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In higher-dimensional spaces, there are more components, but in general they are all part of the so-called Riemann tensor—a rank-4 tensor introduced by Bernhard Riemann in 1854. (In Mathematica, the explicit form of such a tensor can be represented as a nested list for which TensorRank[list]  4 .) … (The parts of the Riemann tensor not captured by the Ricci tensor correspond to the so-called Weyl tensor; for d = 2 the Ricci tensor has only one independent component, equal to the negative of the Gaussian curvature.)
One can characterize the symmetry of a pattern by taking the list v of positions of cells it contains, and looking at tensors of successive ranks n : Apply[Plus, Map[Apply[Outer[Times, ##] &, Table[#, {n}]] &, v]] For circular or spherical patterns that are perfectly isotropic in d dimensions these tensors must all be proportional to (d - 2)!!… But when n = 4 isotropy requires the {1, 1, 1, 1} and {1, 1, 2, 2} tensor components to have ratio β = 3 —while square symmetry allows these components to have any ratio. (In general there will be more than one component unless the representation of the lattice symmetry group carried by the rank n tensor is irreducible.)
But in normal coordinates the first non-trivial term in the expansion of the metric is proportional to the Riemann tensor, yet the symmetry of a spherical volume makes it inevitable that the Ricci scalar is the only combination of components that can appear at lowest order. … Inverse[#2], RotateLeft[ Range[TensorRank[t]]]] &, t, Reverse[gl]] Laplacian[f_] := Inner[D, Sqrt[Det[g]] (Inverse[g] . … It is also known that if the Ricci tensor is non-negative, then the volume never grows faster than r d .
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