On the strongly spherical Sasakian metric of a spherical tangent bundle

Abstract

Let Mn denote an n-dimensional Riemannian manifold. Its metric is called ν -strongly spherical if at every point Q ∈ Mn there exists a ν -dimensional subspace ℒQ ν ⊂ TQMn such that the curvature operator of the metric of Mn satisfies R(X, Y) Z = k(< Y, Z > X < X, Z > Y), where k = const > 0, Y ∈ ℒQ ν, X, Z #x2208; TQMn. The number ν is called the index of sphericity and k the exponent of sphericity. The following theorems are proved in the paper.THEOREM 1. Let the Sasakian metric of T1Mn be ν -strongly spherical with exponent of sphericity k. The following assertions hold: a) ν = 1 if and only if M2 has constant Gaussian curvature K ≠ 1 and k = K2/4; b) ν = 3 if and only if M2 has constant curvature K = 1 and k = 1/4; c) ν = 0, otherwise.THEOREM 2. Let the Sasakian metric of T1Mn (n ≥ Mn) be ν -strongly spherical with exponent of sphericity k. If k > 1/3 and k ≠ 1, then ν = 0. Let us denote by (Mn, K) a space of constant curvatureK. THEOREM 3. Let the Sasakian metric of T1(Mn, K) (n ≥ 3) be ν -strongly spherical with exponent of sphericity k. The following assertions hold: a) ν = 1 if and only if K = 1/4; b) ν = 0, otherwise. In dimension n = 3 Theorem 2 is true for k ∉ {1/4, 1}.