53 Differential geometry -- 53C Global differential geometry 53C15 General geometric structures on manifolds
53 Differential geometry -- 53D Symplectic geometry, contact geometry 53D15 Almost contact and almost symplectic manifolds
Language
English
Online Access
초록
Since B.-Y. Chen introduced sharp bounds for certain Riemannian invariants, involving the mean curvature vector, the sectional curvature, the scalar curvature and the shape operator [Glasgow Math. J. {\bf 38} (1996), no.~1, 87--97; MR1373963 (96m:53066); Glasg. Math. J. {\bf 41} (1999), no.~1, 33--41; MR1689730 (2000c:53072)], many authors have studied similar inequalities for submanifolds in different types of ambient spaces. In this paper, a Chen-type inequality is proved for a submanifold of a locally conformal almost cosymplectic manifold $(\overline{M}(c),\phi,\xi,\eta,g)$ with pointwise constant $\phi$-sectional curvature, tangent to the structure vector field $\xi.$ \par When the submanifold is a $\theta$-slant submanifold [J.~L. Cabrerizo et al., Glasg. Math. J. {\bf 42} (2000), no.~1, 125--138; MR1739684 (2001a:53083)] isometrically immersed in $\overline{M}(c),$ the authors obtain the corresponding inequality in terms of $\theta,$ and similarly for the case of a proper semi-slant submanifold [N. Papaghiuc, An. Ştiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat. {\bf 40} (1994), no.~1, 55--61; MR1328947 (96d:53066)]. \par This paper is closely related to one by K. Arslan et al. [Bull. Inst. Math. Acad. Sinica {\bf 29} (2001), no.~3, 231--242; MR1864013 (2002i:53109)].