DIFFERENTIAL GEOMETRY AND APPLICATIONS Proc. Conf., Aug. 28 – Sept. 1, 1995, Brno, Czech Republic Masaryk University, Brno 1996, 141–150

LIFTINGS OF TENSOR FIELDS

TO THE COTANGENT BUNDLE

Miroslav Doupovec and Jan Kurek

Abstract. We study geometrical constructions transforming tensor fields of type (r, s) to the cotangent bundle. We present the full classification of all such natural operators in the cases (r, s) = (0, 1), (r, s) = (0, 2) and (r, s) = (1, 1). We also study some geometrical constructions of (p, q)–tensor fields on T^{∗}M by means of (r, s)–tensor fields on M, (r, s) ≠ (p, q).

1. Introduction

Constructions transforming tensor fields to natural bundles play an important role in differential geometry. From a general point of view, geometrical constructions are natural differential operators, cf. [5]. In some particular cases it is possible to give the full classification of all natural operators of certain type, i.e. to give the full list of all geometrical constructions in question.

By a tensor field of type (r, s) we shall understand a smooth section of the vector bundle T^{(r,s)}M = ⊗^{r}T M ⊗ ⊗^{s}T^{∗}M over a smooth manifold M. There are many papers which are devoted to the geometrical constructions transforming tensor fields of type (r, s) on M into tensor fields of type (p, q) on F M, where F is a natural bundle, cf. [2], [3] and [4]. Such geometrical constructions are in fact natural operators T^{(r,s)} ⇝ T^{(p,q)}F. The aim of this paper is to study natural operators T^{(r,s)} ⇝ T^{(r,s)}T^{∗}, where T^{∗} is a cotangent bundle. The full classification of all such natural operators was given by Mikulski in the case of (r, s) = (0, 0) (functions) and by Kobak and Kolář in the case (r, s) = (1, 0) (vector fields). We shall present classification theorems for other important cases of (r, s): (r, s) = (0, 1) (1–forms), (r, s) = (0, 2) (2–forms, Riemannian and symplectic structures) and (r, s) = (1, 1) (almost tangent structures, almost complex structures and f– structures). In some particular cases we shall also study geometrical constructions of the type T^{(r,s)} ⇝ T^{(p,q)}T^{∗}, (r, s) ≠ (p, q), i.e. liftings which do not preserve the type of a tensor field.

1991 Mathematics Subject Classification. 53A55, 58A20.

Key words and phrases. Natural operator, tensor field, cotangent bundle.

Supported by the GA CR, Grant No. 201/93/2125 and by the Maria Curie Skłodowska University.

This paper is in final form and no version of it will be submitted for publication elsewhere.

We remark that liftings of tensor fields to the cotangent bundle are used in the geometric approach to analytical mechanics, see [1] and [7]. All natural operators transforming classical linear connections on M into classical linear connections on T^{∗}M were determined by Kureš in [6]. We shall use the concept of a natural bundle and of a natural operator from [4]. All manifolds and maps are assumed to be infinitely differentiable and all manifolds are paracompact.

2. Liftings of functions and vector fields

We shall denote by q_{M} : T^{∗}M → M the cotangent bundle of a smooth mdimensional manifold M. The canonical coordinates on T^{∗}M will be denoted by (x^{i}, p_{i}). Further, denote by

(1)  λ_{M} = p_{i}dx^{i}

the classical Liouville 1–form on T^{∗}M and by

(2)  Λ_{M} = dλ_{M} = dp_{i} ∧ dx^{i}

the canonical symplectic 2–form on T^{∗}M. Then Λ_{M} defines a bijection S_{M} : T^{(1,0)}T^{∗}M → T^{(0,1)}T^{∗}M transforming A ∈ T T^{∗}M into its inner product with Λ_{M}. This isomorphism can be extended into a vector bundles isomorphism (also denoted by S_{M})

(3)  S_{M} : T^{(1,k−1)}T^{∗}M → T^{(0,k)}T^{∗}M

by

(S_{M}K)(X_{1}, …, X_{k}) = ⟨K(X_{1}, …, X_{k−1}), S_{M}(X_{k})⟩

for all vector fields X_{1}, …, X_{k} on T^{∗}M, where <, >: T E ×_{E} T^{∗}E → ℝ means the evaluation mapping. Further, each vector field X on M defines a function ⟨X, −⟩ : T^{∗}M → ℝ. If X = X^{i} \frac{∂}{∂x^{i}} and α = p_{i}dx^{i}, then ⟨X, α⟩ = X^{i}p_{i}. In general, we have an operator

