑚𝑗 , 𝑆𝑚𝑗 ∈ R: 𝑖𝑙 𝑗 𝑘 𝑎𝑘 𝑥̇ (𝑆𝑠𝑗

+

𝑗 𝑠 𝑇𝑠𝑗𝑖𝑙 𝑎𝑘 𝑧̇ 𝑘 ) 𝑎𝑚

(46)

𝑗

𝑖𝑙 𝑡 𝑘 𝑘𝑡 𝑟 𝑎𝑡 𝑎𝑘𝑠 = 𝛿𝑗 𝛿𝑠𝑖 (𝑆𝑚𝑟 or, which is the same, (𝑆𝑠𝑗 𝑥̇ + 𝑇𝑠𝑗𝑖𝑙 𝑧̇ 𝑡 )𝛿𝑚 𝑥̇ + 𝑗

𝑡𝑘 𝑟 𝑧̇ )𝑎𝑡 𝑎𝑘𝑠 ; that is, 𝑇𝑚𝑟 𝑗

𝑗

𝑖𝑙 𝑝 𝑠𝑘 𝑆𝑝𝑗 𝑎𝑡 𝑎𝑚 = 𝑎𝑘𝑙 𝑎𝑠𝑖 𝑆𝑚𝑡 ,

𝑖𝑙 𝑝 𝑠𝑘 𝑇𝑝𝑗 𝑎𝑡 𝑎𝑚 = 𝑎𝑘𝑙 𝑎𝑠𝑖 𝑇𝑚𝑡 .

𝑠𝑘 𝑎𝑘 𝑎𝑠𝑝 𝛿𝑗𝑙 𝛿𝑝𝑖 𝑆𝑚𝑡 from which it follows that

+

𝑖𝑙 𝑘 𝑠 𝑆𝑗𝑝 𝛿𝑡 𝛿𝑚

=

𝑠𝑘 𝛿𝑗𝑙 𝛿𝑝𝑖 𝑆𝑚𝑡

+

𝑞 𝑞 + (𝑅 (𝛿𝑗 𝛿𝑝𝑖 + 𝛿𝑝𝑞 𝛿𝑗𝑖 ) + 𝑆 (𝛿𝑗 𝛿𝑝𝑖 − 𝛿𝑝𝑞 𝛿𝑗𝑖 )) 𝑧̇ 𝑗 , 𝑠𝑘 𝑠 𝑘 𝑠 𝑠 𝑘 𝑠 𝑄𝑚 = (𝑃 (𝛿𝑗𝑘 𝛿𝑚 + 𝛿𝑚 𝛿𝑗 ) + 𝑄 (𝛿𝑗𝑘 𝛿𝑚 − 𝛿𝑚 𝛿𝑗 )) 𝑥̇ 𝑗

𝑝 𝑘 𝑄𝑝𝑖𝑞 (𝑎𝑠𝑗 𝑥̇ 𝑠 , 𝑎𝑠𝑗 𝑧̇ 𝑠 ) 𝑎𝑚 𝑏𝑞

𝑖 𝑘 𝑘 𝑖 𝑡 𝑖 𝑘 𝑘 𝑖 𝑡 𝑧̇ + 𝑅𝛿𝑚 𝑧̇ − 𝑆𝛿𝑚 + 𝑅𝑎𝑚 𝑎𝑡 𝑧̇ + 𝑆𝑎𝑚 𝑎𝑡 𝑧̇ , 𝑠𝑘 (𝑥̇ 𝑗 , 𝑧̇ 𝑗 ) 𝑎𝑠𝑖 𝑄𝑚

(48)

𝑘𝑠 𝛿𝑝𝑙 𝛿𝑗𝑖 𝑆𝑚𝑠 and

𝑖 𝑘 𝑘 𝑖 𝑠 𝑖 𝑘 𝑘 𝑖 𝑠 𝑧̇ + 𝑅𝛿𝑚 𝑧̇ − 𝑆𝛿𝑚 + 𝑅𝑎𝑚 𝑎𝑠 𝑧̇ + 𝑆𝑎𝑚 𝑎𝑠 𝑧̇ ,

proving (13). Now, we are in position to give a solution to (42). 𝑖

𝑖𝑙 𝑖𝑙 𝑠𝑚 𝑚𝑠 𝑛2 𝑆𝑝𝑗 + 𝑛𝑆𝑗𝑝 = 𝛿𝑗𝑙 𝛿𝑝𝑖 𝑆𝑚𝑠 + 𝛿𝑝𝑙 𝛿𝑗𝑖 𝑆𝑚𝑠 .

(49)

Lemma 10. The functions 𝐹̇ satisfy condition (42) if and only if 𝐴 = 0,

(55)

𝑖 𝑗 𝐹̇ (𝑥̇ 𝑗 , 𝑧̇ 𝑗 , 𝑧̇ 𝑘 ) = 𝐾𝑥̇ 𝑖 + 𝐿𝑧̇ 𝑖 + 𝐵𝑥̇ 𝑟 𝑧̇ 𝑖𝑟

(56)

𝑖𝑙 𝑘 𝑖𝑙 𝑘 𝛿𝑡 + 𝑛𝑆𝑗𝑝 𝛿𝑡 = Contractions in 𝑠, 𝑚 and in 𝑘, 𝑡 yield 𝑆𝑝𝑗 𝑘𝑚 𝛿𝑝𝑙 𝛿𝑗𝑖 𝑆𝑚𝑡

𝑖𝑙 𝑛𝑆𝑝𝑗

+ and + From these formulas, we find

𝑖𝑙 𝑛2 𝑆𝑗𝑝

=

𝑚𝑡 𝛿𝑗𝑙 𝛿𝑝𝑖 𝑆𝑚𝑡

+

𝑡𝑚 𝛿𝑝𝑙 𝛿𝑗𝑖 𝑆𝑚𝑡 .

for some constants 𝐾, 𝐿 ∈ R.

