Torsion tensor and critical metrics on contact (2n+1)-manifolds

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Contact Riemannian manifolds (M, Ω,g) satisfying the condition (1) ∇ξτ=0, where τ is the torsion introduced byChern andHamilton [6] and ξ is the characteristic vector field, have interesting geometric properties (see [6], [9], [11]). In this paper we give a variational characterization of compact contact Riemannian manifolds which satisfy (1). Moreover we study the tangent sphere bundles (T1M, ω, g), where (ω,g) is the standard contact Riemannian structure, which satisfy the condition (1); in particular in the 3-dimensional case we find a surprising result (see Corollary 5.3).