(4)  i_{M} : T^{(1,k)}M → T^{(0,k)}T^{∗}M

defined by

(i_{M}K)(X_{1}, …, X_{k}) = ⟨K_{M}(T q_{M}X_{1}, …, T q_{M}X_{k}), −⟩

for all vector fields X_{1}, …, X_{k} on T^{∗}M. In coordinates, if

K = K_{j_{1},...,j_{k}}^{i} \frac{∂}{∂x^{i}} ⊗ dx^{j_{1}} ⊗ ⋯ ⊗ dx^{j_{k}}, then i_{M}K = K_{j_{1},...,j_{k}}^{i}p_{i}dx^{j_{1}} ⊗ ⋯ ⊗ dx^{j_{k}}.

Finally, we have an operator

(5)  γ_{M} : T^{(1,k)}M → T^{(1,k−1)}T^{∗}M

given by γ_{M} = S_{M}^{−1} ∘ i_{M}. If K = K_{j_{1},...,j_{k}}^{i} \frac{∂}{∂x^{i}} ⊗ dx^{j_{1}} ⊗ ⋯ ⊗ dx^{j_{k}}, then γ_{M}K = K_{j_{1},...,j_{k}}^{i}p_{i} \frac{∂}{∂p_{j_{1}}} ⊗ dx^{j_{2}} ⊗ ⋯ ⊗ dx^{j_{k}}. In particular, if F = F_{j}^{i} \frac{∂}{∂x^{i}} ⊗ dx^{j}, then i_{M}F =

F_{j}^{i}p_{i}dx^{j} and γ_{M}F = F_{j}^{i}p_{i} \frac{∂}{∂p_{j}}.

Let f : M → ℝ be a function on M. The vertical lift f^{V} of f to T^{∗}M is a function f^{V} : T^{∗}M → ℝ defined by f^{V} = f ∘ q_{M}. We have (fg)^{V} = f^{V} g^{V} and (f + g)^{V} = f^{V} + g^{V}. Further, Mikulski has in [8] proved

Proposition 1. All natural operators T^{(0,0)} ⇝ T^{(0,0)}T^{∗} transforming functions on M into functions on T^{∗}M are of the form f ↦ ϕ ∘ f^{V}, where ϕ : ℝ → ℝ is an arbitrary smooth function.

For comparison’s sake, in the case of a tangent bundle T M we have also a complete lift f^{C} defined by f^{C}(y) = df_{x}(y), x = p_{M}(y), where p_{M} : T M → M is the canonical projection.

Now let X be an arbitrary vector field on M. The vertical lift X^{V} of X to T^{∗}M is defined as the vector field on T^{∗}M generated by the homotheties of the vector bundle T^{∗}M. In other words, X^{V} is the well-known Liouville vector field L_{M} on T^{∗}M. The complete lift X^{C} of X to T^{∗}M is the flow prolongation of X (denoted also by T^{∗}X). Taking into account the operator i_{M} defined in (4), if X is a vector field on M, then i_{M}X is a function on T^{∗}M, so that the exterior differential d(i_{M}X) is a 1–form on T^{∗}M. Then S_{M}^{−1}(d(i_{M}X)) is exactly the complete lift X^{C}. We have (cf. [4] and [5])

Proposition 2. All natural operators T^{(1,0)} ⇝ T^{(1,0)}T^{∗} are of the form

X ↦ f(⟨X, λ_{M}⟩)X^{V} + g(⟨X, λ_{M}⟩)X^{C}

where f, g ∈ C^{∞}(ℝ, ℝ) are arbitrary smooth functions.

In coordinates, if X = X^{i} \frac{∂}{∂x^{i}}, then L_{M} = X^{V} = p_{i} \frac{∂}{∂p_{i}} and

(6)  X^{C} = X^{i} \frac{∂}{∂x^{i}} − \frac{∂X^{j}}{∂x^{i}} p_{j} \frac{∂}{∂p_{i}} .

