Volume 2013 (2013), Article ID 364301, 10 pages
Research Article

The Geometry of Tangent Bundles: Canonical Vector Fields

1Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China
2Department of Mathematics, Faculty of Science, The University of Ostrava, 30. dubna 22, 70103 Ostrava, Czech Republic
3Department of Mathematics, La Trobe University, Melbourne, Bundoora, VIC 3086, Australia

Received 14 December 2012; Accepted 13 March 2013

Academic Editor: Anna Fino

Copyright © 2013 Tongzhu Li and Demeter 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 [37]. 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, 1012]).

This paper is devoted to the structure theory of vector fields on the tangent bundle of a manifold ; our aim will be to classify all canonical vector fields on , independent of any geometric structure or on the topology of . 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 ), (b) the Liouville vector field, and (c) the vertical lift of a vector field on ; 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 naturally define a Lie group , where , the differential group, and its left action on the type fibre of any natural bundle over [1317]. The general theory gives us equations that determine a canonical vector field on as a differential invariant of the second differential group .

The proof of our result is not straightforward; it relies on the semidirect product structure of 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, is a smooth real -dimensional manifold, is the tangent bundle of , and is the tangent bundle projection. The second tangent bundle of is the tangent bundle over with the tangent bundle projection ; its elements are second-order tangent vectors on . The mapping is the differential of , satisfying . The 1-jet prolongation of is denoted by ; elements of the set are the 1-jets with source and target ; the source and target jet projections are and .

2. Second-Order Vectors and Jets of a Vector Field

In this section, we fix basic notation, used throughout this paper. If , , is a chart on , we denote by , , the associated chart on the tangent bundle ; the associated chart on is denoted by , . The tangent bundle projection has the chart expression , and is expressed by . The tangent mapping is expressed in coordinates as .

In the following two lemmas, we recall standard transformation formulas, needed in proofs.

Lemma 1. Let a second-order tangent vector be expressed in two charts , , and , , as Then,

The chart on , associated with the chart , , on is denoted by , . The coordinates of a 1-jet are , , and .

Lemma 2. For any two charts , , and , , on , such that , the transformation equations on are

We need the pullback fibration over . We have the commutative diagram xy(4) in which the left vertical arrow is the restriction of the first Cartesian projection and the upper horizontal arrow is the restriction of the second Cartesian projection . We denote Any chart , , on induces a chart on the pullback manifold . Denoting by the coordinates of a vector at and by , , the coordinates of a 1-jet , then the induced chart on , , , is defined by In these coordinates, is the mapping and the second Cartesian projection is .

3. Differential Groups and Differential Invariants

Recall that for any positive integer , the th differential group is the group of invertible -jets with source and target at the origin , endowed with its natural Lie group structure; the group multiplication in is the jet composition . The first differential group is just the group . For all , we denote by the canonical jet projection and by the canonical injective Lie group morphism. The normal subgroup of is nilpotent, and is the interior semidirect product of and [13, 17]; we denote .

The first canonical coordinates on are the functions , where , , defined as follows. If is an -jet and its representative, then . Similarly, the formula defines the second canonical coordinates ; clearly, these coordinates satisfy Equations of the subgroup are , and equations of the subgroup are ,, .

By a differential invariant, we mean an -equivariant mapping of left -manifolds [13]. Given the actions of on and , we get the equation for the differential invariant : where and . Equation (8) splits to an equivalent system: where and (the orbit reduction method [18]).

The problem we consider in this paper reduces to solving (9) for some specific left -manifolds and . In our case, the orbit reduction method simplifies (8) and allows us to obtain its complete solution.

4. Canonical Vector Fields on Tangent Bundles

Any diffeomorphism , defined on an open set in , induces the corresponding lifted diffeomorphisms , , and . By a canonical vector field on , we mean a morphism such that and for all diffeomorphisms of , Condition (10) means that the diagram xy(12) commutes or, which is the same, for all , is a second-order vector on at the point , Since , condition (11) can also be written as

We express conditions (13) and (14) in coordinates. We write in a chart , , Then for fixed , the components are functions of the coordinates , and ; that is,

Theorem 3. Let be a morphism over the identity . The following two conditions are equivalent:(1) is a canonical vector field.(2)For any points , any charts , , at and , , at , and any diffeomorphism such that , where

Proof. Consider condition (14). With the abbreviations (18), the coordinates of the vector and the 1-jet are The coordinates of the vector are determined by Writing we can express condition (14), with help of (19) and (20), as

In the well-known sense, the canonical vector fields are completely determined by certain differential invariants, that is, equivariant mappings from the type fibre of into the type fibre of over with respect to the canonical actions of the differential group , induced by diffeomorphisms of . We can characterize these actions explicitly in terms of the first and second canonical coordinates , and , on the differential group . Note that If is an element of the differential group , denote by (resp., ) the transformation of (resp., ), defined by . For any points and , we denote The following lemma defines the points (24) explicitly.

