Homotopy connectedness theorems for submanifolds of Sasakian manifolds

Abstract

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.

Keywords

Sasakian manifold / invariant submanifold / transversal bisectional curvature

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

[2] Blair D E. Riemannian Geometry of Contact and Symplectic Manifolds. 2nd ed. Progress in Mathematics, Vol 203. Boston: Birkhäuser, 2010

[3] Borisenko A, Rovenski V. About Topology of Saddle Submanifolds. Differential Geom Appl, 2007, 25: 220-233

[4] Boyer C, Galicki K. Sasakian Geometry. Oxford: Oxford University Press, 2008

[5] Frankel T. Manifolds with positive curvature. Pacific J Math, 1961, 11: 165-174

[6] Frankel T. On the fundamental groups of a compact minimal submanifold. Ann Math, 1966, 83: 68-73

[7] Fang F, Mendonça S. Complex immersion in Kähler manifolds of positive holomorphic k-Ricci curvature. Trans Amer Math Soc, 2005, 357: 3725-3738

[8] Fang F, Mendonça S, Rong X. A connectedness principle in the geometry of positive curvature. Comm Anal Geom, 2005, 13: 479-501

[9] Kenmotsu K, Xia C. Hadamadra-Frankel type theorems for manifolds with partially positive curvature. Pacific J Math, 1996, 176: 129-139

[10] Milnor J. Morse Theory. Princeton: Princeton University Press, 1963

[11] Pitiş G. On the topology of Sasakian manifolds. Math Scand, 2003, 93: 99-108

[12] Schoen R, Wolfson J. Theorems of Bath-Lefschetz type and Morse theory on the space of paths. Math Z, 1998, 229: 77-89

[13] Wilking B. Torus actions on manifolds of positive sectional curvature. Acta Math, 2003, 191: 259-297

[14] Yano K, Kon M. Structures on Manifolds. Series in Pure Mathematics, Vol 3. Singapore: World Scientific Pub Co, 1984

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg