Abstract
In the present study, we consider isometric immersions \({f : M \rightarrow \tilde{M}(c)}\) of (2n + 1)-dimensional invariant submanifold M 2n+1 of (2m + 1) dimensional Sasakian space form \({\tilde{M}^{2m+1}}\) of constant \({ \varphi}\)-sectional curvature c. We have shown that if f satisfies the curvature condition \({\overset{\_}{R}(X, Y) \cdot \sigma =Q(g, \sigma)}\) then either M 2n+1 is totally geodesic, or \({||\sigma||^{2}=\frac{1}{3}(2c+n(c+1)),}\) or \({||\sigma||^{2}(x) > \frac{1}{3}(2c+n(c+1)}\) at some point x of M 2n+1. We also prove that \({\overset{\_ }{R}(X, Y)\cdot \sigma = \frac{1}{2n}Q(S, \sigma)}\) then either M 2n+1 is totally geodesic, or \({||\sigma||^{2}=-\frac{2}{3}(\frac{1}{2n}\tau -\frac{1}{2}(n+2)(c+3)+3)}\), or \({||\sigma||^{2}(x) > -\frac{2}{3}(\frac{1}{2n} \tau (x)-\frac{1}{2} (n+2)(c+3)+3)}\) at some point x of M 2n+1.
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This paper was supported by the Dumlupınar University research found (No: 2004-9).
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Yıldız, A., Murathan, C. Invariant submanifolds of Sasakian space forms. J. Geom. 95, 135–150 (2009). https://doi.org/10.1007/s00022-009-0011-9
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DOI: https://doi.org/10.1007/s00022-009-0011-9