Projective Connections and the Algebra of Densities

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Here we present a natural definition of projective connection and give the correspon- dence to the Thomas-Weyl-Veblen definition of projective equivalence classes. Most of this account may be extracted, with a little work, from [3]. Recall that a map φ : Rn → Rn is said to be projective or fractional linear if it is of the form α β α β (v) + n n ∗ φ(v) = , where α ∈ GLn(R),β ∈ R ,γ ∈ (R ) ,δ ∈ (R),det 6= 0 γ(v) + δ γ δ With this in mind, we make the following natural definition. Definition. A projective connection on a vector bundle E → M is an Ehresmann connection such that the associated parallel transport induces a projective map between fibres. This intuitive definition is a reformulation, in modern language, of Cartan’s idea as proposed in his seminal paper [4]. In the special case of the tangent bundle TM → M, projective connections are closely related to the following notion. Definition. Two torsion free linear connections on TM → M are said to be projectively equivalent if any of the following hold. 1. ∇ ∼ ∇¯ if they define the same geodesics up to reparametrisation. 2. ∇ ≈ ∇¯ if there is a 1-form ϑ s.t. ∇¯ XY −∇XY = ϑ(X)Y +ϑ(Y )X for all vector fields X,Y. ∇ ∇¯ Γk Γ¯ k 3. ≃ if their coefficients ij and ij satisfy 1 1 Πk := Γk − (δ kΓl + δ kΓl ) = Γ¯ k − (δ kΓ¯ l + δ kΓ¯ l ) =: Π¯ k ij ij n + 1 i lj j il ij n + 1 i lj j il ij This would not be a well defined notion had these conditions not been equivalent. Details of the proof of (1)⇔(2) can be found in [3]; a brute force calculation yields (2)⇔(3). Before relating these two notions, we must rephrase the former analytically. Recall that an Ehresmann connection in a fibre bundle π : E → M is a distribution H of horizontal linear subspaces of TE such that TE = H ⊕ kerπ∗. By horizontal curves we will mean integral curves of H . We express H as the kernel of a collection of 1- forms on E. In terms of local coordinates xi on the base and ξ a on the fibres, these forms Ψa ξ a a i may be written (after a careful choice of coordinates) in the form = d + fi dx for a some functions fi on M. These functions together constitute a local connection 1-form on M. The condition that a curve σ be horizontal is then simply that Ψa(σ ′) = 0 for each a a, or explicitly in terms of fi , d d ξ a(σ(t)) = f a xi(σ(t)). (1) dt i dt In this context, we require that the parallel translation defines a projective map between fibres. This is equivalent to requiring that locally, the horizontal curves are projective flows on the typical fibre, that is, of the form α(σ) + β σ(t) = γ(σ) + δ with the aforementioned non-degeneracy condition on the coefficients. Differentiating this, we have the equation in fibre coordinates d ξ a(σ(t)) = Aa(t) + Ba(t)ξ b(σ(t)) +C ξ a(σ(t))ξ b(σ(t)) dt b b a a for some coefficient functions A , Bb and Cb. Comparing this with (1) we come to a description of projective connection in a vector bundle E → M as an Ehresmann connection defined by the annihilating 1-forms

Ψa ξ a φ a ψa ξ b η ξ aξ b i = d − ( i (x) + bi(x) + bi(x) )dx φ a ψa η for some functions i , bi and bi. Remark. With respect to (linear) transition functions in the vector bundle E, the forms ψa i ψa i η i bidx define a connection 1-form for a linear connection in E, while bidx and bidx define 1-forms taking values in sections of E and the dual bundle E∗ respectively. This will be important in the following section. φ j δ j In the case of the tangent bundle, there is a distinguishedchoice of i , namely i . This Ψi ξ i i ωiξ j ω0ξ iξ j ξ i gives = d − dx − j − j where now are the fibre coordinates naturally related to the local coordinates xi. Projective connections also have a related notion of geodesics. These are given in this case by the equations

d2σ k dσ a dσ i dσ j dσ i dσ j dσ k = + Γk + γ0 dt2 dt ij dt dt ij dt dt dt ωk Γk j ω0 γ0 j where i =: ijdx and i =: ijdx . Having established this notion, we may mimic our previous definition and analogously define projective equivalence classes of projective connections. For arbitrary Ehresmann connections, there is a natural notion of curvature which in this case amounts to a collection of 2-forms: Ωi ωi ωi ωk ω0 i δ iω0 k Ω0 ω0 ω j j = d j − k ∧ j − j ∧ dx − j k ∧ dx and i = j ∧ i . i Ωi i k l i Definition. Define the numbers A jkl and A jk by j =: A jkldx ∧ dx and A jk := A jki.A projective connection is said to be normal if A jk = 0. ωi ωi ω p i k l We also consider the Ricci tensor R jk, where d j − p ∧ j equals R jkldx ∧ dx and i R jk := R jki. The notion of normality was introduced by Cartan and derives its utility from the ωi ω0 following observations. Let ( j, j ) define a projective connection in TM.