𝑖𝑙 𝑖𝑙 𝑚𝑠 𝑡𝑚 𝑛 (𝑛 + 1) (𝑆𝑝𝑗 + 𝑆𝑗𝑝 ) = (𝛿𝑗𝑙 𝛿𝑝𝑖 + 𝛿𝑝𝑙 𝛿𝑗𝑖 ) (𝑆𝑚𝑠 + 𝑆𝑚𝑡 ),

𝑛 (𝑛 − 𝑖𝑙 𝑆𝑝𝑗

+

𝑖𝑙 1) (𝑆𝑝𝑗

−

𝑖𝑙 𝑆𝑗𝑝 )

=

(𝛿𝑗𝑙 𝛿𝑝𝑖

−

𝑡𝑚 𝛿𝑝𝑙 𝛿𝑗𝑖 ) (𝑆𝑚𝑡

+

𝑖 Proof. (1) Suppose that 𝐹̇ satisfy (42). Then if 𝑎𝑠𝑝 = 𝛿𝑠𝑝 , we have

𝑚𝑡 𝑆𝑚𝑡 ),

1 𝑚𝑠 𝑡𝑚 = + 𝑆𝑚𝑡 ), (𝛿𝑙 𝛿𝑖 + 𝛿𝑝𝑙 𝛿𝑗𝑖 ) (𝑆𝑚𝑠 𝑛 (𝑛 + 1) 𝑗 𝑝

𝑖𝑙 𝑆𝑗𝑝

𝑖𝑙 𝑖𝑙 𝑆𝑝𝑗 − 𝑆𝑗𝑝 =

(50)

𝑖 𝑝 𝑚 𝑧̇ + 𝑧̇ 𝑝𝑞 ) 𝐹̇ (𝑥̇ 𝑝 , 𝑧̇ 𝑝 , 𝑎𝑞𝑚 𝑖 𝑖 = 𝑎𝑘𝑝 𝑥̇ 𝑝 (𝐴𝑥̇ 𝑘 + 𝐵𝑧̇ 𝑘 ) + 𝐹̇ (𝑥̇ 𝑠 , 𝑧̇ 𝑠 , 𝑧̇ 𝑚 𝑠 ).

1 𝑡𝑚 𝑚𝑡 (𝛿𝑙 𝛿𝑖 − 𝛿𝑝𝑙 𝛿𝑗𝑖 ) (𝑆𝑚𝑡 + 𝑆𝑚𝑡 ), 𝑛 (𝑛 − 1) 𝑗 𝑝

𝑖

1 𝑚𝑠 𝑡𝑚 = + 𝑆𝑚𝑡 ), (𝛿𝑙 𝛿𝑖 + 𝛿𝑝𝑙 𝛿𝑗𝑖 ) (𝑆𝑚𝑠 2𝑛 (𝑛 + 1) 𝑗 𝑝 +

1 𝑡𝑚 𝑚𝑡 (𝛿𝑙 𝛿𝑖 − 𝛿𝑝𝑙 𝛿𝑗𝑖 ) (𝑆𝑚𝑡 + 𝑆𝑚𝑡 ). 2𝑛 (𝑛 − 1) 𝑗 𝑝

The same computation applies to 𝑖𝑙 𝑆𝑝𝑗

=

𝑃 (𝛿𝑗𝑙 𝛿𝑝𝑖

+

𝛿𝑝𝑙 𝛿𝑗𝑖 )

+

𝑖𝑙 . 𝑇𝑝𝑗

(

−

𝛿𝑝𝑙 𝛿𝑗𝑖 ) ,

𝑖𝑙 = 𝑅 (𝛿𝑗𝑙 𝛿𝑝𝑖 + 𝛿𝑝𝑙 𝛿𝑗𝑖 ) + 𝑆 (𝛿𝑗𝑙 𝛿𝑝𝑖 − 𝛿𝑝𝑙 𝛿𝑗𝑖 ) , 𝑇𝑝𝑗

𝜕𝐹̇ ) 𝜕𝑧̇ 𝑟𝑠 (𝑥̇ 𝑝 ,𝑧̇𝑝 ,𝑎𝑝

𝑝 𝑚 𝑞𝑚 𝑧̇ +𝑧̇ 𝑞 )

(51)

𝑖

=(

𝜕𝐹̇ ) , 𝜕𝑧̇ 𝑟𝑠 (𝑥̇ 𝑝 ,𝑧̇𝑝 ,𝑧̇𝑝 )

(58)

𝑞

𝑖 which shows that the derivative 𝜕𝐹̇ /𝜕𝑧̇ 𝑟𝑠 does not depend on 𝑧̇ 𝑝𝑞 . Then, however, 𝑖 𝑗 𝐹̇ (𝑥̇ 𝑗 , 𝑧̇ 𝑗 , 𝑧̇ 𝑘 ) = 𝑃𝑖 (𝑥̇ 𝑗 , 𝑧̇ 𝑗 ) + 𝑄𝑠𝑖𝑟 (𝑥̇ 𝑗 , 𝑧̇ 𝑗 ) 𝑧̇ 𝑠𝑟 ,

Thus,

𝑄 (𝛿𝑗𝑙 𝛿𝑝𝑖

(57)

Differentiating with respect to 𝑧̇ 𝑟𝑠 , we have

hence 𝑖𝑙 𝑆𝑝𝑗

(54)

𝑖 𝑘 𝑘 𝑖 𝑠 𝑖 𝑘 𝑘 𝑖 𝑠 𝑎𝑠 𝑥̇ + 𝑄𝑎𝑚 𝑎𝑠 𝑥̇ = 𝑃𝑎𝑚 𝑥̇ + 𝑃𝛿𝑚 𝑥̇ − 𝑄𝛿𝑚

𝑘𝑠 𝛿𝑝𝑙 𝛿𝑗𝑖 𝑆𝑚𝑡 .

We apply to this formula various trace operations. Contrac𝑘 𝑖𝑙 𝑘 𝑠𝑘 tions in 𝑠, 𝑡 and then in 𝑘, 𝑚 yield 𝑛𝑆𝑖𝑙𝑝𝑗 𝛿𝑚 + 𝑆𝑗𝑝 𝛿𝑚 = 𝛿𝑗𝑙 𝛿𝑝𝑖 𝑆𝑚𝑠 +

𝑚𝑘 𝛿𝑗𝑙 𝛿𝑝𝑖 𝑆𝑚𝑡

(53)

and substituting these expressions into (44), we have

(47)

𝑖𝑙 𝑖𝑙 𝑠 𝑘 𝑗 , we write 𝑆𝑝𝑗 𝛿𝑡 𝛿𝑚 𝑎𝑠 𝑎𝑘 = To determine the constants 𝑆𝑝𝑗

𝑖𝑙 𝑠 𝑘 𝑆𝑝𝑗 𝛿𝑡 𝛿𝑚

𝑞

𝑖 𝑘 𝑘 𝑖 𝑡 𝑖 𝑘 𝑘 𝑖 𝑡 = 𝑃𝑎𝑚 𝑎𝑡 𝑥̇ + 𝑄𝑎𝑚 𝑎𝑡 𝑥̇ 𝑥̇ + 𝑃𝛿𝑚 𝑥̇ − 𝑄𝛿𝑚

𝑝

𝑗

𝑞

𝑄𝑝𝑖𝑞 = (𝑃 (𝛿𝑗 𝛿𝑝𝑖 + 𝛿𝑝𝑞 𝛿𝑗𝑖 ) + 𝑄 (𝛿𝑗 𝛿𝑝𝑖 − 𝛿𝑝𝑞 𝛿𝑗𝑖 )) 𝑥̇ 𝑗

𝑠 𝑘 𝑠 𝑠 𝑘 𝑠 + (𝑅 (𝛿𝑗𝑘 𝛿𝑚 + 𝛿𝑚 𝛿𝑗 ) + 𝑆 (𝛿𝑗𝑘 𝛿𝑚 − 𝛿𝑚 𝛿𝑗 )) 𝑧̇ 𝑗

𝑘𝑡 𝑗 𝑡𝑘 𝑗 𝑧̇ ) , = 𝑎𝑡𝑙 𝑎𝑘𝑖 (𝑆𝑚𝑗 𝑥̇ + 𝑇𝑚𝑗 𝑗

𝑖𝑘 satisfies (44). Writing (2) Conversely, suppose that 𝑄𝑚

(59)

where the functions 𝑃𝑖 and 𝑄𝑠𝑖𝑟 do not depend on 𝑧̇ 𝑝𝑞 . Substituting from (59) back to (57), we have (52)

for some constants 𝑃, 𝑄, 𝑅, 𝑆 ∈ R. Formula (44) now follows from (52) and (45).

