The homotopy connectedness theorem for invariant immersions in Sasakian manifolds with nonnegative transversal q-bisectional curvature is proved. Some Barth-Lefschetz type theorems for minimal submanifolds and (k, ϵ)-saddle submanifolds in Sasakian manifolds with positive transversal q-Ricci curvature are proved by using the weak (ϵ-)asymptotic index. As a corollary, the Frankel type theorem is proved.
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