On the strongly spherical Sasakian metric of a spherical tangent bundle

Let Mⁿ denote an n-dimensional Riemannian manifold. Its metric is called -strongly spherical if at every point Q Mⁿ there exists a -dimensional subspace _Q T_QMⁿ 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 _Q , X, Z #x2208; T_QMⁿ. The number 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 -strongly spherical with exponent of sphericity k. The following assertions hold: a) = 1 if and only if M² has constant Gaussian curvature K 1 and k = K²/4; b) = 3 if and only if M² has constant curvature K = 1 and k = 1/4; c) = 0, otherwise.THEOREM 2. Let the Sasakian metric of T₁Mⁿ (n Mⁿ) be -strongly spherical with exponent of sphericity k. If k > 1/3 and k 1, then = 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 -strongly spherical with exponent of sphericity k. The following assertions hold: a) = 1 if and only if K = 1/4; b) = 0, otherwise. In dimension n = 3 Theorem 2 is true for k {1/4, 1}.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 150–159, 1992.