Submanifolds of Sasakian Space Forms with Parallel Mean Curvature Vector Fields and Equal Wirtinger Angles
New Zealand Journal of Mathematics
Vol. 36, (2007), Pages 23-34
School of Mathematical Science
School of Mathematics and Computer Science
Abstract We study closed submanifolds M of dimension 2n + 1, immersed into a (4n + 1) − dimensional Sasakian space form with constant sectional curvature c, such that the reeb vector field ξ is tangent to M. Under the assumption that M has equal Wirtinger angles and parallel mean curvature vector fields, we prove that for any positive integer n, M is either an invariant or an anti-invariant submanifold of N if c > − 3, and the common Wirtinger angle must be constant if c = − 3. Moreover, without assuming it being closed, we show that such a conclusion also holds for a slant submanifold M (Wirtinger angles are constant along M) in the first case, which is very different from cases in Kahler geometry.
Keywords Sasakian space, parallel mean curvature, Wirtinger angle, invariant submanifold, slant
Classification (MSC2000) 53C40, 53C42, 53C55