𝑠 𝑧̇ 𝑚 + 𝑧̇ 𝑠𝑟 ) 𝑃𝑖 (𝑥̇ 𝑝 , 𝑧̇ 𝑝 ) + 𝑄𝑠𝑖𝑟 (𝑥̇ 𝑝 , 𝑧̇ 𝑝 ) (𝑎𝑟𝑚 𝑖 = 𝑎𝑘𝑝 𝑥̇ 𝑝 (𝐴𝑥̇ 𝑘 + 𝐵𝑧̇ 𝑘 ) + 𝑃𝑖 (𝑥̇ 𝑝 , 𝑧̇ 𝑝 ) + 𝑄𝑠𝑖𝑟 (𝑥̇ 𝑝 , 𝑧̇ 𝑝 ) 𝑧̇ 𝑠𝑟 , (60)

Geometry

7

𝑠 𝑠 𝑧̇ 𝑚 = 𝛿𝑠𝑖 𝑎𝑟𝑚 hence 𝑄𝑠𝑖𝑟 (𝑥̇ 𝑝 , 𝑧̇ 𝑝 )𝑎𝑟𝑚 𝑥̇ 𝑟 (𝐴𝑥̇ 𝑚 + 𝐵𝑧̇ 𝑚 ). Thus, the 𝑖 𝑖𝑟 functions 𝑃 and 𝑄𝑠 in (59) must satisfy

𝑄𝑠𝑖𝑟 (𝑥̇ 𝑝 , 𝑧̇ 𝑝 ) 𝑧̇ 𝑚 + 𝑄𝑠𝑖𝑚 (𝑥̇ 𝑝 , 𝑧̇ 𝑝 ) 𝑧̇ 𝑟 = 𝛿𝑠𝑖 𝑥̇ 𝑟 (𝐴𝑥̇ 𝑚 + 𝐵𝑧̇ 𝑚 ) + 𝛿𝑠𝑖 𝑥̇ 𝑚 (𝐴𝑥̇ 𝑟 + 𝐵𝑧̇ 𝑟 ) .

(𝑃 (𝑛 + 1) + 𝑄 (𝑛 − 1)) (𝑥̇ 𝑟 𝑧̇ 𝑚 + 𝑥̇ 𝑚 𝑧̇ 𝑟 ) = 𝑛𝐵 (𝑥̇ 𝑚 𝑧̇ 𝑟 + 𝑥̇ 𝑟 𝑧̇ 𝑚 ) (𝑅 (𝑛 + 1) + 𝑆 (𝑛 − 1)) 𝑧̇ 𝑚 𝑧̇ 𝑟 = 0,

(61)

(70) hence

Note that the trace operation in 𝑖 and 𝑠 yields 1 𝑠𝑟 𝑝 𝑝 𝑚 (𝑄 (𝑥̇ , 𝑧̇ ) 𝑧̇ + 𝑄𝑠𝑠𝑚 (𝑥̇ 𝑝 , 𝑧̇ 𝑝 ) 𝑧̇ 𝑟 ) 𝑛 𝑠

Consequently, 𝐴 = 0 and

𝐴 = 0, 𝑃 (𝑛 + 1) + 𝑄 (𝑛 − 1) = 𝑛𝐵,

(62)

𝑅 (𝑛 + 1) + 𝑆 (𝑛 − 1) = 0.

= 𝑥̇ 𝑟 (𝐴𝑥̇ 𝑚 + 𝐵𝑧̇ 𝑚 ) + 𝑥̇ 𝑚 (𝐴𝑥̇ 𝑟 + 𝐵𝑧̇ 𝑟 ) . 𝑖 (2) We now use (56) for the group parameters 𝑎𝑗𝑘 = 0. We have the condition 𝑖 𝑘 𝑝 𝑠 𝑚 𝑏𝑞 𝑧̇ 𝑠 ) = 𝑎𝑘𝑖 𝐹̇ (𝑥̇ 𝑠 , 𝑧̇ 𝑠 , 𝑧̇ 𝑚 𝐹̇ (𝑎𝑠𝑝 𝑥̇ 𝑠 , 𝑎𝑠𝑝 𝑧̇ 𝑠 , 𝑎𝑚 𝑠 ),

(63)

and, from (59), 𝑠 𝑡 𝑚 𝑃𝑖 (𝑎𝑠𝑗 𝑥̇ 𝑠 , 𝑎𝑠𝑗 𝑧̇ 𝑠 ) + 𝑄𝑠𝑖𝑟 (𝑎𝑠𝑗 𝑥̇ 𝑠 , 𝑎𝑠𝑗 𝑧̇ 𝑠 ) 𝑎𝑚 𝑏𝑟 𝑧̇ 𝑡

= 𝑎𝑘𝑖 𝑃𝑘 (𝑥̇ 𝑗 , 𝑧̇ 𝑗 ) + 𝑎𝑘𝑖 𝑄𝑠𝑘𝑟 (𝑥̇ 𝑗 , 𝑧̇ 𝑗 ) 𝑧̇ 𝑠𝑟 .

