scalar curvature sentence in Hindi
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- 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.
- 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 .)
- 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 ..
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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 ).
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