Theorems of Barth–Lefschetz type in Sasakian geometry

Abstract

In this paper, we obtain theorems of Barth-Lefschetz type in Sasakian geometry. As a corollary, this gives a new proof of a classical theorem due to J. Milnor. It also implies connectedness principle and Frankel's type theorem.

Theorems of Barth-Lefschetz type and Morse theory on the space of paths in homogeneous spaces

“…To prove these connectedness Theorems [7] and [8] apply Morse Theory locally on the ambient space but in this work we apply Morse theory on the space of paths following the work of [6], [4], [5], [14], [21], [22], [23], [26] and [19]. The basic idea of [4] and [5] is to demonstrate that the index of the critical points in the space of paths joining two submanifolds has the appropriate lower bound for a chosen Morse function.…”