The round torus metric is most easily constructed via its embedding in a Euclidean space of one higher dimension. As with the sphere, this is not necessary for its definition: intrinsically, T^{n} is defined as a tensor product of n circles.
Let us begin with T^{2} embedded in R^{3}. If a is the distance from the center of the hole to the center of the tube of the torus, and b (< a) is the radius of the tube:
then the torus is described by
(a - (x^{2} + y^{2})^{1/2})^{2} + z^{2} = b^{2}.This can be parameterized by
x = cos(u)(a - b cos(v)),where the polar coordinate ρ = a - b cos(v) and u and v run from 0 to 2 π. Transforming to the intrinsic coordinate chart, we havey = sin(u)(a - b cos(v)) and
z = b sin(v).
g_{ij} = δ_{k l} dx^{k} / du^{i} dx^{l} / du^{j}and obtain the metric
ds^{2} = (a - b cos(v))^{2} du^{2} + b^{2} dv^{2}.For T^{3} embedded in R^{4}, we have
(b - ((a - (x^{2} + y^{2})^{1/2})^{2} + z^{2})^{1/2})^{2} + z_{2}^{2} = c^{2}with parameterization
x = cos(u) (a + cos(v)(-b + c cos(w))),which yields the metricy = sin(u) (a + cos(v)(-b + c cos(w))),
z = sin(v) (b - c cos(w)) and
z_{2} = c sin(w),
ds^{2} = (a + cos(v)(-b + c cos(w)))^{2} du^{2} + (b - c cos(w))^{2} dv^{2} + c^{2} dw^{2}.Note that b > c and a > b + c in order to avoid degeneracy singularities.
T^{4} embedded in R^{5} proceeds in exactly the same fashion, with embedding
(c - ((b - ((a - (x^{2} + y^{2})^{1/2})^{2} + z^{2})^{1/2})^{2} + z_{2}^{2})^{1/2})^{2} + z_{3}^{2} = d^{2}and parameterization
x = cos(u) (a - cos(v) (b + cos(w) (-c + d cos(q)))),yielding the metricy = sin(u) (a - cos(v) (b + cos(w) (-c + d cos(q)))),
z = sin(v) (b + cos(w) (-c + d cos(q))),
z_{2} = sin(w) (c - d cos(q)) and
z_{3} = d sin(q)
ds^{2} = (a - cos(v) (b + cos(w) (-c + d cos(q))))^{2} du^{2} + (b - cos(w) (c - d cos(q)))^{2} dv^{2} +Once again, c > d, b > c + d and a > b + c + d in order to avoid degeneracy singularities.(c - d cos(q))^{2} dw^{2} + d^{2} dq^{2}.
We can use the following Mathematica code to generate and verify a metric for T^{n}:
xu = Table[Symbol["t" <> ToString[i]], {i, dim}];Notice that we include code to verify that the metric satisfies the embedding equations. When constructing metrics in this fashion, it is always a good idea to make sure that the code always works. It is all too easy to generalize an algorithm which works fine for small dimensions and in the process introduce bugs which only show up when the dimension is large.rads = {ra, rb, rc, rd, re, rf, rg, rh, ri, rj, rk};
xcu = {rads[[dim]] Sin[xu[[dim]]]};
Do[
PrependTo[xcu, Simplify[ Sin[xu[[j]]] (rads[[j]] - xcu[[1]] Cot[xu[[j + 1]]])]],PrependTo[ xcu, Cot[xu[[1]]] xcu[[1]]];{j, dim - 1, 1, -1}];
chk = xcu[[1]]^2;
Do[
chk = simpler[ Simplify[ (rads[[j - 1]] - Sqrt[chk + xcu[[j]]^2])^2]],Print[ "Check: ", simpler[ Simplify[ chk]]];{j, 2, dim + 1}];
gll = simpler[ Simplify[ Table[ Table[ simpler[ Simplify[
Sum[ D[xcu[[k]], xu[[i]]] D[xcu[[k]], xu[[j]]], {k, dim + 1}]]],{j, dim}], {i, dim}]]];
After transforming from angular to cosine coordinates, we have for n = 2:
ds^{2} = (r_{a} - c_{2} r_{b})^{2} dc_{1}^{2} +for n = 3:r_{b}^{2} / (1 - c_{2}^{2}) dc_{2}^{2}
ds^{2} = (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c})^{2} dc_{1}^{2} +and for n = 4:(r_{b} - c_{3} r_{c})^{2} / (1 - c_{2}^{2}) dc_{2}^{2} +r_{c}^{2} / (1 - c_{3}^{2}) dc_{3}^{2}
