an

Einstein space which is Ricci-flat and consequently has zero scalar curvature, too.

They adequately generalize the

Einstein space endowed with the cosmological constant [LAMBDA] defined as:

The background space is not one an

Einstein space (where [R.sub.[alpha][beta]] = k[g.sub.[alpha][beta]], k = const) due to the fact that Einstein's equations have k=0 in the background space.

If [T.sub.[alpha][beta]] [varies][g.sub.[alpha][beta]], the associated space is called an

Einstein space. Thus,

Einstein spaces include those described by partially degenerate metrics of this form.

Metric (1a) and Euclidean-like structure (1b) are complementary to each other in the

Einstein space. These space-time coordinates form not just a mathematical coordinate system since a light speed ([ds.sup.2] = 0) is defined in terms of dx/dt, dy/dt, and dz/dt [19].

From the purely geometrical perspective, an

Einstein space [5] is described by any metric obtained from

in which case the space, where the gravitational field is located, is called an

Einstein space. If [kappa] = 0, we have an empty space (without gravitating matter).

Equation (3.4) is the Einstein equation for

Einstein spaces in differential geometry [1,2];

In [13] the result for odd dimensional

Einstein spaces was obtained by Dumitru.

The topics are notations and prerequisites from analysis, curves in Rn, the local theory of surfaces, the intrinsic geometry of surfaces, Riemannian manifolds, spaces of constant curvature, and

Einstein spaces. This third English edition incorporates changes in all six German editions, the most recent published in 2013.

Besides, Schwarzschild space is only a very particular case of

Einstein spaces of type I.

In a previous paper the writer treated of particular classes of cosmological solutions for certain

Einstein spaces and claimed that no such solutions exist in relation thereto.