Some Optimal Inequalities for Anti-invariant Submanifolds of the Unit Sphere

In this paper, we study the rigidity phenomena on the (n+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n+1)$$\end{document}-dimensional anti-invariant submanifolds of the unit sphere of dimension (2n+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2n+1)$$\end{document} from the intrinsic and extrinsic aspects, respectively. First of all, we establish a basic inequality for such submanifolds relative to the norm of the covariant differentiation of both the second fundamental form h and mean curvature vector field H. Secondly, the lower bound of the norm of H is further derived by means of a general inequality. Finally, in dealing with those minimal anti-invariant submanifolds with η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document}-Einstein induced metrics, we obtain an inequality in terms of the Weyl curvature tensor, squared norm S of h, and scalar curvature. In particular, these inequalities above are optimal in the sense that all the submanifolds attaining the equalities are completely determined.