Namely, if \ mathcal { M } ^ 3 has nonnegative scalar curvature, then the Hawking mass of \ Sigma is non-decreasing as the surface \ Sigma flows outward at a speed equal to the inverse of the mean curvature.
42.
As an application of the equivariant index theorem, Atiyah and Hirzebruch showed that manifolds with effective circle actions have vanishing ?genus . ( Lichnerowicz showed that if a manifold has a metric of positive scalar curvature then the ?genus vanishes .)
43.
Where " R ?? " s the Ricci tensor, " g ?? " is the metric tensor, " R " the scalar curvature and ? is Einstein's constant we will calculate in the next section ..
44.
He is known for his research on Riemannian manifolds, topology of manifolds, rigidity and convexity of isometric immersions, minimal surfaces, stability of hypersurfaces, isoperimetric problems, minimal submanifolds of a sphere, and manifolds of constant mean curvature and vanishing scalar curvature.
45.
The borderline case ( 2 ) can be described as the class of manifolds with a "'strongly scalar-flat metric "', meaning a metric with scalar curvature zero such that " M " has no metric with positive scalar curvature.
46.
The borderline case ( 2 ) can be described as the class of manifolds with a "'strongly scalar-flat metric "', meaning a metric with scalar curvature zero such that " M " has no metric with positive scalar curvature.
47.
More generally, the injectivity part of the Baum Connes conjecture for a group " G ", which is known in many cases, would imply that a closed aspherical manifold with fundamental group " G " has no metric with positive scalar curvature.
48.
Thus, by applying a single idea : the principle of least action, to a single quantity : the scalar curvature on the bundle ( as a whole ), one obtains simultaneously all of the needed field equations, for both the space-time and the gauge field.
49.
Given a specified distribution of matter and energy in the form of a stress energy tensor, the EFE are understood to be equations for the metric tensor " g ", as both the Ricci tensor and scalar curvature depend on the metric in a complicated nonlinear manner.
50.
Donaldson's recent work centers on a problem in complex differential geometry concerning a conjectural relationship between algebro-geometric " stability " conditions for smooth projective varieties and the existence of " extremal " K�hler metrics, typically those with constant scalar curvature ( see for example cscK metric ).