𝑠 𝑡 𝑏𝑟 = 𝑎𝑘𝑖 𝑄𝑠𝑘𝑟 (𝑥̇ 𝑗 , 𝑧̇ 𝑗 ) . 𝑄𝑠𝑖𝑟 (𝑎𝑠𝑗 𝑥̇ 𝑠 , 𝑎𝑠𝑗 𝑧̇ 𝑠 ) 𝑎𝑚

(64)

(𝑃 (𝛿𝑗𝑟 𝛿𝑠𝑖 + 𝛿𝑠𝑟 𝛿𝑗𝑖 ) + 𝑄 (𝛿𝑗𝑟 𝛿𝑠𝑖 − 𝛿𝑠𝑟 𝛿𝑗𝑖 )) 𝑥̇ 𝑗 𝑧̇ 𝑚 + 𝑆(

(65)

+ 𝑆(

1−𝑛 𝑚 𝑖 (𝛿 𝛿 + 𝛿𝑠𝑚 𝛿𝑗𝑖 ) + (𝛿𝑗𝑚 𝛿𝑠𝑖 − 𝛿𝑠𝑚 𝛿𝑗𝑖 )) 𝑧̇ 𝑗 𝑧̇ 𝑟 𝑛+1 𝑗 𝑠 (72)

The terms containing 𝑧̇ 𝑗 𝑧̇ 𝑚 should vanish separately. Since these terms are 1 𝑆 (− (𝑛 − 1) 𝛿𝑠𝑖 𝑧̇ 𝑟 𝑧̇ 𝑚 − (𝑛 − 1) 𝛿𝑠𝑟 𝑧̇ 𝑖 𝑧̇ 𝑚 𝑛+1 + (𝑛 + 1) 𝛿𝑠𝑖 𝑧̇ 𝑟 𝑧̇ 𝑚 − (𝑛 + 1) 𝛿𝑠𝑟 𝑧̇ 𝑖 𝑧̇ 𝑚 )

(66) +

for some constants 𝐾, 𝐿 ∈ R (Lemma 8), and 𝑄𝑠𝑖𝑟 (𝑥̇ 𝑗 , 𝑧̇ 𝑗 )

1 𝑆 (− (𝑛 − 1) 𝛿𝑠𝑖 𝑧̇ 𝑚 𝑧̇ 𝑟 − (𝑛 − 1) 𝛿𝑠𝑚 𝑧̇ 𝑖 𝑧̇ 𝑟 𝑛+1

(73)

+ (𝑛 + 1) 𝛿𝑠𝑖 𝑧̇ 𝑚 𝑧̇ 𝑟 − (𝑛 + 1) 𝛿𝑠𝑚 𝑧̇ 𝑖 𝑧̇ 𝑟 )

= (𝑃 (𝛿𝑗𝑟 𝛿𝑠𝑖 + 𝛿𝑠𝑟 𝛿𝑗𝑖 ) + 𝑄 (𝛿𝑗𝑟 𝛿𝑠𝑖 − 𝛿𝑠𝑟 𝛿𝑗𝑖 )) 𝑥̇ 𝑗

(67)

+ (𝑅 (𝛿𝑗𝑟 𝛿𝑠𝑖 + 𝛿𝑠𝑟 𝛿𝑗𝑖 ) + 𝑆 (𝛿𝑗𝑟 𝛿𝑠𝑖 − 𝛿𝑠𝑟 𝛿𝑗𝑖 )) 𝑧̇ 𝑗 ,

+

𝛿𝑠𝑟 𝛿𝑗𝑖 )

+

𝑄 (𝛿𝑗𝑟 𝛿𝑠𝑖

−

𝛿𝑠𝑟 𝛿𝑗𝑖 )) 𝑥̇ 𝑗 𝑧̇ 𝑚

= 𝛿𝑠𝑖 (2𝐴𝑥̇ 𝑚 𝑥̇ 𝑟 + 𝐵 (𝑥̇ 𝑚 𝑧̇ 𝑟 + 𝑥̇ 𝑟 𝑧̇ 𝑚 )) .

=

−2𝑛 𝑆 (𝛿𝑠𝑟 𝑧̇ 𝑖 𝑧̇ 𝑚 + 𝛿𝑠𝑚 𝑧̇ 𝑖 𝑧̇ 𝑟 ) , 𝑛+1

we have, from (71) and (73),

where 𝑃, 𝑄, 𝑅, 𝑆 ∈ R (Lemma 9). These functions satisfy (61); that is, (𝑃 (𝛿𝑗𝑟 𝛿𝑠𝑖

1−𝑛 𝑟 𝑖 (𝛿 𝛿 + 𝛿𝑠𝑟 𝛿𝑗𝑖 ) + (𝛿𝑗𝑟 𝛿𝑠𝑖 − 𝛿𝑠𝑟 𝛿𝑗𝑖 )) 𝑧̇ 𝑗 𝑧̇ 𝑚 𝑛+1 𝑗 𝑠

= 𝛿𝑠𝑖 𝐵 (𝑥̇ 𝑚 𝑧̇ 𝑟 + 𝑥̇ 𝑟 𝑧̇ 𝑚 ) .

Then, however, 𝑃𝑖 (𝑥̇ 𝑗 , 𝑧̇ 𝑗 ) = 𝐾𝑥̇ 𝑖 + 𝐿𝑧̇ 𝑖 ,

Then from (68),

+ (𝑃 (𝛿𝑗𝑚 𝛿𝑠𝑖 + 𝛿𝑠𝑚 𝛿𝑗𝑖 ) + 𝑄 (𝛿𝑗𝑚 𝛿𝑠𝑖 − 𝛿𝑠𝑚 𝛿𝑗𝑖 )) 𝑥̇ 𝑗 𝑧̇ 𝑟

These equations split to the system 𝑃𝑖 (𝑎𝑠𝑗 𝑥̇ 𝑠 , 𝑎𝑠𝑗 𝑧̇ 𝑠 ) = 𝑎𝑘𝑖 𝑃𝑘 (𝑥̇ 𝑗 , 𝑧̇ 𝑗 ) ,

(71)

(68)

𝑆 = 0,

𝑅 = 0.

(74)

Analogously, the terms with 𝑥̇ 𝑗 𝑧̇ 𝑚 should vanish separately; that is, (𝑃 (𝛿𝑗𝑟 𝛿𝑠𝑖 + 𝛿𝑠𝑟 𝛿𝑗𝑖 ) + 𝑄 (𝛿𝑗𝑟 𝛿𝑠𝑖 − 𝛿𝑠𝑟 𝛿𝑗𝑖 )) 𝑥̇ 𝑗 𝑧̇ 𝑚 + (𝑃 (𝛿𝑗𝑚 𝛿𝑠𝑖 + 𝛿𝑠𝑚 𝛿𝑗𝑖 ) + 𝑄 (𝛿𝑗𝑚 𝛿𝑠𝑖 − 𝛿𝑠𝑚 𝛿𝑗𝑖 )) 𝑥̇ 𝑗 𝑧̇ 𝑟 (75)

The trace in 𝑖 and 𝑠 yields (𝑃 (𝑛 + 1) + 𝑄 (𝑛 − 1)) 𝑥̇ 𝑟 𝑧̇ 𝑚 + (𝑅 (𝑛 + 1) + 𝑆 (𝑛 − 1)) 𝑧̇ 𝑟 𝑧̇ 𝑚 + (𝑃 (𝑛 + 1) + 𝑄 (𝑛 − 1)) 𝑥̇ 𝑚 𝑧̇ 𝑟