Each vector field is completely determined by means of its action on functions. Moreover, we have

Lemma 1. Let Y_{1} and Y_{2} be vector fields on T^{∗}M such that Y_{1}(i_{M}X) = Y_{2}(i_{M}X) for all vector fields X on M. Then Y_{1} = Y_{2}.

and

Lemma 2. Let G_{1} and G_{2} be tensor fields on T^{∗}M of the type (0, k) or (1, k), k > 0 such that

G_{1}(X_{1}^{C}, …, X_{k}^{C}) = G_{2}(X_{1}^{C}, …, X_{k}^{C})

for all vector fields X_{1}, …, X_{k} on M. Then G_{1} = G_{2}.

For the proofs of Lemma 1 and Lemma 2 see e.g. [9].

3. Liftings of 1–forms

We first draw our attention to one geometrical construction transforming 1– forms on M into vector fields on T^{∗}M. This is a lifting which changes the type of an original tensor field. Let ω = ω_{i}dx^{i} be a 1–form on M and q_{M}^{∗}ω = ω_{i}dx^{i} be its pull-back to T^{∗}M. Then S_{M}^{−1}(q_{M}^{∗}ω) is a vertical vector field on T^{∗}M, called the vertical lift of ω to T^{∗}M and denoted by ω^{V}. In coordinates,

(7)  ω^{V} = ω_{i} \frac{∂}{∂p_{i}} ,

so that also (dx^{i})^{V} = \frac{∂}{∂p_{i}}. One verifies easily

Lemma 3. Let ω, θ be 1–forms on M and f be a function on M. Then I. ω^{V} (f^{V}) = 0, II. (ω + θ)^{V} = ω^{V} + θ^{V}, III. (fω)^{V} = f^{V} ω^{V}, IV. [ω^{V} , θ^{V} ] = 0.

Obviously,

(8)  ω^{V} (i_{M}X) = (⟨ω, X⟩)^{V} ,

so that by Lemma 1 one can also define the vertical lift ω^{V} by means of the condition (8). In the following assertion we describe the main properties of the complete lift of vector fields.

Lemma 4. Let X, Y be vector fields on M, ω be a 1–form on M, F be a (1, 1)– tensor field on M and let S be a (1, 2)–tensor field on M. Then

I. (X + Y)^{C} = X^{C} + Y^{C}, II. (fX)^{C} = f^{V} X^{C} − (i_{M}X)(df)^{V}, III. [X^{C}, ω^{V} ] = (L_{X}ω)^{V}, where L_{X} denotes the Lie derivative with respect

to X, IV. [X^{C}, γF ] = γ(L_{X}F), V. [X^{C}, Y^{C}] = [X, Y]^{C}, VI. (γS)(X^{C}) = γ(S_{X}), where S_{X} is the tensor field of the type (1, 1) defined

by S_{X}(Z) = S(X, Z) for any vector field Z on M, VII. X^{C}f^{V} = (Xf)^{V}, VIII. X^{C}(i_{M}Y) = i_{M} [X, Y].

By Lemma 1, the complete lift X^{C} can be also defined by VIII. The aim of this section is to prove

Proposition 3. All natural operators T^{(0,1)} ⇝ T^{(0,1)}T^{∗} transforming 1–forms on M into 1–forms on T^{∗}M are of the form

(9)  ω ↦ k_{1}λ_{M} + k_{2}q_{M}^{∗}ω

where λ_{M} is the Liouville 1–form (1), q_{M}^{∗}ω is the pull-back of ω to T^{∗}M with respect to the bundle projection q_{M} : T^{∗}M → M and k_{1} and k_{2} are arbitrary real numbers.

Proof. Let G_{m}^{r} be the group of all invertible r–jets of ℝ^{m} into ℝ^{m} with source and target zero. We shall denote by (a_{j}^{i}, a_{jk}^{i}, …, a_{j_{1},...,j_{r}}^{i}) the canonical coordinates on G_{m}^{r} and by tilde the coordinates of the inverse element. By [5], r–th order natural operators T^{(0,1)} ⇝ T^{(0,1)}T^{∗} are in a canonical bijection with G_{m}^{r+1}–equivariant maps (J^{r}T^{(0,1)})_{0}ℝ^{m} ⊕ T_{0}^{∗}ℝ^{m} → (T^{(0,1)}T^{∗})ℝ^{m} of the corresponding standard fibres. Each 1–form on T^{∗}M has the coordinate expression A_{i}dx^{i} + B^{i}dp_{i}. The canonical coordinates on the standard fibre (J^{r}T^{(0,1)})_{0}ℝ^{m} will be denoted by ω_{i}, ω_{ij}, …, ω_{i_{1},...,i_{r}}. The rest of the proof will proceed in three steps.

