Scalar curvature
It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls.This is significant in general relativity, where scalar curvature of a Lorentzian metric is one of the key terms in the Einstein field equations.On noncompact spaces, this is the context of the positive mass theorem proved by Richard Schoen and Shing-Tung Yau in the 1970s, and reproved soon after by Edward Witten with different techniques.Schoen and Yau, and independently Mikhael Gromov and Blaine Lawson, developed a number of fundamental results on the topology of closed manifolds supporting metrics of positive scalar curvature.In combination with their results, Grigori Perelman's construction of Ricci flow with surgery in 2003 provided a complete characterization of these topologies in the three-dimensional case.It is a straightforward consequence of the first variation formulas that, to first order, a Ricci-flat Riemannian metric on a closed manifold cannot be deformed so as to have either positive or negative scalar curvature.Also to first order, an Einstein metric on a closed manifold cannot be deformed under a volume normalization so as to increase or decrease scalar curvature.In the 1960s, André Lichnerowicz found that on a spin manifold, the difference between the square of the Dirac operator and the tensor Laplacian (as defined on spinor fields) is given exactly by one-quarter of the scalar curvature.As a consequence, if a Riemannian metric on a closed manifold has positive scalar curvature, then there can exist no harmonic spinors.It is then a consequence of the Atiyah–Singer index theorem that, for any closed spin manifold with dimension divisible by four and of positive scalar curvature, the  genus must vanish.This (in a precise formulation) in turn would be a special case of the strong Novikov conjecture for the fundamental group, which deals with the K-theory of C*-algebras.Similarly to Lichnerowicz's analysis, the key is an application of the maximum principle to prove that solutions to the Seiberg–Witten equations must be trivial when scalar curvature is positive.[33] By contrast to the above nonexistence results, Lawson and Yau constructed Riemannian metrics of positive scalar curvature from a wide class of nonabelian effective group actions.For example, it immediately shows that the connected sum of an arbitrary number of copies of spherical space forms and generalized cylinders Sm × Sn has a Riemannian metric of positive scalar curvature.Grigori Perelman's construction of Ricci flow with surgery has, as an immediate corollary, the converse in the three-dimensional case: a closed orientable 3-manifold with a Riemannian metric of positive scalar curvature must be such a connected sum.[34] Based upon the surgery allowed by the Gromov–Lawson and Schoen–Yau construction, Gromov and Lawson observed that the h-cobordism theorem and analysis of the cobordism ring can be directly applied.They proved that, in dimension greater than four, any non-spin simply connected closed manifold has a Riemannian metric of positive scalar curvature.[35] Stephan Stolz completed the existence theory for simply-connected closed manifolds in dimension greater than four, showing that as long as the α-genus is zero, then there is a Riemannian metric of positive scalar curvature.Kazdan–Warner's result focuses attention on the question of which manifolds have a metric with positive scalar curvature, that being equivalent to property (1).