Where \ mathbf { R } is the Ricci tensor, \ mathbf { g } is the metric tensor and R is the scalar curvature.
22.
He is best known for his work in conformal geometry, his study of extremal metrics and his research on scalar curvature and Q-curvature.
23.
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 ).
24.
Unlike the Riemann curvature tensor or the Ricci tensor, both of which can be defined for any affine connection, the scalar curvature requires a metric of some kind.
25.
For example, the Ricci tensor is a non-metric contraction of the Riemann curvature tensor, and the scalar curvature is the unique metric contraction of the Ricci tensor.
26.
In relativity, if translations mix up nontrivially with rotations, but the universe is still homogeneous and isotropic, the only option is that spacetime has a uniform scalar curvature.
27.
As a consequence, a metric-affine f ( R ) gravity, whose Lagrangian is an arbitrary function of a scalar curvature R of \ Gamma, is considered.
28.
Where ?" " x " } } is the Riemannian norm in the tangent space at, is the divergence of at, and is the scalar curvature at.
29.
In 1997, together with Tom Ilmanen ( ETH Zurich ), he was able to prove the Riemannian manifold with positive scalar curvature, in the presence of a single black hole.
30.
For example, Gromov and Lawson showed that a closed manifold that admits a metric with sectional curvature d " 0, such as a torus, has no metric with positive scalar curvature.