I. Consider first the zero order natural operators. We have to find all G_{m}^{1}– equivariant maps of the form A_{i} = A_{i}(p_{i}, ω_{i}), B^{i} = B^{i}(p_{i}, ω_{i}) which express the coordinate form of the natural operators in question. Using standard evaluations one computes easily the following transformations laws

\overline{p}_{i} = ã_{i}^{j}p_{j},
\overline{ω}_{i} = ã_{i}^{j}ω_{j}, \overline{B}^{i} = a_{j}^{i}B^{j},
\overline{A}_{i} = ã_{i}^{j}A_{j} + a_{jk}^{ℓ}ã_{i}^{k}p̄_{ℓ}B^{j}.

Then the homotheties ã_{i}^{j} = tδ_{i}^{j}, ã_{jk}^{i} = 0 yield

tA_{i} = A_{i}(tp_{i}, tω_{i}) and \frac{1}{t}B^{i} = B^{i}(tp_{i}, tω_{i}).

Hence A_{i} are linear in p_{i} and in ω_{i}, so that

(10)  A_{i} = k_{1}p_{i} + k_{2}ω_{i}

which is the coordinate expression of (9). Further, B^{i} = tB^{i}(tp_{i}, tω_{i}). Setting t → 0 we get B^{i} = 0. Finally, the full equivariance does not change the expression (10).

II. The r–th order operators correspond to the equivariant maps of the form

A_{i} = A_{i}(p_{i}, ω_{i}, ω_{ij}, …, ω_{i_{1},...,i_{r}})
B^{i} = B^{i}(p_{i}, ω_{i}, ω_{ij}, …, ω_{i_{1},...,i_{r}}).

Applying homotheties, one verifies easily that

tA_{i} = A_{i}(tp_{i}, tω_{i}, t^{2}ω_{ij}, …, t^{r}ω_{i_{1},...,i_{r}})
\frac{1}{t}B^{i} = B^{i}(tp_{i}, tω_{i}, t^{2}ω_{ij}, …, t^{r}ω_{i_{1},...,i_{r}}).

Therefore r–th order natural operators are reduced to the case I. for any r > 0. III. By [5], every natural operator T^{(0,1)} ⇝ T^{(0,1)}T^{∗} has finite order. □

4. Natural operators T^{(r,s)} ⇝ T^{(p,q)}T^{∗}

In this section we shall solve the following problem: For which r, s, p, q does there exist a natural operator T^{(r,s)} ⇝ T^{(p,q)}T^{∗}? Such natural operators are examples of geometrical constructions which change the type of a tensor field (if (p, q) ≠ (r, s)).

Definition. An operator A : T^{(r,s)} ⇝ T^{(p,q)}T^{∗} is said to be absolute, if A_{M}X = A_{M}O_{M} for all tensor fields X : M → T^{(r,s)}M, where O_{M} means the zero tensor field of type (r, s).

By the following assertion, if (p, q) ≠ (0, 0), then we can associate an absolute (p, q)–tensor field on T^{∗}M to each (r, s)–tensor field on M.

Lemma 5. If (p, q) ≠ (0, 0), then there is an absolute operator T^{(r,s)} ⇝ T^{(p,q)}T^{∗} for all r, s, p, q.

Proof. Obviously, X ↦ L_{M} is an absolute operator T^{(r,s)} ⇝ T^{(1,0)}T^{∗} and X ↦ λ_{M} is an absolute operator T^{(r,s)} ⇝ T^{(0,1)}T^{∗}. Evaluating the tensor products of L_{M} and λ_{M}, we can construct a tensor of an arbitrary type (p, q) on T^{∗}M. □

Lemma 6. A natural operator T^{(r,s)} ⇝ T^{(0,0)}T^{∗} transforming (r, s)–tensor fields on M into functions on T^{∗}M exists if and only if r ≥ s.

