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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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-01-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/1/68 |
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| Summary: | 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. |
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| ISSN: | 2075-1680 |