Lobachevskii Journal of Mathematics

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ISSN 1818-9962

Vol. 27, 2007, 41–46 \begin{matrix}c \\ ◯ \end{matrix} J. Kurek and W.M. Mikulski

J. Kurek and W.M. Mikulski

RIEMANNIAN STRUCTURES ON HIGHER ORDER FRAME BUNDLES OVER RIEMANNIAN MANIFOLDS

(submitted by M. A. Malakhaltsev)

Abstract. We describe all Mf_{m}-natural operators A : Riem ⇝ RiemP^{r} transforming Riemannian structures g on m-dimensional manifolds M into Riemannian structures A(g) on the r-th order frame bundle P^{r}M = invJ_{0}^{r}(R^{m}, M) over M.

Manifolds and maps are assumed to be of class C^{∞}. Manifolds assumed to be finite dimensional and without boundaries.

Let Mf_{m} denote the category of m-dimensional manifolds and their embeddings (i.e. diffeomorphisms onto open subsets) and FM denote the category of fibred manifolds and their fibred maps.

For any m-manifold M we have the r-th order frame bundle P^{r}M = invJ_{0}^{r}(R^{m}, M) of M. This is a principal bundle with the corresponding Lie group G_{m}^{r} = J_{0}^{r}(R^{m}, R^{m})_{0} acting on the right on P^{r}M via compositions of jets. Every Mf_{m}-map ψ : M_{1} → M_{2} induces P^{r}ψ : P^{r}M_{1} → P^{r}M_{2} by P^{r}ψ(j_{0}^{r}φ) = j_{0}^{r}(ψ ∘ φ), where φ : R^{m} → M_{1} is a Mf_{m}-map. The correspondence P^{r} : Mf_{m} → FM is a bundle functor in the sense of [2].

For any n-manifold N we have the Riemannian bundle Riem(N) = ⋃_{y∈N} Met(T_{y}N) over N, where given a vector space V we denote the set

natural operator.

2000 Mathematical Subject Classification. 58A20, 58A32.

Key words and phrases. Riemannian structure, higher order frame bundle,

of scalar multiplications G : V ×V → R on V by Met(V). (We recall that G : V × V → R is a scalar multiplication if it is symmetric bilinear and positive define.) Clearly, Riem(N) is an open subbundle in the vector bundle T^{∗}N ⊙ T^{∗}N of symmetric tensors of type (0, 2) over N. Sections g : N → Riem(N) are the so called Riemannian structures on N. Every embedding ψ : N_{1} → N_{2} induces Riem(ψ) : Riem(N_{1}) → Riem(N_{2}) being the restriction of T^{∗}ψ ⊙ T^{∗}ψ : T^{∗}N_{1} ⊙ T^{∗}N_{1} → T^{∗}N_{2} ⊙ T^{∗}N_{2}. The correspondence Riem : Mf_{n} → FM is also a bundle functor in the sense of [2].

In the present short note we study the problem how a Riemannian structure g on an m-dimensional manifold M can induce (canonically) a Riemannian structure A(g) on P^{r}M. This problem is reflected in the concept of Mf_{m}-natural operators A : Riem ⇝ RiemP^{r}. In the note we describe explicitly all Mf_{m}-natural operators A in question.

A general concept of natural operators can be found in the fundamental monograph [2]. We need only the following partial case of the definition of natural operators.

An Mf_{m}-natural operator A : Riem ⇝ RiemP^{r} is a family of Mf_{m}- invariant regular operators (functions)

A = A_{M} : \underline{Riem}(M) → \underline{Riem}(P^{r}M)

for any Mf_{m}-object M, where \underline{Riem}(N) is the set of all Riemannian structures on N (sections of Riem(N) → N) for any manifold N. The invariance means that if g_{1} ∈ \underline{Riem}(M_{1}) and g_{2} ∈ \underline{Riem}(M_{2}) are related by an Mf_{m}-map ψ : M_{1} → M_{2} (i.e. Riem(ψ) ∘ g_{1} = g_{2} ∘ ψ) then A(g_{1}) and A(g_{2}) are P^{r}ψ-related. The regularity means that A transforms smoothly parametrized families of Riemannian structures into smoothly parametrized ones.

For r = 1, P^{1}M is equivalent with the bundle of linear frames over M. In this case we have the following example basing on a very important classical construction presented in the proof of Theorem 1.5 in [1].

Example 1. Let g be a Riemannian structure on an m-manifold M. Let ∇ be the Levi-Civita connection of g and let ω = (ω_{k}^{j}) : T P^{1}M → gl(m) be its connection form. Let θ = (θ^{i}) : T P^{1}M → R^{m} be the canonical form on P^{1}M. We put

g̃(X^{∗}, Y^{∗}) = ∑_{i} θ^{i}(X^{∗})θ^{i}(Y^{∗}) + ∑_{j,k} ω_{k}^{j}(X^{∗})ω_{k}^{j}(Y^{∗}) , X^{∗}, Y^{∗} ∈ T_{u}P^{1}M.

Then g̃ is a Riemannian structure on P^{1}M, see the proof of Theorem 1.5 in [1]. Clearly, the correspondence A^{1} : Riem ⇝ RiemP^{1} given by A^{1}(g) = g̃ for all g ∈ \underline{Riem}(M) is an Mf_{m}-natural operator.

To generalize Example 1 on all r we firstly reformulate it as follows.

Example 2. Let g be a Riemannian structure on M. Let Let (θ^{i}, ω_{k}^{j}) be the basis of 1-forms on P^{1}M, where θ = (θ^{i}) and ω = (ω_{k}^{j}) is as in Example 1 for g. Let X^{1}(g), …, X^{L_{1}}(g), where L_{1} = dim(P^{1}R^{m}) = m + m^{2}, be the dual (to (θ^{i}, ω_{k}^{j})) basis of vector fields on P^{1}M. Then g̃ = ∑_{s=1}^{L_{1}}(X^{s}(g))^{∗} ⊙ (X^{s}(g))^{∗} is a Riemannian structure on P^{1}M (the same as in Example 1).

So, to generalize Example 1 (or Example 2) on all r we need to construct an absolute parallelism (basis of global vector fields) on P^{r}M canonically dependent on a given Riemannian structure g on M.

From now on let A_{1}, …, A_{dim(g_{m}^{r})} be the standard basis in g_{m}^{r} = Lie(G_{m}^{r}) (i.e. the basis j_{0}^{r}(x^{α} \frac{∂}{∂x^{i}}) ∈ (J_{0}^{r}T R^{m})_{0} = g_{m}^{r} for i = 1, …, m, 1 ≤ |α| ≤ r).

Example 3. Construction of an absolute parallelism on P^{r}M from a Riemannian structure on M. Let g be a Riemannian structure on an m-manifold M. Let ∇ be the Levi-Civita connection of g. Let i = 1, …, m. We have a vector field Y^{i}(g) on P^{r}M defined as follows. Let σ = j_{0}^{r}φ ∈ (P^{r}M)_{x}, x ∈ M. Let v^{i} = T φ(\frac{∂}{∂x^{i}} _{|0}) ∈ T_{x}M. We extend v^{i} to the constant vector field ṽ^{i} on T_{x}M. Then on some neighborhood of x we have the vector field V^{i}(g) = (Exp_{x}^{∇})_{∗}ṽ^{i}, where Exp_{x}^{∇} : T_{x}M ⊃ U_{0_{x}} → Ũ_{x} ⊂ M is the exponent of ∇. We define Y^{i}(g)_{σ} := P^{r}(V^{i}(g))_{σ}, where P^{r}V is the flow lifting of a vector field V on M to P^{r}M (if {φ_{t}} is the flow of V then {P^{r}φ_{t}} is the flow of P^{r}V). It is easy to see that Y^{i}(g)_{σ} projects onto v^{i} by the bundle projection P^{r}M → M. So, it is a simple observation that Y^{i}(g), A_{j}^{∗} for i = 1, …, m, j = 1, …, dim(g_{m}^{r}) is an absolute parallelism on P^{r}M (canonically depending on g), where given A ∈ Lie(G_{m}^{r}) we denote the fundamental vector field on the principal bundle P^{r}M by A^{∗}.

Now, we are in position to extend Example 1 (or 2) on all r.

Example 4. Let g be a Riemannian structure on an m-manifold M. Let (Y^{i}(g), A_{j}^{∗}) be the parallelism from Example 3. Let ω^{s}(g) for s = 1, …, dim(P^{r}R^{m}) be the dual basis of 1-form on P^{r}M. We put

g̃^{r} := ∑_{s} ω^{s}(g) ⊙ ω^{s}(g) .

Clearly, g̃^{r} is a Riemannian structure on P^{r}M. Clearly, the correspondence A^{[r]} : Riem ⇝ RiemP^{r} given by A^{[r]}(g) = g̃^{r} for all g in question is an Mf_{m}-natural operator. One can observe easily that A^{1} = A^{[1]}.

To present a general example of Mf_{m}-natural operators Riem ⇝ RiemP^{r} we need some preparation and notations.

According to the global basis of vector fields Y^{i}(g), A_{j}^{∗} on P^{r}M from Example 3, given g ∈ \underline{Riem}(M) we have a canonical (in g) fibred diffeomorphism

(∗)  I_{g} : P^{r}M × Met(R^{L_{r}}) → Riem(P^{r}M)

covering id_{P^{r}M} defined by the condition that the matrix of I_{g}(σ, G) in the basis (Y^{i}(g)(σ), A_{j}^{∗}(σ)) is the same as the one of G in the usual canonical basis of R^{L_{r}}.

Given g ∈ \underline{Riem}(M) we have the projection

Ort(g) : P^{1}M = LM → O(M, g)

given by the Gramm orthonormalization with respect to g (for l = (l_{i}) ∈ L_{x}M, Ort(g)(l) is the orthonormalization of l with respect to g_{x}).