ds^{2} = (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c} - c_{2} c_{3} c_{4} r_{d})^{2} dc_{1}^{2} +Note that in each case we have introduced coordinate singularities at c_{i > 1} = +- 1. These can be covered by translation of the original angle.(r_{b} - c_{3} r_{c} + c_{3} c_{4} r_{d})^{2} / (1 - c_{2}^{2}) dc_{2}^{2} +(r_{c} - c_{4} r_{d})^{2} / (1 - c_{3}^{2}) dc_{3}^{2} +
r_{d}^{2} / (1 - c_{4}^{2}) dc_{4}^{2}
The nonzero Christoffel Symbols for n = 2, 3 and 4 are
n = 2 | n = 3 | n = 4 | |
Γ^{1}_{1 2} | r_{b} / (- r_{a} + c_{2} r_{b}) | (r_{b} - c_{3} r_{c}) / | (r_{b} - c_{3} r_{c} + c_{3} c_{4} r_{d}) / |
(- r_{a} + c_{2} r_{b} - c_{2} c_{3} r_{c}) | (- r_{a} + c_{2} r_{b} - c_{2} c_{3} r_{c} + c_{2} c_{3} c_{4} r_{d}) | ||
Γ^{1}_{1 3} | c_{2} r_{c} / (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c}) | c_{2} (r_{c} - c_{4} r_{d}) / | |
(r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c} - c_{2} c_{3} c_{4} r_{d}) | |||
Γ^{1}_{1 4} | c_{2} c_{3} r_{d} / (- r_{a} + c_{2} r_{b} - c_{2} c_{3} r_{c} + c_{2} c_{3} c_{4} r_{d}) | ||
Γ^{2}_{1 1} | - (1 - c_{2}^{2}) (- r_{a} + c_{2} r_{b}) / | - (1 - c_{2}^{2}) (- r_{a} + c_{2} r_{b} - c_{2} c_{3} r_{c}) / | - (1 - c_{2}^{2}) (- r_{a} + c_{2} r_{b} - c_{2} c_{3} r_{c} + |
r_{b} | (r_{b} - c_{3} r_{c}) | c_{2} c_{3} c_{4} r_{d}) / (r_{b} - c_{3} r_{c} + c_{3} c_{4} r_{d}) | |
Γ^{2}_{2 2} | c_{2} / (1 - c_{2}^{2}) | c_{2} / (1 - c_{2}^{2}) | c_{2} / (1 - c_{2}^{2}) |
Γ^{2}_{2 3} | r_{c} / (- r_{b} + c_{3} r_{c}) | (r_{c} - c_{4} r_{d}) / (- r_{b} + c_{3} r_{c} - c_{3} c_{4} r_{d}) | |
Γ^{2}_{2 4} | c_{3} r_{d} / (r_{b} - c_{3} r_{c} + c_{3} c_{4} r_{d}) | ||
Γ^{3}_{1 1} | - c_{2} (1 - c_{3}^{2}) (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c}) / | - c_{2} (1 - c_{3}^{2}) (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c} - | |
r_{c} | c_{2} c_{3} c_{4} r_{d}) / (r_{c} - c_{4} r_{d}) | ||
Γ^{3}_{2 2} | - (1 - c_{3}^{2}) (- r_{b} + c_{3} r_{c}) / | - (1 - c_{3}^{2}) (- r_{b} + c_{3} r_{c} - c_{3} c_{4} r_{d}) / | |
((1 - c_{2}^{2}) r_{c}) | ((1 - c_{2}^{2}) (r_{c} - c_{4} r_{d})) | ||
Γ^{3}_{3 3} | c_{3} / (1 - c_{3}^{2}) | c_{3} / (1 - c_{3}^{2}) | |
Γ^{3}_{3 4} | r_{d} / (- r_{c} + c_{4} r_{d}) | ||
Γ^{4}_{1 1} | - c_{2} c_{3} (1 - c_{4}^{2}) (- r_{a} + c_{2} r_{b} - c_{2} c_{3} r_{c} + | ||
c_{2} c_{3} c_{4} r_{d}) / r_{d} | |||
Γ^{4}_{2 2} | - c_{3} (1 - c_{4}^{2}) (r_{b} - c_{3} r_{c} + c_{3} c_{4} r_{d}) / | ||
((1 - c_{2}^{2}) r_{d}) | |||
Γ^{4}_{3 3} | - (1 - c_{4}^{2}) (- r_{c} + c_{4} r_{d}) / ((1 - c_{3}^{2}) r_{d}) | ||
Γ^{4}_{4 4} | c_{4} / (1 - c_{4}^{2}) |
Naively, we see that the c_{n} direction is geodesic, and there are n - 2 geodesic hypersurfaces defined by {c_{i} ... c_{n}} for i from 2 to n - 1. But note that for specific values of c_{i}, other directions are geodesic as well. In general, c_{i} is a geodesic direction when c_{i+1} through c_{n} are equal to +-1. But since the origin of the c_{i} can be translated by a constant without changing the Christoffel Symbols, we see that in fact every direction is a geodesic direction. This is of course consistent with the topology of the torus as a tensor product of circles, and indicates that the business of analyzing the Christoffel Symbols for geodesic behavior is a mildly dangerous one, since they are not invariant with respect to diffeomorphisms.
