Notice that the cosmological constant is carefully chosen to partially cancel the matter density of the dust. The Gödel spacetime is a rare example of a "regular" (singularity-free) solution of the Einstein field equation. Gödel's original chart (given here) is geodesically complete and singularity free; therefore, it is a global chart, and the spacetime is homeomorphic to R4, and therefore, simply connected. In any Lorentzian spacetime, the fourth-rank Riemann tensor is a multilinear operator on the four-dimensional space of tangent vectors (at some event), but a linear operator on the six-dimensional space of bivectors at that event. Accordingly, it has a characteristic polynomial, whose roots are the eigenvalues. In the Gödel spacetime, these eigenvalues are very simple:  This spacetime admits a five-dimensional Lie algebra of Killing vectors, which can be generated by "time translation" formula_17, two "spatial translations" formula_18, plus two further Killing vector fields: and The isometry group acts "transitively" (since we can translate in formula_21, and using the fourth vector we can move along formula_22 as well), so the spacetime is "homogeneous". However, it is not "isotropic", as we shall see. It is obvious from the generators just given that the slices formula_23 admit a transitive abelian three-dimensional transformation group, so a quotient of the solution can be reinterpreted as a stationary cylindrically symmetric solution. Less obviously, the slices formula_24 admit an SL(2,R) action, and the slices formula_25 admit a Bianchi III (c.f. the fourth Killing vector field). We can restate this by saying that our symmetry group includes as three-dimensional subgroups examples of Bianchi types I, III and VIII. Four of the five Killing vectors, as well as the curvature tensor, do not depend upon the coordinate y. Indeed, the Gödel solution is the Cartesian product of a factor R with a three-dimensional Lorentzian manifold (signature −++). It can be shown that the Gödel solution is, up to local isometry, the "only" perfect fluid solution of the Einstein field equation admitting a five-dimensional Lie algebra of Killing vectors. The Weyl tensor of the Gödel solution has Petrov type D.