It is known that Mendonça and Tojeiro (2013) [19] have established a complete classification of parallel submanifolds in the product manifold Qk1n1×Qk2n2, where Qk1n1 (resp. Qk2n2) is an n1-dimensional (resp. n2-dimensional) real space form with constant curvature k1 (resp. k2). In this paper, motivated by this result with considering further generalization, we study those semi-parallel hypersurfaces in case Qk1n1=Sk1n1 and Qk2n2=Sk2n2 with k1,k2>0. As the main result, we classify semi-parallel hypersurfaces of Sk1n1×Sk2n2 for n1,n2≥2. 

Classification of semi-parallel hypersurfaces of the product of two spheres

On the Geometry of Hypersurfaces in $${\mathbb {S}}^{2} \times {\mathbb {S}}^{2}$$

On the Geometry of Hypersurfaces in S2×S2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {S}}^{2} \times {\ma

A Gap in the Morse Index of Minimal Hypersurfaces in the Product of Spheres

Structures on Manifolds

Given $$n_1\ge n_2\ge 2$$
 , let $$\Sigma $$
 be an orientable, minimal hypersurface of $$\mathbb {S}^{n_1}(1)\times \mathbb {S}^{n_2}(a)$$
 with index one. Assume $$\Sigma $$
 is closed and either smooth or singular with a singular set $${{\,\mathrm{sing}\,}}(\Sigma )$$
 satisfying $$\mathcal {H}^{n-2}({{\,\mathrm{sing}\,}}(\Sigma ))=0$$
 . By using the almost product structure, we prove that, when $$a^{2}\ge \frac{n_2}{n_1-1}$$
 or $$a^{2}\le \frac{n_2-1}{n_1}$$
 , such $$\Sigma $$
 must be totally geodesic. As an application, combining with the results of X. Zhou [37], we compute the width of $$\mathbb {S}^{n_1}(1)\times \mathbb {S}^{n_2}(a)$$
 .

Minimal hypersurfaces in the product of two spheres with index one

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Minimal hypersurfaces in the product of two spheres with index one