On the topology of Sasakian manifolds

Abstract

The notion of q-bisectional curvature of a Sasakian manifold M is defined. It is proved that if M has lower bounded q-bisectional curvature and contains a compact invariant submanifold tangent to the structure vector field then M is compact. Myers and Frankel type theorems for Sasakian manifolds with lower bounded and positive q-bisectional curvature, respectively, are also given.

Theorems of Barth–Lefschetz type in Sasakian geometry

“…Note this Frankel's type theorem was also obtained by other people, see [17,24]. The proof of our Main theorem is based on Morse theory on path space.…”

A Frankel type theorem for CR submanifolds of Sasakian manifolds

“…In the following we shall deal with Sasakian manifolds with nonnegative ϕ-bisectional curvature. So, for the convenience of the reader, we recall the definition of this kind of curvature which was introduced by Tanno and Baik in [13], and used in [3] and [10] to obtain Frankel type theorems about the intersection of two invariant submanifolds. It is an adaptation to the Sasakian case of the notion of holomorphic bisectional curvature introduced by Goldberg and Kobayashi in [8] for Kähler manifolds.…”

“…We shall also consider the case when one of the submanifolds is invariant. In [3] and [10] this kind of results were discussed for the case of two invariant submanifolds.…”

A Frankel type theorem for generic submanifolds of Sasakian manifolds

“…In the following we will deal with generic and invariant submanifolds of Sasakian manifolds with nonnegative ϕ-bisectional curvature. So we recall the definition of this kind of curvature which was introduced by Tanno and Baik in [10] and used by in [1] and [7] to obtain Frankel type theorems about the intersection of two invariant submanifolds. It is an adaptation to the Sasakian case of the notion of holomorphic bisectional curvature introduced by Goldberg and Kobayashi in [5] for Kähler manifolds.…”

“…We shall also consider the case when one of the submanifolds is generic and the other one is invariant. In [1] and [7] this kind of results were discussed for the case of two invariant submanifolds.…”