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](Gaussian%20curvature)
- Defined as the metric trace of the [Ricci curvature](Ricci%20curvature.md) :
$$
S = \tr_g \Ric
$$
- Can be expressed in terms of [Christoffel symbols](Christoffel%20symbols).
- Requires a [metric](metric.md) $g$.
- Big question: which smooth closed manifolds have metrics with positive scalar curvature?
	- A lot is known!
	- Gromov and Lawson: every simply connected [metric](metric.md) with positive scalar curvature.
		- Uses an $\alpha$ invariant taking values in $\K O_n$ 
			- See [alpha invariant](alpha%20invariant.md).
	- Dimensions 3 and 4: as a cconsequence of [aspherical space](aspherical%20space.md) 3-manifolds and copies of $S^2 \cross S^1$.
	- In dimension 4, positive scalar curvature has stronger implications than in higher dimensions using [Seiberg-Witten invariants](Seiberg-Witten%20invariants)
		-  If $X$ is a compact [Kahler](Kahler.md)manifold of complex dimension 2 which is not rational or ruled, then X (as a smooth 4-manifold) has no [Riemannian metric](Riemannian%20metric) with positive scalar curvature.[8]