Regularity of the Geodesic Equation in the Space of Sasakian Metrics

Sasakian manifolds provide rich source of constructing new Einstein manifolds in odd dimensions [2]. They play some important role in the superstring theory in mathematical physics [19, 20]. There is a renewed interest on Sasakian manifolds recently. The present paper is devoted to the regularity analysis of a geodesic equation in the space of Sasakian metrics H (definition in (1.2)) and some of geometric applications. This equation was introduced in [16]. We believe it encodes important geometric information. This geodesic approach is modeled in Kähler case [18, 25, 8, 4, 3, 22]. The C w (see definition 1) regularity proved by Chen [4] for the geodesic equation in the space of Kähler metrics has significant geometric consequences. Here we will deduce the parallel results in Sasakian geometry. A Sasakian manifold (M, g) is Riemannian manifold with the property that the cone manifold (C(M), g̃) = (M × R, rg + dr) is Kähler. A Sasakian structure on M consists of a Reeb field ξ of unit length on M , a (1, 1) type tensor field Φ(X) = ∇Xξ and a contact 1-form η (which is the dual 1-form of ξ with respect to g). (ξ, η,Φ, g). Φ defines a complex structure on the contact sub-bundle D = ker{η}. (D,Φ|D, dη) provides M a transverse Kähler structure with Kähler form 1 2dη and metric g T defined by g (·, ·) = 1 2dη(·,Φ·). The complexification DC of the sub-bundle D can be decomposed it into its eigenspaces with respect to Φ|D as DC = D1,0 ⊕ D0,1. A p-form θ on Sasakian manifold (M, g) is called basic if iξθ = 0, Lξθ = 0 where iξ is the contraction with the Reeb field ξ, Lξ is the Lie derivative with respect to ξ. The exterior differential preserves basic forms. There is a natural splitting of the complexification of the bundle of the sheaf of germs of basic p-forms ∧pB(M) on M , ∧pB (M)⊗ C = ⊕i+j=p ∧ i,j B (M), (1.1)