Sphere Theorems for <i>σ<sub>k</sub></i>-Einstein Manifolds
A problem that geometers have always been concerned with is when a closed manifold is isometric to a round sphere. A classical result shows that a closed locally conformally flat Einstein manifold is always isometric to a quotient of a round sphere. In this note, we provide the definitions of <in...
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2025-01-01
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| author | Jingyang Zhong Xinran Mu |
| author_facet | Jingyang Zhong Xinran Mu |
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| description | A problem that geometers have always been concerned with is when a closed manifold is isometric to a round sphere. A classical result shows that a closed locally conformally flat Einstein manifold is always isometric to a quotient of a round sphere. In this note, we provide the definitions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>σ</mi><mi>k</mi></msub></semantics></math></inline-formula>-curvatures and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>σ</mi><mi>k</mi></msub></semantics></math></inline-formula>-Einstein manifolds, and we show that a closed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>σ</mi><mi>k</mi></msub></semantics></math></inline-formula>-Einstein manifold under certain pinching conditions of a Weyl curvature and Einstein curvature is isometric to a quotient of a round sphere. |
| format | Article |
| id | doaj-art-03e6cb217efa4a17a84cbfd481697a37 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-01-01 |
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| spelling | doaj-art-03e6cb217efa4a17a84cbfd481697a372025-01-24T13:22:19ZengMDPI AGAxioms2075-16802025-01-011416810.3390/axioms14010068Sphere Theorems for <i>σ<sub>k</sub></i>-Einstein ManifoldsJingyang Zhong0Xinran Mu1School of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, ChinaSchool of Mathematics and Statistics, Fuzhou University, Fuzhou 350108, ChinaA problem that geometers have always been concerned with is when a closed manifold is isometric to a round sphere. A classical result shows that a closed locally conformally flat Einstein manifold is always isometric to a quotient of a round sphere. In this note, we provide the definitions of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>σ</mi><mi>k</mi></msub></semantics></math></inline-formula>-curvatures and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>σ</mi><mi>k</mi></msub></semantics></math></inline-formula>-Einstein manifolds, and we show that a closed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>σ</mi><mi>k</mi></msub></semantics></math></inline-formula>-Einstein manifold under certain pinching conditions of a Weyl curvature and Einstein curvature is isometric to a quotient of a round sphere.https://www.mdpi.com/2075-1680/14/1/68scalar curvatureEinstein manifoldsphere theorem<i>σ<sub>k</sub></i>-Einstein manifolds<i>σ<sub>k</sub></i>-curvatures |
| spellingShingle | Jingyang Zhong Xinran Mu Sphere Theorems for <i>σ<sub>k</sub></i>-Einstein Manifolds Axioms scalar curvature Einstein manifold sphere theorem <i>σ<sub>k</sub></i>-Einstein manifolds <i>σ<sub>k</sub></i>-curvatures |
| title | Sphere Theorems for <i>σ<sub>k</sub></i>-Einstein Manifolds |
| title_full | Sphere Theorems for <i>σ<sub>k</sub></i>-Einstein Manifolds |
| title_fullStr | Sphere Theorems for <i>σ<sub>k</sub></i>-Einstein Manifolds |
| title_full_unstemmed | Sphere Theorems for <i>σ<sub>k</sub></i>-Einstein Manifolds |
| title_short | Sphere Theorems for <i>σ<sub>k</sub></i>-Einstein Manifolds |
| title_sort | sphere theorems for i σ sub k sub i einstein manifolds |
| topic | scalar curvature Einstein manifold sphere theorem <i>σ<sub>k</sub></i>-Einstein manifolds <i>σ<sub>k</sub></i>-curvatures |
| url | https://www.mdpi.com/2075-1680/14/1/68 |
| work_keys_str_mv | AT jingyangzhong spheretheoremsforissubksubieinsteinmanifolds AT xinranmu spheretheoremsforissubksubieinsteinmanifolds |