= 𝛿𝑠𝑖 𝐵 (𝑥̇ 𝑚 𝑧̇ 𝑟 + 𝑥̇ 𝑟 𝑧̇ 𝑚 ) . Since this equation can be written as (𝑃 + 𝑄) 𝛿𝑠𝑖 (𝑥̇ 𝑟 𝑧̇ 𝑚 + 𝑥̇ 𝑚 𝑧̇ 𝑟 ) + (𝑃 − 𝑄) (𝛿𝑠𝑟 𝑥̇ 𝑖 𝑧̇ 𝑚 + 𝛿𝑠𝑚 𝑥̇ 𝑖 𝑧̇ 𝑟 )

+ (𝑅 (𝑛 + 1) + 𝑆 (𝑛 − 1)) 𝑧̇ 𝑚 𝑧̇ 𝑟 = 𝑛 (2𝐴𝑥̇ 𝑚 𝑥̇ 𝑟 + 𝐵 (𝑥̇ 𝑚 𝑧̇ 𝑟 + 𝑥̇ 𝑟 𝑧̇ 𝑚 )) . (69)

= 𝛿𝑠𝑖 𝐵 (𝑥̇ 𝑚 𝑧̇ 𝑟 + 𝑥̇ 𝑟 𝑧̇ 𝑚 ) ,

(76)

8

Geometry

we get 𝑃 + 𝑄 = 𝐵 and 𝑃 − 𝑄 = 0; thus, 1 𝑃 = 𝑄 = 𝐵. 2

(77)

such that for any points 𝑥, 𝑦 ∈ 𝑋, any charts (𝑈, 𝜑), 𝜑 = (𝑥𝑖 ), at 𝑥 and (𝑉, 𝜓), 𝜓 = (𝑦𝑖 ), at 𝑦, and any diffeomorphism 𝛼 : 𝑈 → 𝑉 such that 𝛼(𝑥) = 𝑦, 𝑖 𝑘 𝑏𝑗𝑚 𝜉𝑘 + 𝑎𝑘𝑖 𝑏𝑗𝑚 𝜉𝑚 ) Ξ𝑙(𝑉,𝜓) (𝑎𝑘𝑖 𝜁𝑘 , 𝑎𝑘𝑖 𝜉𝑘 , 𝑎𝑘𝑚

Summarizing, we see that condition (42) implies, from (59), (66), (67), (71), (74), and (77), 𝐴 = 0,

(78)

𝑘 ), = 𝑎𝑘𝑙 Ξ𝑘(𝑈,𝜑) (𝜁𝑘 , 𝜉𝑘 , 𝜉𝑚 𝑙 𝑖 𝑘 Ξ̇ (𝑉,𝜓) (𝑎𝑘𝑖 𝜁𝑘 , 𝑎𝑘𝑖 𝜉𝑘 , 𝑎𝑘𝑚 𝑏𝑗𝑚 𝜉𝑘 + 𝑎𝑘𝑖 𝑏𝑗𝑚 𝜉𝑚 ) 𝑘

𝑙 𝑗 𝑘 𝑘 = 𝑎𝑗𝑘 ) + 𝑎𝑘𝑙 Ξ̇ (𝑈,𝜑) (𝜁𝑘 , 𝜉𝑘 , 𝜉𝑚 ). 𝑥̇ (𝜁) Ξ𝑘(𝑈,𝜑) (𝜁𝑘 , 𝜉𝑘 , 𝜉𝑚 (84)

(48) and 𝑖 𝑗 𝐹̇ (𝑥̇ 𝑗 , 𝑧̇ 𝑗 , 𝑧̇ 𝑘 ) = 𝐾𝑥̇ 𝑖 + 𝐿𝑧̇ 𝑖 + 𝐵𝑥̇ 𝑟 𝑧̇ 𝑖𝑟 .

(79)

(3) It remains to prove that conditions (55) and (56) imply (42). The left-hand side of (42) is 𝑖 𝑝 𝑠 𝑚 𝑝 𝑠 𝑚 𝑏𝑞 𝑧̇ + 𝑎𝑚 𝑏𝑞 𝑧̇ 𝑠 ) 𝐹̇ (𝑎𝑠𝑝 𝑥̇ 𝑠 , 𝑎𝑠𝑝 𝑧̇ 𝑠 , 𝑎𝑠𝑚 𝑖 𝑖 𝑚 𝑧̇ 𝑚 + 𝑎𝑚 𝑧̇ 𝑡 ) , = 𝐾𝑎𝑠𝑖 𝑥̇ 𝑠 + 𝐿𝑎𝑠𝑖 𝑧̇ 𝑠 + 𝐵𝑥̇ 𝑡 (𝑎𝑡𝑚

(80)

and the right-hand side is 𝑘 𝑖 𝑎𝑘𝑝 𝑥̇ 𝑝 (𝐴𝑥̇ 𝑘 + 𝐵𝑧̇ 𝑘 ) + 𝑎𝑘𝑖 𝐹̇ (𝑥̇ 𝑠 , 𝑧̇ 𝑠 , 𝑧̇ 𝑚 𝑠 ) 𝑖 𝑘 = 𝑎𝑘𝑖 𝐾𝑥̇ 𝑘 + 𝑎𝑘𝑖 𝐿𝑧̇ 𝑘 + 𝐵𝑥̇ 𝑝 (𝑎𝑘𝑝 𝑧̇ + 𝑎𝑘𝑖 𝑧̇ 𝑘𝑝 ) .

(81)

We can now summarize our results in the following theorem. 2

→ 𝑇𝑋 Theorem 11. Let 𝑋 be a manifold and let Ξ : be a morphism over id𝑇𝑋 . The following two conditions are equivalent:

(b) For any chart (𝑈, 𝜑), 𝜑 = (𝑥𝑖 ), on 𝑋 𝜕 𝜕 𝜕 + 𝑥̇ 𝑟 𝑧̇ 𝑖𝑟 𝑖 ) + (𝐾𝑥̇ 𝑖 + 𝐿𝑧̇ 𝑖 ) 𝑖 , 𝑖 𝜕𝑥 𝜕𝑥̇ 𝜕𝑥̇

𝜕 𝜕 𝜕 = 𝐵 (𝑢̇ 𝑖 + 𝑦̇ 𝑟 𝑢̇ 𝑖𝑟 𝑖 ) + (𝐾𝑦̇ 𝑖 + 𝐿𝑢̇ 𝑖 ) 𝑖 ; 𝜕𝑦 𝜕𝑦̇ 𝜕𝑦̇

(85)

that is, in the notation of Lemma 1, 𝐵𝑢̇ 𝑙 =

𝜕𝑦𝑙 𝑘 𝐵𝑧̇ , 𝜕𝑥𝑘

𝐾𝑦̇ 𝑙 + 𝐿𝑢̇ 𝑙 + 𝐵𝑦̇ 𝑟 𝑢̇ 𝑙𝑟 =

(86)

