| 1. | In general relativity, the scalar curvature is the Einstein metrics.
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| 2. | Two more generalizations of curvature are the scalar curvature and Ricci curvature.
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| 3. | This has motivated the work of Simon Donaldson on scalar curvature and stability.
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| 4. | This difference ( in a suitable limit ) is measured by the scalar curvature.
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| 5. | Since S is a scalar, the scalar curvature is the appropriate measure of curvature.
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| 6. | Where is the Ricci tensor, is the metric tensor and is the scalar curvature.
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| 7. | It has constant negative scalar curvature.
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| 8. | Each of the scalar curvature and Ricci curvature are defined in analogous ways in three and higher dimensions.
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| 9. | Related results give an almost complete understanding of which manifolds have a Riemannian metric with positive scalar curvature.
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| 10. | In other words, does a smooth function exist on for which the metric has constant scalar curvature?
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