Geometry and topology of submanifolds immersed in space forms and ellipsoids

Zhang Xue-Shan

doi:10.2996/kmj/1138039965

Geometry and topology of submanifolds immersed in space forms and ellipsoids

DOI 被引用文献2件参考文献5件

Zhang Xue-Shan

DEPARTMENT OF MATHEMATICS XIAN UNIVERSITY OF ARCHITECTURE AND TECHNOLOGY

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説明

Let M^m be a compact submanifold of a simply connected space form Nⁿ(c) with c{≥}0. Denote by s and H the square length of the second fundamental form and the mean curvature vector field of M respectively. By introducing a selfadjoint linear operator Q^A associated with the shape operator of M, we show that there are no stable currents in M and topologically, M is a sphere if s<H²/(m−1). For an immersed submanifold of the ellipsoid we show that appropriate assumption on Q^A implies the vanishing of a given homology group.