From now on we denote

Q^{r} = (Ort(g^{o}) ∘ π_{1}^{r})^{−1}(l^{o}) ⊂ (P^{r}R^{m})_{0} ,

where g^{o} is the usual flat Riemannian structure on R^{m} and l^{o} is the usual canonical basis in R^{m} = T_{0}R^{m} and π_{1}^{r} : P^{r}M → P^{1}M is the jet projection. Of course, Q^{r} is a submanifold in (P^{r}R^{m})_{0}.

For s = 0, 1, …, ∞, let Z^{s} = J_{0}^{s}(Riem(R^{m})) be the set of all s-jets j_{0}^{s}g of Riemannian structures g on R^{m}. If s is finite, Z^{s} is a finite dimensional manifold (as the fibre of the s-jet prolongation J^{s}(Riem(R^{m})) of the bundle Riem(R^{m}) → R^{m}). Z^{∞} is a topological space with respect to the inverse limit topology given by the inverse system … → Z^{s+1} → Z^{s} → … → Z^{0} of jet projections.

Now, we are in position to present the following general construction.

Example 5. General construction. Let μ : Z^{∞} × Q^{r} → Met(R^{L_{r}}), where L_{r} = dim(P^{r}R^{m}), be a map satisfying the following local finite determination property (a_{r}):

(a_{r}) For any ρ ∈ Z^{∞} and σ ∈ Q^{r} we can find an open neighborhood U ⊂ Z^{∞} of ρ, an open neighborhood V ⊂ Q^{r} of σ, a natural number s and a smooth map f : π_{s}(U)×V → Met(R^{L_{r}}) such that μ = f ∘(π_{s}×id_{V}) on U × V, where π_{s} : Z^{∞} → Z^{s} is the jet projection.

(A simple example of such μ is μ = f ∘ (π_{s} × id_{Q^{r}}) for smooth f : Z^{s} × Q^{r} → Met(R^{L_{r}}) for finite s.) Given a Riemannian structure g on an m-manifold M we define a Riemannian structure A^{<μ>}(g) on P^{r}M as follows. Let σ ∈ (P^{r}M)_{x}, x ∈ M. Choose a g-normal coordinate system ψ on M with center x such that P^{r}ψ(σ) ∈ Q^{r}. Of course, such ψ exists. Then germ_{x}(ψ) is uniquely determined. We put

(∗∗) A^{<μ>}(g)_{σ} = Riem(P^{r}(ψ^{−1}))(I_{ψ_{∗}g}(P^{r}ψ(σ), μ(j_{0}^{∞}(ψ_{∗}g), P^{r}ψ(σ)))) .

Since germ_{x}(ψ) is uniquely determined the definition (**) is correct. The family A^{<μ>} : Riem ⇝ RiemP^{r} is an Mf_{m}-natural operator.

The main result of the present note is the following theorem.

Theorem 1. Any Mf_{m}-natural operator A : Riem ⇝ RiemP^{r} is A^{<μ>} for some μ : Z^{∞} × Q^{r} → Met(R^{L_{r}}) satisfying the property (a_{r}).

Proof. Let A : Riem ⇝ RiemP^{r} be an Mf_{m}-natural operator. Define μ : Z^{∞} × Q^{r} → Met(R^{L_{r}}) by

(σ, μ(j_{0}^{∞}g, σ)) = I_{g}^{−1}(A(g)(σ)) .

Then by the non-linear Peetre theorem [2], μ satisfies the property (a_{r}). Then by the definition of μ and A^{<μ>} we see that A(g)(σ) = A^{<μ>}(g)(σ) for any Riemannian structure g on R^{m} such that the identity map id_{R^{m}} is a g-normal coordinate system with center 0 and any σ ∈ Q^{r}. Then by the invariance of A and A^{<μ>} with respect to normal coordinates we deduce that A = A^{<μ>}. □

Remark 1. One can observe that A^{[r]} = A^{<μ>} for constant μ : Z^{∞} ×Q^{r} → Met(R^{L_{r}}) equal to the standard scalar multiplication, where A^{[r]} is as in Example 4.

Remark 2. The map μ from Theorem 1 is not uniquely determined by A. One can observe that A^{<μ_{1}>} = A^{<μ_{2}>} iff μ_{1}(j_{0}^{∞}g, σ) = μ_{2}(j_{0}^{∞}g, σ) for all Riemannian structures g on R^{m} such that the identity map id_{R^{m}} is a g-normal coordinate system with center 0 and all σ ∈ Q^{r}.

References

[1] Kobayashi S., Nomizu K., Foundations of Differential Geometry, Moskow (Mir) 1981 (in Russian).

46 J. KUREK AND W.M. MIKULSKI

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

Institute of Mathematics, Maria Curie-Sklodowska University, Pl. Marii Curie Sklodowska 1, Lublin, Poland E-mail address: kurekgolem.umcs.lublin.pl Institute of Mathematics, Jagiellonian University, ul. Reymonta 4, Kraków, Poland E-mail address: mikulskiim.uj.edu.pl

Received August 13, 2007