𝜕 2 𝑦𝑙 𝜕𝑦𝑙 𝑝 𝑘 ̇ ̇ 𝑧 𝐵 𝑥 + (𝐾𝑥̇ 𝑘 + 𝐿𝑧̇ 𝑘 + 𝑥̇ 𝑟 𝑧̇ 𝑘𝑟 ) . 𝜕𝑥𝑝 𝜕𝑥𝑘 𝜕𝑥𝑘

𝐹𝑖 = 𝐵𝑧̇ 𝑖 , (82)

where 𝐵, 𝐾, 𝐿 ∈ R are arbitrary constants. Proof. (1) We show that (a) implies (b). Suppose that we have a canonical vector field Ξ. Then, in any chart on 𝑋 the 2ndorder vector field Ξ has an expression Ξ (𝜁, 𝐽𝑥1 𝜉) = Ξ𝑘(𝑈,𝜑) (𝜁, 𝐽𝑥1 𝜉) (

𝜕 𝜕 𝜕 + 𝑥̇ 𝑟 𝑧̇ 𝑖𝑟 𝑖 ) + (𝐾𝑥̇ 𝑖 + 𝐿𝑧̇ 𝑖 ) 𝑖 𝑖 𝜕𝑥 𝜕𝑥̇ 𝜕𝑥̇

Each element of the family of vector fields (82) defines a 𝑖 differential invariant 𝐹 = (𝐹𝑖 , 𝐹̇ ), where

(a) Ξ is a canonical vector field.

Ξ = 𝐵 (𝑧̇ 𝑖

𝐵 (𝑧̇ 𝑖

𝑖

These formulas already verify condition (42). The proof is complete.

𝜏𝑋∗ 𝐽1 𝑇𝑋

(Theorem 3). If 𝑥 = 𝑦 and (𝑈, 𝜑) = (𝑉, 𝜓), the components 𝑙 𝑖 Ξ𝑙(𝑈,𝜑) and Ξ̇ (𝑈,𝜑) define a differential invariant 𝐹 = (𝐹𝑖 , 𝐹̇ ) (Theorem 5); then, however, Ξ must be of the form (82) (Lemmas 9 and 10). (2) To prove that (b) implies (a), we first show that any two members of the family of vector fields (82) agree on intersection of their domains. Let (𝑈, 𝜑), 𝜑 = (𝑥𝑖 ), and (𝑉, 𝜓), 𝜓 = (𝑦𝑖 ), be two charts on 𝑋 such that 𝑈 ∩ 𝑉 ≠ 0, let (𝑥𝑖 , 𝑥̇ 𝑖 , 𝑧̇ 𝑖 , 𝑧̇ 𝑖𝑗 ) and (𝑦𝑖 , 𝑦̇ 𝑖 , 𝑢̇ 𝑖 , 𝑢̇ 𝑖𝑗 ) be the corresponding coordinates on 𝜏𝑋∗ 𝐽1 𝑇𝑋. We want to show that

𝜕 𝜕 𝑘 ) + Ξ̇ (𝑈,𝜑) (𝜁, 𝐽𝑥1 𝜉) ( 𝑘 ) 𝑘 𝜕𝑥 𝜁 𝜕𝑥̇ 𝜁 (83)

𝑖 𝐹̇ = 𝐾𝑥̇ 𝑖 + 𝐿𝑧̇ 𝑖 + 𝐵𝑥̇ 𝑟 𝑧̇ 𝑖𝑟 .

(87)

Recall that 𝑝 𝑠 𝑚 𝑝 𝑠 𝑚 𝐹𝑖 (𝑎𝑠𝑝 𝑥̇ 𝑠 , 𝑎𝑠𝑝 𝑧̇ 𝑠 , 𝑎𝑠𝑚 𝑏𝑞 𝑧̇ + 𝑎𝑚 𝑏𝑞 𝑧̇ 𝑠 )

= 𝑎𝑘𝑖 𝐹𝑘 (𝑥̇ 𝑠 , 𝑧̇ 𝑠 , 𝑧̇ 𝑚 𝑠 ),

(88)

𝑖 𝑝 𝑠 𝑚 𝑝 𝑠 𝑚 𝑏𝑞 𝑧̇ + 𝑎𝑚 𝑏𝑞 𝑧̇ 𝑠 ) 𝐹̇ (𝑎𝑠𝑝 𝑥̇ 𝑠 , 𝑎𝑠𝑝 𝑧̇ 𝑠 , 𝑎𝑠𝑚 𝑖 𝑝 𝑘 𝑖 ̇𝑘 𝑠 𝑠 𝑚 𝜁 𝐹 (𝑥̇ 𝑠 , 𝑧̇ 𝑠 , 𝑧̇ 𝑚 = 𝑎𝑘𝑝 𝑠 ) + 𝑎𝑘 𝐹 (𝑥̇ , 𝑧̇ , 𝑧̇ 𝑠 ) ,

(Theorem 5). Now the right-hand sides of (86) can be written as 𝑎𝑘𝑙 𝐹𝑘 (𝑥̇ 𝑖 , 𝑧̇ 𝑖 , 𝑧̇ 𝑖𝑟 ) ,

𝑘

𝑙 𝐹𝑘 (𝑥̇ 𝑖 , 𝑧̇ 𝑖 , 𝑧̇ 𝑖𝑟 ) 𝑧̇ 𝑝 + 𝑎𝑘𝑙 𝐹̇ (𝑥̇ 𝑖 , 𝑧̇ 𝑖 , 𝑧̇ 𝑖𝑟 ) , 𝑎𝑘𝑝 (89)

Geometry

9

and the left-hand sides are

We substitute these expressions into formulas (17) of Theorem 3,

𝐹𝑙 (𝑦̇ 𝑖 , 𝑢̇ 𝑖 , 𝑢̇ 𝑖𝑟 ) =𝐹

𝑙

𝑖 (𝑎𝑠𝑖 𝑥̇ 𝑠 , 𝑎𝑠𝑖 𝑧̇ 𝑠 , (𝑎𝑠𝑚 𝑥̇ 𝑚

+

𝑖 𝑘 𝑏𝑗𝑚 𝜉𝑘 + 𝑎𝑘𝑖 𝑏𝑗𝑚 𝜉𝑚 ) Ξ𝑙(𝑉,𝜓) (𝑎𝑘𝑖 𝜁𝑘 , 𝑎𝑘𝑖 𝜉𝑘 , 𝑎𝑘𝑚

𝑖 𝑚 𝑎𝑚 𝑥̇ 𝑠 ) 𝑏𝑟𝑠 ) ,

= 𝑎𝑘𝑙 Ξ𝑘(𝑈,𝜑) (𝜁𝑖 , 𝜉𝑖 , 𝜉𝑗𝑖 ) ,

(90)

