Tags: ##differential_geometry
- The scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the Gaussian curvature
- Defined as the metric trace of the Ricci curvature : \begin{align*} S = {\mathrm{tr}}_g \operatorname{Ric} \end{align*}
- Can be expressed in terms of Christoffel symbols.
- Requires a metric \(g\).
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Big question: which smooth closed manifolds have metrics with positive scalar curvature?
- A lot is known!
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Gromov and Lawson: every simply connected metric with positive scalar curvature.
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Uses an \(\alpha\) invariant taking values in \({\mathsf{K}}O_n\)
- See alpha invariant.
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Uses an \(\alpha\) invariant taking values in \({\mathsf{K}}O_n\)
- Dimensions 3 and 4: as a cconsequence of aspherical space 3-manifolds and copies of \(S^2 \times S^1\).
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In dimension 4, positive scalar curvature has stronger implications than in higher dimensions using Seiberg-Witten invariants
- If \(X\) is a compact Kahlermanifold of complex dimension 2 which is not rational or ruled, then X (as a smooth 4-manifold) has no Riemannian metric with positive scalar curvature.[8]