New Characterizations of the Clifford Torus and the Calabi Torus.

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Bibliographic Details
Title:	New Characterizations of the Clifford Torus and the Calabi Torus.
Authors:	Dehe Li¹ lidehehe@163.com, Shujie Zhai² zhaishujie@zzu.edu.cn
Source:	IAENG International Journal of Applied Mathematics. Aug2025, Vol. 55 Issue 8, p2373-2378. 6p.
Subjects:	Geometric rigidity, Curvature, Submanifolds, Calabi-Yau manifolds, Clifford algebras
Abstract:	Let Mn be a compact submanifold minimally immersed into the unit sphere Sn+p with codimension p, and denote by h the second fundamental form. As our main results, we first establish two rigidity theorems in terms of the geometric quantity σ(u) = \|\|h(u, u)\|\|² for any unit vector u tangent to Mn, where \|\| · \|\|² denotes the squared norm with respect to the standard metric g on Sn+p. Furthermore, we establish an optimal inequality for the conformally flat minimal Legendrian submanifolds in S2n+1 with constant scalar curvature, involving the normalized scalar curvature and the squared norms of the traceless Ricci tensor and second fundamental form. In particular, our first theorem related to the hypersurfaces of Sn+1 gives a new characterization of the Clifford torus, whereas the other theorems are about the Legendrian submanifolds such that new characterizations of the Calabi torus can be presented. [ABSTRACT FROM AUTHOR]
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Database:	Engineering Source

Description
Abstract:	Let Mn be a compact submanifold minimally immersed into the unit sphere Sn+p with codimension p, and denote by h the second fundamental form. As our main results, we first establish two rigidity theorems in terms of the geometric quantity σ(u) = \|\|h(u, u)\|\|² for any unit vector u tangent to Mn, where \|\| · \|\|² denotes the squared norm with respect to the standard metric g on Sn+p. Furthermore, we establish an optimal inequality for the conformally flat minimal Legendrian submanifolds in S2n+1 with constant scalar curvature, involving the normalized scalar curvature and the squared norms of the traceless Ricci tensor and second fundamental form. In particular, our first theorem related to the hypersurfaces of Sn+1 gives a new characterization of the Clifford torus, whereas the other theorems are about the Legendrian submanifolds such that new characterizations of the Calabi torus can be presented. [ABSTRACT FROM AUTHOR]
ISSN:	19929978