For the same reason, the components of the Riemann and Ricci Tensors are less useful than the invariants which we are examining for the purpose of building geometric intuition. We will nonetheless continue to present them in "natural coordinates" for the sake of completeness, and in order to allow students to check their work. The nonzero components of the Riemann Tensor for n = 2, 3 and 4 are
n = 2 | n = 3 | n = 4 | |
R_{1 2 1 2} | c_{2} r_{b} (- r_{a} + c_{2} r_{b}) / | c_{2} c_{3}^{2} (- r_{b} + c_{3} r_{c}) * | c_{2} c_{3}^{2} c_{4}^{2} (r_{b} - c_{3} r_{c} + c_{3} c_{4} r_{d}) * |
(1 - c_{2}^{2}) | (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c}) / (1 - c_{2}^{2}) | (- r_{a} + c_{2} r_{b} - c_{2} c_{3} r_{c} + c_{2} c_{3} c_{4} r_{d}) / (1 - c_{2}^{2}) | |
R_{1 3 1 3} | c_{2} c_{3} r_{c} (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c}) / | c_{2} c_{3} c_{4}^{2} (- r_{c} + c_{4} r_{d}) * | |
(1 - c_{3}^{2}) | (- r_{a} + c_{2} r_{b} - c_{2} c_{3} r_{c} + c_{2} c_{3} c_{4} r_{d}) / (1 - c_{3}^{2}) | ||
R_{1 4 1 4} | c_{2} c_{3} c_{4} r_{d} (- r_{a} + c_{2} r_{b} - c_{2} c_{3} r_{c} + c_{2} c_{3} c_{4} r_{d}) / (1 - c_{4}^{2}) | ||
R_{2 3 2 3} | c_{3} r_{c} (- r_{b} + c_{3} r_{c}) / | c_{3} c_{4}^{2} (- r_{c} + c_{4} r_{d}) (r_{b} - c_{3} r_{c} + c_{3} c_{4} r_{d}) / | |
((1 - c_{2}^{2}) (1 - c_{3}^{2})) | ((1 - c_{2}^{2}) (1 - c_{3}^{2})) | ||
R_{2 4 2 4} | c_{3} c_{4} r_{d} (r_{b} - c_{3} r_{c} + c_{3} c_{4} r_{d}) / ((1 - c_{2}^{2}) (1 - c_{4}^{2})) | ||
R_{3 4 3 4} | c_{4} r_{d} (- r_{c} + c_{4} r_{d}) / ((1 - c_{3}^{2}) (1 - c_{4}^{2})) |
The nonzero components of the Ricci Tensor for n = 2, 3 and 4 are
n = 2 | n = 3 | n = 4 | |
R_{1 1} | c_{2} (- r_{a} + c_{2} r_{b}) / rb | c_{2} c_{3} (- r_{b} + 2 c_{3} r_{c}) (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c}) / (r_{c} (- r_{b} + c_{3} r_{c})) | c_{2} c_{3} c_{4} (- r_{a} + c_{2} r_{b} - c_{2} c_{3} r_{c} + c_{2} c_{3} c_{4} r_{d}) (- r_{b} r_{c} + c_{3} r_{c}^{2} + 2 c_{4} r_{b} r_{d} - 4 c_{3} c_{4} r_{c} r_{d} + 3 c_{3} c_{4}^{2} r_{d}^{2}) / (r_{d} (- r_{c} + c_{4} r_{d}) (r_{b} - c_{3} r_{c} + c_{3} c_{4} r_{d})) |
R_{2 2} | c_{2} r_{b} / ((1 - c_{2}^{2}) (- r_{a} + c_{2} r_{b})) | c_{3} (- r_{b} + c_{3} r_{c}) (r_{a} - c_{2} r_{b} + 2 c_{2} c_{3} r_{c}) / ((1 - c_{2}^{2}) r_{c} (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c})) | c_{3} c_{4} (r_{b} - c_{3} r_{c} + c_{3} c_{4} r_{d}) (r_{a} r_{c} - c_{2} r_{b} r_{c} + c_{2} c_{3} r_{c}^{2} - 2 c_{4} r_{a} r_{d} + 2 c_{2} c_{4} r_{b} r_{d} - 4 c_{2} c_{3} c_{4} r_{c} r_{d} + 3 c_{2} c_{3} c_{4}^{2} r_{d}^{2}) / ((1 - c_{2}^{2}) r_{d} (- r_{c} + c_{4} r_{d}) (- r_{a} + c_{2} r_{b} - c_{2} c_{3} r_{c} + c_{2} c_{3} c_{4} r_{d})) |
