Pinching rigidity theorems for normal scalar curvature
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866915869898047488 |
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| author | Ge, Jianquan Li, Fagui Zhang, Yunheng |
| author_facet | Ge, Jianquan Li, Fagui Zhang, Yunheng |
| contents | Let $M^n$ be an $n$-dimensional closed minimal submanifold immersed in the unit sphere $\mathbb{S}^{n+m}$. Denote by $S$ and $ρ^{\perp}$ the squared norm of the second fundamental form and the normal scalar curvature of $M^n$, respectively. Let $\{A^α\}_{α=n+1}^{n+m}$ be the shape operators of $M^n$ with respect to a local orthonormal normal frame. Denote by $λ_{1}$ the largest eigenvalue of the positive semi-definite symmetric matrix $\mathcal{A}=(\langle A^α,A^β\rangle)_{m\times m}$. We show that if $λ_{1}\leqslant n$ and $ρ^{\perp}\leqslant \left[{\sqrt{2}n(n-1)}\right]^{-1} \mathop{\inf}\limits_{p\in M}(n-λ_{1})(p)$, then $ρ^{\perp}\equiv 0$, which means the normal bundle of $M^n$ is flat, and further we give the classification of $M^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16504 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Pinching rigidity theorems for normal scalar curvature Ge, Jianquan Li, Fagui Zhang, Yunheng Differential Geometry 53C20, 53C24, 53C42 Let $M^n$ be an $n$-dimensional closed minimal submanifold immersed in the unit sphere $\mathbb{S}^{n+m}$. Denote by $S$ and $ρ^{\perp}$ the squared norm of the second fundamental form and the normal scalar curvature of $M^n$, respectively. Let $\{A^α\}_{α=n+1}^{n+m}$ be the shape operators of $M^n$ with respect to a local orthonormal normal frame. Denote by $λ_{1}$ the largest eigenvalue of the positive semi-definite symmetric matrix $\mathcal{A}=(\langle A^α,A^β\rangle)_{m\times m}$. We show that if $λ_{1}\leqslant n$ and $ρ^{\perp}\leqslant \left[{\sqrt{2}n(n-1)}\right]^{-1} \mathop{\inf}\limits_{p\in M}(n-λ_{1})(p)$, then $ρ^{\perp}\equiv 0$, which means the normal bundle of $M^n$ is flat, and further we give the classification of $M^n$. |
| title | Pinching rigidity theorems for normal scalar curvature |
| topic | Differential Geometry 53C20, 53C24, 53C42 |
| url | https://arxiv.org/abs/2603.16504 |