H-Contact Unit Tangent Sphere Bundles of Einstein Manifolds

A Remark on H-Contact Unit Tangent Sphere Bundles

“…We shall recall the following fact ( [8], Lemma 4.1) which plays an important role in the proof of Main Theorem.…”

“…Then we see that (M, g) is Ricci flat and 2-stein, and hence, the unit tangent sphere bundle T 1 M equipped with the standard contact metric structure is Hcontact. However, we may also check that the square norm of the curvature tensor is not constant [8]. Reflecting on Main Theorem, the following question will naturally arise.…”

“…Calvaruso and Perrone [5] obtained the same result in the case of the n(≥ 4)-dimensional conformally flat manifold. The authors proved that the unit tangent sphere bundle T 1 M of an n(≥ 3)-dimensional Einstein manifold is H-contact if and only if the base manifold is 2-stein ( [8], Main Theorem). The main purpose of the present paper is to prove the following: This Main Theorem together with Theorem 2 in Section 5 can be comparable with the results ( [7], Theorem 1 and Theorem 2).…”

H-CONTACT UNIT TANGENT SPHERE BUNDLES OF FOUR-DIMENSIONAL RIEMANNIAN MANIFOLDS

“…The authors [8] have also worked on the problem of determining the base space when the unit tangent bundle of a Riemannian manifold is η-Einstein. In [7] we raised the question: 'If the unit tangent sphere bundle T 1 M equipped with the standard contact metric structure on n-dimensional Riemannian manifold is H-contact, where n ≥ 3, then is the base Riemannian manifold M Einstein?' In this paper we answer this question when n = 4 by proving the following theorem.…”

HOMOGENEOUS AND H-CONTACT UNIT TANGENT SPHERE BUNDLES

“…Questions 1.1 and 1.2 referred in [11] to the Sasaki metric and in [17] to the standard contact metric structure on T 1 M, respectively. However, they also make sense for more general Riemannian metrics and contact metric structures.…”