© Hindawi Publishing Corp. CHERN-SIMONS FORMS OF PSEUDO-RIEMANNIAN HOMOGENEITY ON THE OSCILLATOR GROUP

We consider forms of Chern-Simons type associated to homogeneous 
pseudo-Riemannian structures. The corresponding secondary classes 
are a measure of the lack of a homogeneous pseudo-Riemannian 
space to be locally symmetric. In the present paper, we compute 
these forms for the oscillator group and the corresponding 
secondary classes of the compact quotients of this group.

In the present paper, we calculate the forms of pseudo-Riemannian homogeneity on the oscillator group equipped with the above-mentioned leftinvariant metrics.Further, we obtain the corresponding secondary classes of the compact quotients of this group.
2. Chern-Simons forms associated to a homogeneous pseudo-Riemannian structure.Let (M, g) be a connected C ∞ pseudo-Riemannian manifold of dimension n and signature (k, n−k).Let ∇ be the Levi-Civita connection of g and R its curvature tensor field.A homogeneous pseudo-Riemannian structure on (M, g) is a tensor field S of type (1,2) on M such that the connection ∇ = ∇−S satisfies ∇g = 0, ∇R = 0, ∇S = 0. (2.1) If g is a Lorentzian metric (k = 1), we say that S is a homogeneous Lorentzian structure.In [3], we proved that if (M, g) is connected, simply connected, and geodesically complete, then it admits a homogeneous pseudo-Riemannian structure if and only if it is a reductive homogeneous pseudo-Riemannian manifold.This result extends the well-known characterization of Ambrose and Singer [1] of homogeneous connected, simply connected, and complete Riemannian manifolds in terms of homogeneous Riemannian structures.Now, let P = (P ,M,G) be a principal bundle over the n-dimensional C ∞ manifold M. Let Ᏽ r (G) be the space of Ad(G)-invariant polynomials of degree r .Let D, D be connections on P , with respective connection 1-forms ω, ω and curvature forms Ω = dω + ω ∧ ω and Ω.If I ∈ Ᏽ r (G), we can consider for each r the 2r -form I(Ω r ) = I(Ω,...,Ω) on P , which projects to a (unique) 2rform on M, say again I(Ω r ), which is closed.Consider the connection given, for t ∈ [0, 1], by ω t = ω+t(ω− ω), with curvature form Ω t .Then one has the transgression formula where We consider the bundle of pseudo-orthonormal frames p : (2.4) Let Ω be the curvature form of a connection ω in ᏻ k,n−k (M).Then, for each f r , r = 1,...,n, there exists a unique closed 2r -form υ r on M such that p * (υ r ) = f r (Ω).One has det(I + Ω) = p * (1 + υ 1 + ••• + υ n ); hence having characteristic forms υ r of degree 2r , and a total form The forms f r (Ω) are the elementary symmetric functions s r (Ω), r = 1,...,n, of the eigenvalues of Ω so that det(I (Ω).By using Newton's recursive formulas, one can further compute the functions s r (Ω) in terms of the traces of the powers of Ω from the expressions and since tr Ω = 0, we have, after computation, If we consider here the Levi-Civita connection ∇ and the linear connection ∇ = ∇−S with connection form ω and curvature form Ω, where S is a homogeneous pseudo-Riemannian structure on (M, g), the general equation (2.2) can be written in this case as s r (Ω) − s r Ω = dQ ω, ω . (2.7) If s r (Ω) = s r ( Ω), then Q(ω, ω) is closed, thus determining a secondary class.
In particular, if r = 2 or 3, then this happens if tr(Ω r ) = tr( Ω r ).We denote by Definition 2.1.Let (M, g) be a pseudo-Riemannian manifold and S a homogeneous pseudo-Riemannian structure on M. The forms

called Chern-Simons forms of pseudo-Riemannian homogeneity (or simply forms of homogeneity) on (M,g,S). The corresponding real cohomology classes [Q S 2r −1 ](M, g) are called secondary classes of pseudo-Riemannian homogeneity (or simply secondary classes of homogeneity).
The case r = 1 in (2.7) is trivial.For r = 2, one has the formula where σ = ω − ω.One can obtain similar formulas for any r with 2r ≤ dim M.
We have the following proposition.
Proof.The proof follows immediately from (2.3).

Forms Q S
3 on the oscillator group.The (4-dimensional) oscillator group is the simply connected Lie group Os with Lie algebra os = P,X,Y ,Q having nonzero brackets 3.1.The metrics g ε , ε ≠ 0. We first endow Os with the family of left-invariant Lorentzian metrics given at os by Let {η, α, β, ξ} denote the basis dual to {P, X, Y, Q}.Integrating Ambrose-Singer's equations (2.1), we obtain (see [4]) the family of homogeneous Lorentzian structures corresponding to (Os,g ε ): We have

.4)
Assuming that we can write Then, after some calculations from (2.8), we obtain the following proposition.

.14)
We have and for the first family (3.9), we get which lead to a nontrivial form Q S (x,y,z,w) 3 . However, for the other five families of homogeneous Lorentzian structures, the corresponding matrices ω and Ω give and we have the following proposition.
Proposition 3.2.The forms of homogeneity on the oscillator group (Os,g 0 ) corresponding to the previous six families of homogeneous Lorentzian structures are null except for the first family S (x,y,z,w) , for which (3.18)

Classes [Q S
3 ] of the compact quotients of Os.Now, we determine the secondary classes [Q S  3 ] of the compact quotients of the oscillator group.For this, we first note that given a left-invariant form α on a Lie group G, then it is invariant under the action of a discrete subgroup Γ of G, so that there exists a form α on the quotient Γ \G such that π * ( α) = α, where π denotes the natural projection π : G → Γ \G.In the sequel, we will denote by α such a projected form of a left-invariant form α on G onto a compact quotient Γ \G.If g is a left-invariant metric on G, then it projects to a metric g on Γ \G such that the map π : (G, g) → (Γ \G, g) is a local pseudo-Riemannian isometry.Moreover, the Levi-Civita connection ∇ projects to the Levi-Civita connection ∇ on Γ \G and each homogeneous pseudo-Riemannian structure S projects to a homogeneous pseudo-Riemannian structure S on Γ \G, where Γ is a uniform discrete subgroup of G.The computation of the Chevalley-Eilenberg cohomology of the oscillator algebra os gives us The Lie algebra os is a semidirect product of the Heisenberg algebra h 3 and R, and the oscillator group is a semidirect product Os = R H 3 .If Γ is a uniform discrete subgroup of the Heisenberg group H 3 , we can consider the solvmanifold (Z Γ )\ Os.The cohomology of such solvmanifold is known to be isomorphic to that of S 1 × S 3 (see [9]); that is, H k (os, R) ≈ H k ((Z Γ )\ Os, R) for each k.The forms project to the quotients and we have the following proposition.
Thus these two pseudo-Riemannian compact quotients of the oscillator group, endowed with those homogeneous pseudo-Riemannian structures are "more symmetric" than the spaces corresponding to the rest of values of a.

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