Which smooth compact 4-manifolds admit an Einstein metric with non-negative Einstein constant? A complete answer is provided in the special case of 4-manifolds that also happen to admit either a complex structure or a symplectic structure. A Riemannian manifold (M, g) is said to be Einstein if it has constant Ricci curvature, in the sense that the function v −→ r(v, v) on the unit tangent bundl...

Einstein vacuum equations, that is a system of nonlinear partial differential equations (PDEs) are derived from Weyl metric by using relation between Einstein tensor and metric tensor. The symmetries of Einstein vacuum equations for static axisymmetric gravitational fields are obtained using the Lie classical method. We have examined the optimal system of vector fields which is further used to ...

We prove that a Sasakian 3-manifold admitting a non-trivial solution to the Einstein-Dirac equation has necessarily constant scalar curvature. In the case when this scalar curvature is non-zero, their classi cation follows then from a result by Th. Friedrich and E.C. Kim. We also prove that a scalarat Sasakian 3-manifold admits no local Einstein spinors.

The inclusion of cosmological constant into Einstein field equation was done primarily by Einstein himself with a view to model a static universe. Later he discarded the cosmological constant (and commented [1] “...... the biggest blunder of my life”), on the basis of the observational data supporting the expansion of the universe. Subsequently, it was reintroduced and then omitted to match wit...