Xiong, Yueshan Homotopy connectedness theorems for submanifolds of Sasakian manifolds. (English) Zbl 1329.53081 Front. Math. China 10, No. 2, 395-414 (2015). Summary: 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 \((k,\varepsilon)\)-saddle submanifolds in Sasakian manifolds with positive transversal \(q\)-Ricci curvature are proved by using the weak (\(\varepsilon\)-)asymptotic index. As a corollary, a Frankel type theorem is proved. MSC: 53C40 Global submanifolds 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 55Q05 Homotopy groups, general; sets of homotopy classes Keywords:Sasakian manifold; invariant submanifold; transversal bisectional curvature PDFBibTeX XMLCite \textit{Y. Xiong}, Front. Math. China 10, No. 2, 395--414 (2015; Zbl 1329.53081) Full Text: DOI References: [1] Binh T Q, Ornea L, Tamássy L. Intersections of Riemannian submanifolds variations on a theme by T. J. Frankel. Rend Mat, 1999, 19: 107-121 · Zbl 0947.53027 [2] Blair D E. Riemannian Geometry of Contact and Symplectic Manifolds. 2nd ed. Progress in Mathematics, Vol 203. Boston: Birkhäuser, 2010 · Zbl 1246.53001 · doi:10.1007/978-0-8176-4959-3 [3] Borisenko A, Rovenski V. About Topology of Saddle Submanifolds. Differential Geom Appl, 2007, 25: 220-233 · Zbl 1119.53036 · doi:10.1016/j.difgeo.2006.08.001 [4] Boyer C, Galicki K. Sasakian Geometry. Oxford: Oxford University Press, 2008 · Zbl 1155.53002 [5] Frankel T. Manifolds with positive curvature. Pacific J Math, 1961, 11: 165-174 · Zbl 0107.39002 · doi:10.2140/pjm.1961.11.165 [6] Frankel T. On the fundamental groups of a compact minimal submanifold. Ann Math, 1966, 83: 68-73 · Zbl 0189.22401 · doi:10.2307/1970471 [7] Fang F, Mendonça S. Complex immersion in Kähler manifolds of positive holomorphic k-Ricci curvature. Trans Amer Math Soc, 2005, 357: 3725-3738 · Zbl 1088.32011 · doi:10.1090/S0002-9947-05-03675-5 [8] Fang F, Mendonça S, Rong X. A connectedness principle in the geometry of positive curvature. Comm Anal Geom, 2005, 13: 479-501 · Zbl 1131.53306 · doi:10.4310/CAG.2005.v13.n4.a2 [9] Kenmotsu K, Xia C. Hadamadra-Frankel type theorems for manifolds with partially positive curvature. Pacific J Math, 1996, 176: 129-139 · Zbl 0865.53053 [10] Milnor J. Morse Theory. Princeton: Princeton University Press, 1963 · Zbl 0108.10401 [11] Pitiş G. On the topology of Sasakian manifolds. Math Scand, 2003, 93: 99-108 · Zbl 1064.57030 [12] Schoen R, Wolfson J. Theorems of Bath-Lefschetz type and Morse theory on the space of paths. Math Z, 1998, 229: 77-89 · Zbl 0939.58017 · doi:10.1007/PL00004651 [13] Wilking B. Torus actions on manifolds of positive sectional curvature. Acta Math, 2003, 191: 259-297 · Zbl 1062.53029 · doi:10.1007/BF02392966 [14] Yano K, Kon M. Structures on Manifolds. Series in Pure Mathematics, Vol 3. Singapore: World Scientific Pub Co, 1984 · Zbl 0557.53001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.