Proof. The coordinate expression of a natural operator T^{(r,s)} ⇝ T^{(0,0)}T^{∗} is f(X_{j_{1},...,j_{s}}^{i_{1},...,i_{r}}, p_{i}). By homotheties,

f (X_{j_{1},...,j_{s}}^{i_{1},...,i_{r}}, p_{i}) = f (k^{r−s}X_{j_{1},...,j_{s}}^{i_{1},...,i_{r}}, kp_{i}) .

Hence if s > r, then the only function f is zero. For r ≥ s we can get a function on T^{∗}M by contraction. □

Lemma 5 and Lemma 6 describe the most trivial constructions of a (p, q)–tensor field on T^{∗}M by means of an (r, s)–tensor field on M. There are also nontrivial natural operators T^{(r,s)} ⇝ T^{(p,q)}T^{∗} in many particular cases. For example, X ↦ d(i_{M}X) is an operator T^{(1,0)} ⇝ T^{(0,1)}T^{∗}. On the other hand, f ↦ S_{M}^{−1}(df^{V}) is an operator T^{(0,0)} ⇝ T^{(1,0)}T^{∗}. In what follows we give the full classification of natural operators transforming 1–forms on M into vector fields and into (1, 1)–tensor fields on T^{∗}M. Considering the isomorphism S_{M} : T^{(1,0)}T^{∗}M → T^{(0,1)}T^{∗}M we see that (S_{M}^{−1})λ_{M} is exactly the Liouville vector field L_{M} on T^{∗}M. We have

Proposition 4. All natural operators T^{(0,1)} ⇝ T^{(1,0)}T^{∗} transforming 1–forms on M into vector fields on T^{∗}M are of the form

(11)  ω ↦ k_{1}L_{M} + k_{2}ω^{V}

where k_{1}, k_{2} are arbitrary real numbers.

Proof. Each vector field on T^{∗}M is of the form Y^{i} \frac{∂}{∂x^{i}} +P_{i} \frac{∂}{∂p_{i}}. The transformation formulas for Y^{i} and P_{i} are

\overline{Y}^{i} = a_{j}^{i}Y^{j}, \overline{P}_{i} = ã_{i}^{j}P_{j} + ã_{iℓ}^{j}a_{k}^{ℓ}Y^{k}p_{j}.

The rest of the proof is quite analogous to that of Proposition 3. □

Proposition 5. All natural operators T^{(0,1)} ⇝ T^{(1,1)}T^{∗} transforming 1–forms on M into (1, 1)–tensor fields on T^{∗}M are of the form

(12) ω ↦ k_{1}Id_{T^{∗}M} + k_{2}L_{M} ⊗ λ_{M} + k_{3}ω^{V} ⊗ q_{M}^{∗}ω + k_{4}L_{M} ⊗ q_{M}^{∗}ω + k_{5}ω^{V} ⊗ λ_{M}

where Id_{T^{∗}M} is the identity (1, 1)–tensor field on T^{∗}M and k_{1}, …, k_{5} are arbitrary real numbers.

Proof. The coordinate expression of a (1, 1)–tensor field on T^{∗}M is

(13)  A_{j}^{i} \frac{∂}{∂x^{i}} ⊗ dx^{j} + B^{ij} \frac{∂}{∂x^{i}} ⊗ dp_{j} + C_{ij} \frac{∂}{∂p_{i}} ⊗ dx^{j} + D_{i}^{j} \frac{∂}{∂p_{i}} ⊗ dp_{j}.

The action of G_{m}^{2} on the standard fibre of T^{(1,1)}T^{∗} is

(14)  \begin{aligned}\overline{A}_{j}^{i} & = a_{k}^{i}ã_{j}^{ℓ}A_{ℓ}^{k} + a_{k}^{i}ã_{ℓs}^{r}ã_{j}^{s}\overline{p}_{r}B^{kℓ} \\ \overline{B}^{ij} & = a_{k}^{i}a_{ℓ}^{j}B^{kℓ} \\ \overline{C}_{ij} & = ã_{is}^{r}a_{k}^{s}ã_{j}^{ℓ}p_{r}A_{ℓ}^{k} + ã_{is}^{r}a_{k}^{s}a_{ℓu}^{t}ã_{j}^{u}p_{r}\overline{p}_{t}B^{kℓ} + ã_{i}^{k}ã_{j}^{ℓ}C_{kℓ} + ã_{i}^{k}a_{ℓs}^{r}ã_{j}^{s}\overline{p}_{r}D_{k}^{ℓ} \\ \overline{D}_{i}^{j} & = ã_{is}^{r}a_{k}^{s}a_{ℓ}^{j}p_{r}B^{kℓ} + ã_{i}^{k}a_{ℓ}^{j}D_{k}^{ℓ} \end{aligned}