• Given that the Ricci tensor Rij is symmetric, the projective connection is normal if ω0 2 k and only if j = n−1 R jkdx . ωi • Given any linear connection j in the tangent bundle, there is another with symmetric Ricci tensor with the same geodesics up to reparametrisation. ωi ω0 ωi ω0 • If ( j, j ), ( ¯ j, ¯ j ) are two normal projective connections in the tangent bundle ωi ωi whose associated linear connections j and ¯ j are projectively equivalent, then ωi ω0 ωi ω0 ( j, j ) and ( ¯ j, ¯ j ) are projectively equivalent. Proofs of these facts may be be found in [3]. Given these, we have a one-to-one corre- spondence between projective equivalence classes of linear connections and projective equivalence classes of projective connections on any given manifold M. All of the notions discussed throughout this paper also make sense on supermanifolds. Bearing these relations in mind, we will adopt from now on the (classical) terminology and call a projective equivalence class of linear connections on a manifold a Thomas projective connection.

RELATIONS WITH DENSITIES

Definition. A density of weight λ is a formal expression of the form φ = φ(x)(Dx)λ defined in local coordinates, Dx being the associated local volume form and λ ∈ R. λ There is a natural notion of multiplication for densities given by φ · χ =(φ(x)(Dx) 1) · λ λ λ (χ(x)Dx 2 ) = φ(x)χ(x)(Dx) 1+ 2 . The algebra of densities denoted V(M), is the al- λ gebra of finite formal sums ∑λ φλ (x)(Dx) of densities with the multiplication defined previously (see [1]). A λ-density may equally be thought of as a ‘function’ φ(x) which under a change of coordinates, picks up the modulus of the Jacobian of the transformation to the λ-th power as a factor. Having defined densities, it is natural to define a linear operator on V(M), namely the weight operator w. This is defined as having each λ-density as a λ-eigenvector, that is w(φ(x)(Dx)λ ) = λφ(x)(Dx)λ . Let dimM = n. We now define two (n + 1)-dimensional manifolds M and M which are fibre bundles over M and significant with respect to the algebra of densities and projective connections respectively. b e λ M : Densities as functions. A density ∑φλ (Dx) may be interpreted as a function on the manifold M defined in some sense as the ‘strictly positive half’ of the deter- b minant bundle (see below). Denoting the fibre coordinate by t(> 0), the algebra of b ∞ λ densities is the subalgebra of C (M) consisting of functions of the form ∑φλ t . M : Thomas projective connections asb linear connections. In [5], T. Y. Thomas details the construction of a manifold M from a given manifold M such that any e Πk projective equivalence class ij on M gives rise canonically to linear connection Γ˜ k e coefficients ij on M. From the viewpoint of the previous section, each projective equivalence class ofe projective connections gives rise to a linear connection on M. e If x0,...,xn are local coordinates in some neighbourhood of M, these are given by Γ˜ k Γ˜ k Πk e ij = ji = ij for i, j,k = 1,...,n (2) δ k Γ˜ k = Γ˜ k = − i for i,k = 0,...,n (3) i0 0i n + 1 ∂Πr n + 1 ij Γ˜ 0 = Γ˜ 0 = − Πr Πs for i, j = 1,...,n. (4) ij ji n − 1 ∂xr si rj

