On the Ricci curvature of 3-submanifolds in the unit sphere

Abstract

In this short note, studying 3-dimensional compact and minimal submanifolds of the $$(3+p)$$ -dimensional unit sphere $${\mathbb {S}}^{3+p}(1)$$ , we establish two rigidity theorems in terms of the Ricci curvature. The first theorem related to hypersurfaces of $${\mathbb {S}}^4(1)$$ gives a new characterization of the minimal Clifford torus, whereas the second theorem is about the Legendrian submanifolds of $${\mathbb {S}}^7(1)$$ so that a new characterization of the Calabi torus can be presented.