I. The first order natural operators correspond to the G_{m}^{2}– equivariant maps

A_{j}^{i} = A_{j}^{i}(p_{i}, ω_{i}, ω_{ij}), B^{ij} = B^{ij}(p_{i}, ω_{i}, ω_{ij})
C_{ij} = C_{ij}(p_{i}, ω_{i}, ω_{ij}), D_{i}^{j} = D_{i}^{j}(p_{i}, ω_{i}, ω_{ij})

where \overline{ω}_{ij} = ã_{i}^{k}ã_{j}^{ℓ}ω_{kℓ}+ã_{ij}^{r}ω_{r}. Using homotheties we find A_{j}^{i} = k_{1}δ_{j}^{i}, B^{ij} = 0, C_{ij} = k_{2}p_{i}p_{j} + k_{3}ω_{i}ω_{j} + k_{4}p_{i}ω_{j} + k_{5}ω_{i}p_{j} + k_{6}ω_{ij}, D_{i}^{j} = k_{7}δ_{i}^{j}. Then the full equivariance yields k_{6} = 0 and k_{7} = k_{1}, which is nothing else but the coordinate form of (12). The remaining steps II. and III. are quite similar to those in the proof of Proposition 3. □

5. Liftings of (0, 2)–tensor fields

Let G = g_{ij}dx^{i} ⊗ dx^{j} be a (0, 2)–tensor field on M. Then G induces another (0, 2)–tensor field G′ = g_{ji}dx^{i} ⊗ dx^{j} on M given by ⟨G, X ⊗ Y ⟩ = ⟨G′, Y ⊗ X⟩. We prove

Proposition 6. All natural operators T^{(0,2)} ⇝ T^{(0,2)}T^{∗} transforming (0, 2)– tensor fields on M into (0, 2)–tensor fields on T^{∗}M are of the form

(15)  G ↦ k_{1}Λ_{M} + k_{2}λ_{M} ⊗ λ_{M} + k_{3}q_{M}^{∗}G + k_{4}q_{M}^{∗}G′

where k_{1}, k_{2}, k_{3} and k_{4} are arbitrary real numbers.

Proof. A (0, 2)–tensor field on T^{∗}M is of the form

P_{ij}dx^{i} ⊗ dx^{j} + Q_{i}^{j}dx^{i} ⊗ dp_{j} + R_{j}^{i}dp_{i} ⊗ dx^{j} + S^{ij}dp_{i} ⊗ dp_{j}.

A direct calculation leads to the formulas

\overline{P}_{ij} = ã_{i}^{k}ã_{j}^{ℓ}P_{kℓ} + ã_{j}^{k}a_{ℓr}^{m}ã_{j}^{r}\overline{p}_{m}Q_{k}^{ℓ} + a_{kr}^{m}ã_{i}^{r}ã_{j}^{ℓ}\overline{p}_{m}R_{ℓ}^{k} + a_{kr}^{m}ã_{i}^{r}a_{ℓs}^{n}ã_{j}^{s}\overline{p}_{m}\overline{p}_{n}S^{kℓ},
\overline{Q}_{i}^{j} = ã_{i}^{k}ã_{ℓ}^{j}Q_{k}^{ℓ} + a_{kr}^{m}ã_{i}^{r}a_{ℓ}^{j}\overline{p}_{m}S^{kℓ},
\overline{R}_{j}^{i} = a_{k}^{i}ã_{j}^{ℓ}R_{ℓ}^{k} + a_{k}^{i}a_{ℓs}^{n}ã_{j}^{s}\overline{p}_{n}S^{kℓ},
\overline{S}^{ij} = a_{k}^{i}a_{ℓ}^{j}S^{kℓ}.