Lemma 4. Let be an element of the differential group .(a)The canonical group action of on is given by the equations: (b) The canonical group action of on is given by the equations:

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

Since a canonical vector field is always a morphism over the identity mapping , the corresponding differential invariant has an expression ; we denote .

Theorem 5. A mapping is a differential invariant if and only if for all and .

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 , preserving a given point , such that , and to charts and such that and . Then, the components and of a canonical vector field satisfy where

5. Canonical Vector Fields: Classification

We find all solutions of the equations for differential invariants, associated with canonical vector fields (Theorem 5). These equations can be written in coordinates as The following lemma solves (31).

Lemma 8. The functions satisfy condition (31) if and only if for some constants .

Proof. (1) First we consider (31) for the group elements, belonging to the subgroup of ; equations of are and we get the system This equation is obviously satisfied at all points where . On the other hand, suppose that there exists at least one index such that ; then to every point one can find the group parameters such that . Indeed, if, for example, , we set This choice of the group parameters yields which shows that is independent of .
(2) In view of (37), we can write instead of . Turning back to conditions (31), we have the following equations for the functions : Properties of are completely determined by the subgroup of . If , , then by (38) satisfies the positive homogeneity condition We suppose, however, that the functions are defined at the origin , ; then it is easily seen that (39) also holds for . Indeed, in this case, we have for all , hence . On the other hand, the points and are always defined for ; then , which proves (39) for .
Then, we have from (28) by differentiation with respect to ; we see that the expression on the left does not depend on . For , we get , showing that is linear in and ; that is, Substituting now into (38), we get That is, and . Thus, , hence and by the trace operation, . Analogously, . These expressions together with (40) prove formula (33).
(3) If condition (33) is satisfied, then we get (31) by immediate substitution.

Now, we wish to solve (32). In view of Lemma 8, these equations are of the form

We prove separately the following lemma.

Lemma 9. The following two conditions are equivalent:(a)The functions satisfy (b)The functions are of the form where .

Proof. (1) If , Lemma 9, equation (43) yields for some (Lemma 8). Substituting back to (43), we get conditions for the coefficients : or, which is the same, ; that is,
To determine the constants , we write from which it follows that We apply to this formula various trace operations. Contractions in and then in yield and Contractions in and in yield and . From these formulas, we find hence The same computation applies to . Thus, for some constants . Formula (44) now follows from (52) and (45).
(2) Conversely, suppose that satisfies (44). Writing and substituting these expressions into (44), we have proving (13).

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

Lemma 10. The functions satisfy condition (42) if and only if for some constants .

Proof. (1) Suppose that satisfy (42). Then if , we have Differentiating with respect to , we have which shows that the derivative does not depend on . Then, however, where the functions and do not depend on . Substituting from (59) back to (57), we have hence . Thus, the functions and in (59) must satisfy
Note that the trace operation in and yields
(2) We now use (56) for the group parameters . We have the condition and, from (59), These equations split to the system Then, however, for some constants (Lemma 8), and where (Lemma 9). These functions satisfy (61); that is, The trace in and yields Consequently, and hence Then from (68), The terms containing should vanish separately. Since these terms are we have, from (71) and (73), Analogously, the terms with should vanish separately; that is, Since this equation can be written as we get and ; thus,
Summarizing, we see that condition (42) implies, from (59), (66), (67), (71), (74), and (77), (48) and
(3) It remains to prove that conditions (55) and (56) imply (42). The left-hand side of (42) is and the right-hand side is These formulas already verify condition (42). The proof is complete.

We can now summarize our results in the following theorem.

Theorem 11. Let be a manifold and let be a morphism over . The following two conditions are equivalent:(a) is a canonical vector field.(b)For any chart , , on where are arbitrary constants.

Proof. (1) We show that (a) implies (b). Suppose that we have a canonical vector field . Then, in any chart on the 2nd-order vector field has an expression such that for any points , any charts , , at and , , at , and any diffeomorphism such that , (Theorem 3). If and , the components and define a differential invariant (Theorem 5); then, however, 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 , , and , , be two charts on such that , let and be the corresponding coordinates on . We want to show that that is, in the notation of Lemma 1,
Each element of the family of vector fields (82) defines a differential invariant , where Recall that (Theorem 5). Now the right-hand sides of (86) can be written as and the left-hand sides are where Expressions (89) and (90) prove (86) as well as existence of .
(3) To complete the proof, it remains to show that the vector field is a canonical vector field; to this purpose, we verify condition (2) of Theorem 3. Express the vectors and as in formula (15), Section 4, We have already proved that