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The topological rigidity theorem for submanifolds in space forms
Differential Geometry · Submitted 1 Mar 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We verify that if the scalar curvature of $M$ satisfies $R>n(n-2)(c+H^2)$, and if $Ric_M\geq (n-2-\frac{2\sigma_n}{2n-\sigma_n})(c+H^2)$, then $M$ is homeomorphic to a sphere.
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The abstract, sentence by sentence
Let $M$ be an $n(\geq 4)$-dimensional compact submanifold in the simply connected space form $F^{n+p}(c)$ with constant curvature $c\geq 0$, where $H$ is the mean curvature of $M$. We verify that if the scalar curvature of $M$ satisfies $R>n(n-2)(c+H^2)$, and if $Ric_M\geq (n-2-\frac{2\sigma_n}{2n-\sigma_n})(c+H^2)$, then $M$ is homeomorphic to a sphere. Here $\sigma_n=sgn(n-4)((-1)^n+3)$, and $sgn(\cdot)$ is the standard sign function. This improves our previous sphere theorem \cite{XG2}. It should be emphasized that our pinching conditions above are optimal. We also obtain some new topological sphere theorems for submanifolds with pinched scalar curvature and Ricci curvature.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $M$ be an $n(\geq 4)$-dimensional compact submanifold in the simply connected space form $F^{n+p}(c)$ with constant curvature $c\geq 0$, where $H$ is the mean curvature of $M$. | background 0.98 | 0.11 | 0.02 | unclear 0.52 |
| s1 | We verify that if the scalar curvature of $M$ satisfies $R>n(n-2)(c+H^2)$, and if $Ric_M\geq (n-2-\frac{2\sigma_n}{2n-\sigma_n})(c+H^2)$, then $M$ is homeomorphic to a sphere. | result 0.99 | 0.94 | 0.91 | asserted 1.00 |
| s2 | Here $\sigma_n=sgn(n-4)((-1)^n+3)$, and $sgn(\cdot)$ is the standard sign function. | background 0.61 | 0.48 | 0.02 | asserted 0.87 |
| s3 | This improves our previous sphere theorem \cite{XG2}. | interpretation 0.49 | 0.53 | 0.04 | asserted 1.00 |
| s4 | It should be emphasized that our pinching conditions above are optimal. | interpretation 0.89 | 0.73 | 0.04 | asserted 0.99 |
| s5 | We also obtain some new topological sphere theorems for submanifolds with pinched scalar curvature and Ricci curvature. | contribution 0.54 | 0.95 | 0.11 | asserted 1.00 |
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The paper
Original abstract
Let $M$ be an $n(\geq 4)$-dimensional compact submanifold in the simply connected space form $F^{n+p}(c)$ with constant curvature $c\geq 0$, where $H$ is the mean curvature of $M$. We verify that if the scalar curvature of $M$ satisfies $R>n(n-2)(c+H^2)$, and if $Ric_M\geq (n-2-\frac{2\sigma_n}{2n-\sigma_n})(c+H^2)$, then $M$ is homeomorphic to a sphere. Here $\sigma_n=sgn(n-4)((-1)^n+3)$, and $sgn(\cdot)$ is the standard sign function. This improves our previous sphere theorem \cite{XG2}. It should be emphasized that our pinching conditions above are optimal. We also obtain some new topological sphere theorems for submanifolds with pinched scalar curvature and Ricci curvature.
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