I. Zero order natural operators correspond to the G_{m}^{1}–equivariant maps P_{ij} = P_{ij}(p_{i}, g_{ij}), Q_{i}^{j} = Q_{i}^{j}(p_{i}, g_{ij}), R_{j}^{i} = R_{j}^{i}(p_{i}, g_{ij}), S^{ij} = S^{ij}(p_{i}, g_{ij}). Evaluating equivariance with respect to the homotheties we find S^{ij} = 0, R_{j}^{i} = k_{1}δ_{j}^{i}, Q_{j}^{i} = k_{2}δ_{j}^{i}, P_{ij} = k_{3}g_{ij} + k_{4}g_{ji} + k_{5}p_{i}p_{j}. Then the full equivariance leads to such relations among k_{1}, …, k_{5} which correspond to (15). The rest of the proof is similar to steps II. and III. in the proof of Proposition 2. □

6. Liftings of (1, 1)–tensor fields

Let F be a tensor field of type (1, 1) on M, F = F_{j}^{i} \frac{∂}{∂x^{i}} ⊗ dx^{j}. Then i_{M}F = F_{i}^{k}p_{k}dx^{i} is a 1–form on T^{∗}M, so that the exterior differential d(i_{M}F) is a 2–form on T^{∗}M. The complete lift of F to T^{∗}M is defined as the (1, 1)– tensor field on T^{∗}M given by F^{C} = S_{M}^{−1}(d(i_{M}F)). In coordinates,

(16)  F^{C} = F_{j}^{i} \frac{∂}{∂x^{i}} ⊗ dx^{j} + (\frac{∂F_{j}^{k}}{∂x^{i}} − \frac{∂F_{i}^{k}}{∂x^{j}} ) p_{k} \frac{∂}{∂p_{i}} ⊗ dx^{j} + F_{j}^{i} \frac{∂}{∂p_{i}} ⊗ dp_{j}.

Let Id_{M} be the identity (1, 1)–tensor field on M and F̃ = F_{i}^{i} be the function on T^{∗}M given by the full contraction of F. We prove

Proposition 7. All first order natural ℝ–linear operators T^{(1,1)} ⇝ T^{(1,1)}T^{∗} are of the form

(17) F ↦ k_{1}F^{C} + k_{2}Id_{T^{∗}M} + k_{3}L_{M} ⊗ λ_{M} + k_{4}F̃ · Id_{T^{∗}M} + k_{5}F̃ · L_{M} ⊗ λ_{M}

where k_{1}, …, k_{5} are arbitrary real numbers.

Proof. According to the general theory, we have to find all G_{m}^{2}–equivariant maps A_{j}^{i} = A_{j}^{i}(p_{i}, F_{j}^{i}, F_{jk}^{i}), B^{ij} = B^{ij}(p_{i}, F_{j}^{i}, F_{jk}^{i}), C_{ij} = C_{ij}(p_{i}, F_{j}^{i}, F_{jk}^{i}) and D_{i}^{j} = D_{i}^{j}(p_{i}, F_{j}^{i}, F_{jk}^{i}). The action of G_{m}^{2} on the standard fibre of T^{(1,1}T^{∗} is given by (14). Further,

\overline{F} _{j}^{i} = a_{ℓ}^{i}ã_{j}^{k}F_{k}^{ℓ},
\overline{F} _{jk}^{i} = a_{m}^{i}ã_{j}^{n}ã_{k}^{p}F_{np}^{m} + a_{m}^{i}ã_{jk}^{n}F_{n}^{m} + a_{mp}^{i}a_{j}^{n}ã_{k}^{p}F_{n}^{m}.

Using homotheties we find easily A_{j}^{i} = a_{1}F_{j}^{i} + a_{2}δ_{j}^{i}F_{ℓ}^{ℓ} + a_{9}δ_{j}^{i}, B^{ij} = 0, C_{ij} = (a_{7}F_{ℓ}^{ℓ} + a_{8})p_{i}p_{j} + a_{5}p_{ℓ}F_{ij}^{ℓ} + a_{6}p_{ℓ}F_{ji}^{ℓ}, D_{j}^{i} = a_{3}F_{j}^{i} + a_{4}δ_{j}^{i}F_{ℓ}^{ℓ} + a_{10}δ_{j}^{i}. Finally, the full equivariance reads a_{1} = a_{3} = a_{5} = −a_{6} = k_{1}, a_{2} = a_{4} = k_{4}, a_{9} = a_{10} = k_{2}, a_{8} = k_{3} and a_{7} = k_{5}, which corresponds to (17). □

Some properties of the complete lift F^{C} are expressed in the following assertion (cf. [9]).

