In reference , some properties such as Gaussian curvature
K(t), Ricci scalar curvature R(t), matter and dark energy density parameters ([[OMEGA].
Then K : M [right arrow] R, K(P) = det S(P) function is called the Gaussian curvature
function of M.
The third edition adds sections on eigenvectors, Gaussian curvature
, and electromagnetism.
where g(x) is the Gaussian curvature
of [partial derivative]X at x, i.
2] or equivalently, there is a nontrivial functional relation [PHI] (K, H) = 0 between its Gaussian curvature
K and mean curvature H.
mu]v] = 0 cannot discriminate about these solutions, additional considerations must be examined: in the following it will be shown that, since R(r) is related to the Gaussian curvature
, it cannot be set equal to the radial coordinate r as in (4) because this brings to unphysical consequences.
2] in (8) is the Gaussian curvature
of a spherical volume of the PV equal to 4[pi][r.
1) is the inverse square root of the Gaussian curvature
of the spherically symmetric geodesic surface in the spatial section [16,17,19].
2] are the PPs' Gaussian curvature
and mass-energy density respectively.
As the curvature force, the mass density, and the Gaussian curvature
are intimately related to the PPs in the negative-energy PV, it is easy to conclude that the Einstein equation and General Relativity must also be intimately related to that vacuum state.
These line-elements, although different, are not disjoint, being coupled by quantities that are determined from the line-element for the interior of the star and by a common Gaussian curvature
at the surface boundary of the object, as the study by Schwarzschild  (and my generalisation thereof ) for the ideal case of a homogeneous incompressible sphere of fluid teaches us.
It is in fact the radius of curvature, in that it determines the Gaussian curvature
G = 1/[R.