𝑙 𝐹̇ (𝑦̇ 𝑖 , 𝑢̇ 𝑖 , 𝑢̇ 𝑖𝑟 )

𝑙 𝑖 𝑘 Ξ̇ (𝑉,𝜓) (𝑎𝑘𝑖 𝜁𝑘 , 𝑎𝑘𝑖 𝜉𝑘 , 𝑎𝑘𝑚 𝑏𝑗𝑚 𝜉𝑘 + 𝑎𝑘𝑖 𝑏𝑗𝑚 𝜉𝑚 )

𝑙 𝑖 𝑖 𝑚 = 𝐹̇ (𝑦̇ 𝑖 , 𝑢̇ 𝑖 , (𝑎𝑠𝑚 𝑥̇ 𝑚 + 𝑎𝑚 𝑥̇ 𝑠 ) 𝑏𝑟𝑠 ) ,

(95)

𝑘 𝑙 𝑗 𝑘 𝑖 𝑖 = 𝑎𝑗𝑘 𝜁 Ξ(𝑈,𝜑) (𝜁𝑖 , 𝜉𝑖 , 𝜉𝑚 ) + 𝑎𝑘𝑙 Ξ̇ (𝑈,𝜑) (𝜁𝑖 , 𝜉𝑖 , 𝜉𝑚 ).

where

Then since 𝑎𝑘𝑙 = ( 𝑙 =( 𝑎𝑘𝑝

𝑏𝑟𝑝

𝑙

𝜕𝑦 ) , 𝜕𝑥𝑘 Ψ(𝜁)

𝑖 𝑘 ̃ 𝑙 𝜉𝑘 , 𝑏𝑗𝑚 𝜉𝑘 + 𝑎𝑘𝑖 𝑏𝑗𝑚 𝜉𝑚 ) = 𝐵𝑎 Ξ𝑙(𝑉,𝜓) (𝑎𝑘𝑖 𝜁𝑘 , 𝑎𝑘𝑖 𝜉𝑘 , 𝑎𝑘𝑚 𝑘

𝑎𝑘𝑙 Ξ𝑘(𝑈,𝜑) (𝜁𝑖 , 𝜉𝑖 , 𝜉𝑗𝑖 ) = 𝑎𝑘𝑙 𝐵𝜉𝑘 ,

2 𝑙

𝜕𝑦 ) , 𝜕𝑥𝑝 𝜕𝑥𝑘 Ψ(𝜁)

(91)

𝑙 𝑖 𝑘 Ξ̇ (𝑉,𝜓) (𝑎𝑘𝑖 𝜁𝑘 , 𝑎𝑘𝑖 𝜉𝑘 , 𝑎𝑘𝑚 𝑏𝑗𝑚 𝜉𝑘 + 𝑎𝑘𝑖 𝑏𝑗𝑚 𝜉𝑚 )

𝜕𝑥𝑝 =( 𝑟) . 𝜕𝑦 Φ(𝜁)

̃ 𝑘 𝜉𝑙 + 𝐵𝑎 ̃ 𝑟 𝜁𝑝 (𝑎𝑘 𝑏𝑚 𝜉𝑙 + 𝑎𝑘 𝑏𝑚 𝜉𝑙 ) , ̃ 𝑘 𝜁𝑙 + 𝐿𝑎 = 𝐾𝑎 𝑙 𝑙 𝑝 𝑙𝑚 𝑟 𝑙 𝑟 𝑚

Expressions (89) and (90) prove (86) as well as existence of Ξ. (3) To complete the proof, it remains to show that the vector field Ξ is a canonical vector field; to this purpose, we verify condition (2) of Theorem 3. Express the vectors 1 (𝑇𝛼 ∘ 𝜉 ∘ 𝛼−1 )) as in formula (15), Ξ(𝜁, 𝐽𝑥1 𝜉) and Ξ(𝑇𝑥 𝛼 ⋅ 𝜁, 𝐽𝛼(𝑥) Section 4,

𝑘 𝑙 𝑗 𝑘 𝑖 𝑖 𝜁 Ξ(𝑈,𝜑) (𝜁𝑖 , 𝜉𝑖 , 𝜉𝑚 ) + 𝑎𝑘𝑙 Ξ̇ (𝑈,𝜑) (𝜁𝑖 , 𝜉𝑖 , 𝜉𝑚 ) 𝑎𝑗𝑘

= 𝐵𝑎𝑗𝑙𝑘 𝜁𝑗 𝜉𝑙 + 𝑎𝑙𝑘 (𝐾𝜁𝑙 + 𝐿𝜉𝑙 + 𝐵𝜁𝑟 𝜉𝑟𝑙 ) , we have ̃ 𝑙 𝜉𝑘 = 𝐵𝑎𝑙 𝜉𝑘 , 𝐵𝑎 𝑘 𝑘 ̃ 𝑘 𝜉𝑙 + 𝐵𝑎 ̃ 𝑟 𝜁𝑝 (𝑎𝑘 𝑏𝑚 𝜉𝑙 + 𝑎𝑘 𝑏𝑚 𝜉𝑙 ) ̃ 𝑘 𝜁𝑙 + 𝐿𝑎 𝐾𝑎 𝑙 𝑙 𝑝 𝑙𝑚 𝑟 𝑙 𝑟 𝑚

Ξ (𝜁, 𝐽𝑥1 𝜉) =

Ξ𝑘(𝑈,𝜑)

(𝜁, 𝐽𝑥1 𝜉) (

𝜕 𝜕 𝑘 ) + Ξ̇ (𝑈,𝜑) (𝜁, 𝐽𝑥1 𝜉) ( 𝑘 ) , 𝑘 𝜕𝑥 𝜁 𝜕𝑥̇ 𝜁

1 Ξ (𝑇𝑥 𝛼 ⋅ 𝜁, 𝐽𝛼(𝑥) (𝑇𝛼 ∘ 𝜉 ∘ 𝛼−1 )) 1 = Ξ𝑘(𝑉,𝜓) (𝑇𝑥 𝛼 ⋅ 𝜁, 𝐽𝛼(𝑥) (𝑇𝛼 ∘ 𝜉 ∘ 𝛼−1 )) (

𝜕 ) 𝜕𝑦𝑘 𝜁

𝜕 𝑘 1 + Ξ̇ (𝑉,𝜓) (𝑇𝑥 𝛼 ⋅ 𝜁, 𝐽𝛼(𝑥) (𝑇𝛼 ∘ 𝜉 ∘ 𝛼−1 )) ( 𝑘 ) . 𝜕𝑦̇ 𝜁 (92)