Lemma 7. Let X be a vector field on M, ω be a 1–form and let F and G be (1, 1)–tensor fields on M. Then

I. F^{C}ω^{V} = (ω ∘F)^{V} where ω ∘F is a 1–form defined by (ω ∘F)(Z) = ω(F Z)

for any vector field Z on M, II. F^{C}(γG) = γ(G ∘ F), III. F^{C}(X^{C}) = (F (X))^{C} + γ(L_{X}F), IV. (F^{C})^{2} = (F^{2})^{C} + γN_{F} where N_{F} is the Nijenhuis tensor of F, V. If F is an almost complex structure on M, then F^{C} is an almost complex

structure on T^{∗}M if and only if F is integrable, VI. (Id_{M})^{C} = Id_{T^{∗}M}.

By Lemma 2, the complete lift F^{C} can be also defined by III. from the above assertion.

Remark 1. If E is a product preserving functor, then the definition of the complete lift of a (1, 1)–tensor field F from a manifold M to E(M) is quite different, cf. [3]. In this case we interpret F as a linear mapping F : T M → T M over the identity of M and we define F^{C} = η_{M} ∘ E(F) ∘ η_{M}^{−1}, where η_{M} : E(T M) → T (EM) is a natural isomorphism from [5]. Then the property III from Lemma 7 is of the form

(*)  F^{C}(X^{C}) = (F (X))^{C}.

Moreover, there is another approach to the definition of the complete lift, where the relation (*) even plays an essential role, see [2]. Let E be a natural bundle (not necessarily a product preserving functor) and F be a tensor field of type (1, 1) on M. F is said to admit the complete lift to E(M) if there is a tensor field \overline{F} of type (1, 1) on E(M) such that \overline{F} (X^{C}) = (F (X))^{C} for all vector fields X on M. Gancarzewicz proved in [2] that F admits the complete lift to T^{∗}M (in the sense of the above definition) if and only if F = f · Id_{M} for some function f on M.

Remark 2. There are also nontrivial natural operators T^{(1,1)} ⇝ T^{(p,q)}T^{∗}, (p, q) ≠ (1, 1). For example, F ↦ iF and F ↦ γF are examples of natural operators T^{(1,1)} ⇝ T^{(0,1)}T^{∗} and T^{(1,1)} ⇝ T^{(1,0)}T^{∗}, respectively. Further, the contractions F_{i}^{i}, F_{k}^{i}F_{i}^{k}, … are functions on T^{∗}M naturally generated by a (1, 1)–tensor field F on M.

References

[1] Crampin, M., Cantrijn, J. and Sarlet, W., Lifting geometric objects to a cotangent bundle, and the geometry of the cotangent bundle of a tangent bundle, JGP 4, n. 4 (1987), 469– 492.

[2] Gancarzewicz, J., Complete lifts of tensor fields of type (1, k) to natural bundles, Zeszyty naukowe UJ, Prace matematyczne 23 (1882), 51–84.

[3] Gancarzewicz, J., Mikulski, W. and Pogoda, Z., Lifts of some tensor fields and connections to product preserving bundles, Nagoya Math. J. 135 (1994), 1–41.

[4] Kobak, P., Natural liftings of vector fields to tangent bundles of 1–forms, Mathematica Bohemica 116 (1991), 319–326.

[5] Kolář, I., Michor, P.W. and Slovák, J., Natural Operations in Differential Geometry, Springer-Verlag, 1993.

[6] Kureš, M., Natural lifts of classical linear connections to the cotangent bundle (to appear).

[7] de León, M. and Rodrigues, P.R., Methods of Differential Geometry in Analytical Mechanics, North-Holland Mathematics Studies 158, Amsterdam, 1989.

[8] Mikulski, W., Natural transformations transforming functions to functions on some linear natural bundles (to appear).

[9] Yano, K. and Ishihara, S., Tangent and cotangent bundles, Marcel Dekker Inc., New York, 1973.

Department of Mathematics, Technical University of Brno Technická 2, 616 69 Brno, Czech Republic E-mail: doupovec@kinf.fme.vutbr.cz

Institute of Mathematics, Maria Curie-Skłodowska University Plac Marii Curie-Skłodowskej 1, 20-031 Lublin, Poland E-mail: kurek@golem.umcs.lublin.pl