The manifolds M and M are defined locally in terms of the charts on M. Let us for a moment denote local coordonates on M by t,xˆ1,...,xˆn and on M byx ˜0,...,xñ. Then if x′i = f i(x1,...,xen) is a coordinateb transformation on M, define b e On M On M ′0 0 ′ x˜ = x˜ + logJf ; e t = tJf ; b x˜′i = f i(x˜1,...,xñ), for i = 1,...,n, xˆ′i = f i(xˆ1,...,xˆn) for i = 1,...,n, where Jf denotes the modulus of the Jacobian of f considered as a function of coordinates. We are immediately led to the following observation. Theorem 1. 1. The bundles π˜ : M → M and πˆ : M → M are isomorphic, that is, there is a diffeomorphism F : M → M such that π˜ ◦ F = πˆ. e b 2. The weight operator w is mapped to ∂ under F. b e ∂(x˜0) Πk 3. A projective equivalence class ij on M canonically induces a linear connection on M, in particular allowing us to consider covariant derivatives of densities along vector fields on M. b Proof. 1. In coordinates, simply define

logt for i = 0, and t > 0; x˜i(F(x)) = Fi(t,xˆ1,...,xˆn) = xî for i = 1,...,n. 2. Written in terms of generating functions w takes the form of a logarithmic derivative t ∂ , which is mapped to ∂ . ∂t ∂(x˜0) Πk 3. A projective equivalence class ij canonically defines, via Thomas’ construction, a linear connection ∇ on M whose coefficients are given by (2)-(4). Now define a linear connection on M by pulling ∇ back along F. e e b e

OPERATORS, BRACKETS AND FUTURE DEVELOPMENTS

Let M be a manifold endowed with a tensor field Sij. This defines a map S♯ : T ∗M → TM i ji given in local coordinates by ωidx 7→ ω jS ∂i. Definition. Let E → M be a vector bundle over a manifold M endowed with a tensor field Sij. An upper connection on E over Sij is an R-bilinear map ∇ : Ω1(M) × Γ(E) → Γ(E) satisfying

ω ω ω ω ∞ ∇ f σ = f ∇ σ and ∇ f σ = f ∇ σ +(S♯ω)( f )σ for f ∈ C (M).

If Sij is an invertible matrix, upper and ordinary connections correspond by raising and lowering indices. In [1], symmetric biderivations on the algebra of densities (or more brieﬂy brackets) were considered. In a system of local coordinates x1,...xn, a homogeneous bracket {·, ·} is uniquely deﬁned by a triple of quantities Sij,γi,θ via

λ λ λ {xi,x j} = Sij(Dx) , {xi,Dx} = γi(Dx) +1, {Dx,Dx} = θ(Dx) +2; where λ ∈ R is the weight of the bracket. Here Sij = S ji is symmetric. For a bracket of weight zero, Sij is a tensor and γi defines an upper connection over Sij in the bundle of volume forms on M. Theorem 2. Let M be a manifold endowed with a projective equivalence class (Thomas Πi ij projective connection jk) and a tensor field S . Then: 1. These data define an upper connection over Sij in the bundle of volume forms given by the coefficients n + 1 γi = (∂ Sij + S jkΠi ). (5) n + 3 j jk 2. The following expression defines an invariant second order differential operator acting on functions on M

ij 2 ij n + 1 jk i ∆ = S ∂i∂ j + ∂ jS − S Π ∂i . (6) n + 3 n + 3 jk Γk Remark. In general, a linear connection ij gives rise to a connection on the bundle γ Γk ij of volume forms by taking the trace i := ki. Suppose that S is non-degenerate, i.e., Sij = gij defines a metric. An upper connection may be obtained in two ways: Γk • Take the coefficients ij of the Levi-Civita connection associated with gij. Define γi ijΓk := g kj. Πk Γk • Take the class ij associated with the Levi-Civita connection coefficients ij and use (5). These two constructions give exactly the same coefficients γi. Given a Thomas projective connection on M, the principal symbol of a second order differential operator (of arbitrary weight) on the algebra of densities V(M) may similarly V Πk be extended to an invariant operator on (M). From a projective equivalence class ij Γk on M, Theorem 1 gives a linear connection ij on M. Applying formula (6) to the associated Thomas projective connection on Mbgives theb result. In the presence of a Thomas projective connection on M, this is a method of construct- ing, from a bracket on V(M), a canonical differentialb operator on V(M) generating the bracket. A similar construction using a natural inner product was one of the main results of [1]. Comparing operators from (6) and [1] gives expressions for γi and θ in terms of ij Πk ij S and ij. Given a manifold, we have therefore a map from the space of symbols S to the space of second order differential operators on the algebra of densities, or using Ovsienko’s terminology (see [6]), a projectively equivariant quantisation (in the non-flat case). Indeed it would no doubt prove fruitful to investigate this relation further. Acknowledgments: The author is greatly indebted to H. Khudaverdian and Th. Voronov for their invaluable guidance and many discussions and most grateful to A. Odzijewicz for providing such a wonderful forum in Białowieza˙ in which to present the results of this note.

REFERENCES

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