Ξ = 𝐵𝑢̇ 𝑘

𝜕 𝜕 + (𝐾𝑦̇ 𝑘 + 𝐿𝑢̇ 𝑘 + 𝐵𝑦̇ 𝑟 𝑢̇ 𝑘𝑟 ) 𝑘 , 𝜕𝑦𝑘 𝜕𝑦̇

̃ = 𝐵, 𝐾 ̃ = 𝐾, and But these conditions are equivalent to 𝐵 ̃ 𝐿 = 𝐿 as required. Having in mind that the canonical constructions are geometric constructions independent of charts, we can also state our theorem in an equivalent way as follows. Let 𝑋 be an 𝑛-dimensional manifold. Then for any vector field 𝜁 on 𝑋, there are exactly three independent canonical vector fields 𝑍 on 𝑇𝑋 that can canonically be constructed from 𝜁. If in a chart 𝜕 , 𝜕𝑥𝑖

(98)

𝑖 𝜕 𝜕 𝑟 𝜕𝜁 𝜕 ̇ + 𝑥 ) + (𝐾𝑥̇ 𝑖 + 𝐿𝜁𝑖 ) 𝑖 , 𝜕𝑥𝑖 𝜕𝑥𝑟 𝜕𝑥̇ 𝑖 𝜕𝑥̇

(99)

𝜁 = 𝜁𝑖 then

(93)

on the corresponding coordinate neighbourhoods. Thus, the components of these vector fields are Ξ𝑖(𝑈,𝜑) = 𝐵𝑧̇ 𝑖 ,

𝑘 Ξ̇ (𝑈,𝜑) = 𝐾𝑥̇ 𝑘 + 𝐿𝑧̇ 𝑘 + 𝐵𝑥̇ 𝑟 𝑧̇ 𝑘𝑟 ,

̃ 𝑢̇ 𝑖 , Ξ𝑖(𝑉,𝜓) = 𝐵

𝑘 ̃ 𝑦̇ 𝑘 + 𝐿 ̃ 𝑢̇ 𝑘 + 𝐵 ̃ 𝑦̇ 𝑟 𝑢̇ 𝑘 . Ξ̇ (𝑉,𝜓) = 𝐾 𝑟

(97)

= 𝐵𝑎𝑗𝑙𝑘 𝜁𝑗 𝜉𝑙 + 𝑎𝑙𝑘 (𝐾𝜁𝑙 + 𝐿𝜉𝑙 + 𝐵𝜁𝑟 𝜉𝑟𝑙 ) .

We have already proved that 𝜕 𝜕 Ξ = 𝐵𝑧̇ 𝑘 𝑘 + (𝐾𝑥̇ 𝑘 + 𝐿𝑧̇ 𝑘 + 𝐵𝑥̇ 𝑟 𝑧̇ 𝑘𝑟 ) 𝑘 , 𝜕𝑥 𝜕𝑥̇

(96)

𝑍 = 𝐵 (𝜁𝑖

where 𝐵, 𝐾, 𝐿 ∈ R are arbitrary constants. Taking 𝐾, 𝐿 = 0 and 𝐵 = 1, we get the variational vector field; if 𝐵, 𝐿 = 0 and 𝐾 = 1, we get the Liouville vector field, and, if 𝐵, 𝐾 = 0 and 𝐿 = 1, we have the vertical lift vector field.

Acknowledgments (94)

The first author (T. Li) is grateful for the support of the National Natural Science Foundation of China (Grant no.

10 10801006). The second author (D. Krupka) acknowledges the support of the National Science Foundation of China (Grant no. 10932002) and the Czech Science Foundation (Grant no. 201/09/0981).

References [1] D. Gromoll, W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Grossen, Lecture Notes in Mathematics, no. 55, Springer, Berlin, Germany, 1968. [2] Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001. [3] O. Krupkov´a and G. E. Prince, “Second order ordinary differential equations in jet bundles and the inverse problem of the calculus of variations,” in Handbook of Global Analysis, D. Krupka and D. Saunders, Eds., pp. 837–904, Elsevier Sci. B. V., Amsterdam, The Netherlands, 2008. [4] S. Lang, Fundamentals of Differential Geometry, vol. 191 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1999. [5] R. S. Palais and C. L. Terng, “Natural bundles have finite order,” Topology, vol. 19, no. 3, pp. 271–277, 1977. [6] D. J. Saunders, “Jet fields, connections and second-order differential equations,” Journal of Physics A, vol. 20, no. 11, pp. 3261– 3270, 1987. [7] S. Sternberg, Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, NJ, USA, 1964. [8] H. Goldschmidt and S. Sternberg, “The Hamilton-Cartan formalism in the calculus of variations,” Annales de l’Institut Fourier, vol. 23, no. 1, pp. 203–267, 1973. [9] D. Krupka, “Some geometric aspects of variational problems in fibred manifolds,” Folia Facultatis Scientiarum Naturalium Universitatis Purkynianae. Brunensis, vol. 14, pp. 1–65, 1973. [10] R. Abraham and J. E. Marsden, Foundations of Mechanics, Benjamin/Cummings, Reading, Mass, USA, 2nd edition, 1978. [11] D. Krupka, O. Krupkov´a, G. Prince, and W. Sarlet, “Contact symmetries of the Helmholtz form,” Differential Geometry and Its Applications, vol. 25, no. 5, pp. 518–542, 2007. [12] W. Sarlet, A. Vandecasteele, F. Cantrijn, and E. Mart´ınez, “Derivations of forms along a map: the framework for timedependent second-order equations,” Differential Geometry and Its Applications, vol. 5, no. 2, pp. 171–203, 1995. [13] D. Krupka and J. Janyˇska, Lectures on Differential Invariants, vol. 1 of Folia Facultatis Scientiarum Naturalium Universitatis Purkynianae Brunensis. Mathematica, University J. E. Purkynˇe, Brno, Czech Republic, 1990. [14] D. R. Grigore and D. Krupka, “Invariants of velocities and higher-order Grassmann bundles,” Journal of Geometry and Physics, vol. 24, no. 3, pp. 244–264, 1998. [15] I. Kol´aˇr, P. W. Michor, and J. Slov´ak, Natural Operations in Differential Geometry, Springer, Berlin, Germany, 1993. [16] D. Krupka and M. Krupka, “Jets and contact elements,” in Proceedings of the Seminar on Differential Geometry (Opava, 2000), vol. 2 of Mathematical Publications, pp. 39–85, Silesian Univ. Opava, Opava, Czech Republic, 2000. [17] W. Sarlet and F. Cantrijn, “Special symmetries for Lagrangian systems and their analogues in nonconservative mechanics,” in Proceedings of the Conference on Differential Geometry and Its Applications, D. Krupka, Ed., pp. 247–260, Univ. J. E. Purkynˇe, Brno, Czech Republic, 1984.

Geometry [18] D. Krupka, “Local invariants of a linear connection,” in Differential Geometry (Budapest, 1979), vol. 31 of Colloq. Math. Soc. J´anos Bolyai, pp. 349–369, North-Holland, Amsterdam, The Netherlands, 1982.

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