R_{3 3} | c_{3} r_{c} (r_{a} - 2 c_{2} r_{b} + 2 c_{2} c_{3} r_{c}) / ((1 - c_{3}^{2}) (- r_{b} + c_{3} r_{c}) (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c})) | c_{4} (- r_{c} + c_{4} r_{d}) (- (r_{a} r_{b}) + c_{2} r_{b}^{2} + c_{3} r_{a} r_{c} - 2 c_{2} c_{3} r_{b} r_{c} + c_{2} c_{3}^{2} r_{c}^{2} - 2 c_{3} c_{4} r_{a} r_{d} + 4 c_{2} c_{3} c_{4} r_{b} r_{d} - 4 c_{2} c_{3}^{2} c_{4} r_{c} r_{d} + 3 c_{2} c_{3}^{2} c_{4}^{2} r_{d}^{2}) / ((1 - c_{3}^{2}) r_{d} (r_{b} - c_{3} r_{c} + c_{3} c_{4} r_{d}) (- r_{a} + c_{2} r_{b} - c_{2} c_{3} r_{c} + c_{2} c_{3} c_{4} r_{d})) | |
R_{4 4} | c_{4} r_{d} (- (r_{a} r_{b}) + c_{2} r_{b}^{2} + 2 c_{3} r_{a} r_{c} - 4 c_{2} c_{3} r_{b} r_{c} + 3 c_{2} c_{3}^{2} r_{c}^{2} - 2 c_{3} c_{4} r_{a} r_{d} + 4 c_{2} c_{3} c_{4} r_{b} r_{d} - 6 c_{2} c_{3}^{2} c_{4} r_{c} r_{d} + 3 c_{2} c_{3}^{2} c_{4}^{2} r_{d}^{2}) / ((1 - c_{4}^{2}) (- r_{c} + c_{4} r_{d}) (r_{b} - c_{3} r_{c} + c_{3} c_{4} r_{d}) (- r_{a} + c_{2} r_{b} - c_{2} c_{3} r_{c} + c_{2} c_{3} c_{4} r_{d})) |
The invariants we have chosen to examine, for n = 2 and 3, are
R | 2 c_{2} / | 2 c_{3} (r_{a} - 2 c_{2} r_{b} + 3 c_{2} c_{3} r_{c}) / |
(r_{b} (- r_{a} + c_{2} r_{b})) | (r_{c} (- r_{b} + c_{3} r_{c}) (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c})) | |
R^{a b} R_{a b} | 2 c_{2}^{2} / | (2 c_{3}^{2} (r_{a}^{2} - 3 c_{2} r_{a} r_{b} + 3 c_{2}^{2} r_{b}^{2} + 4 c_{2} c_{3} r_{a} r_{c} - 8 c_{2}^{2} c_{3} r_{b} r_{c} + 6 c_{2}^{2} c_{3}^{2} r_{c}^{2})) / |
(r_{b}^{2} (- r_{a} + c_{2} r_{b})^{2}) | (r_{c}^{2} (- r_{b} + c_{3} r_{c})^{2} (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c})^{2}) | |
R^{a b c d} R_{a b c d} | 4 c_{2}^{2} / | (4 c_{3}^{2} (r_{a}^{2} - 2 c_{2} r_{a} r_{b} + 2 c_{2}^{2} r_{b}^{2} + 2 c_{2} c_{3} r_{a} r_{c} - 4 c_{2}^{2} c_{3} r_{b} r_{c} + 3 c_{2}^{2} c_{3}^{2} r_{c}^{2})) / |
(r_{b}^{2} (- r_{a} + c_{2} r_{b})^{2}) | (r_{c}^{2} (- r_{b} + c_{3} r_{c})^{2} (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c})^{2}) | |
R^{a b} R^{c}_{a} R_{b c} | 2 c_{2}^{3} / | (c_{3}^{3} (2 r_{a}^{3} - 9 c_{2} r_{a}^{2} r_{b} + 15 c_{2}^{2} r_{a} r_{b}^{2} - 10 c_{2}^{3} r_{b}^{3} + 12 c_{2} c_{3} r_{a}^{2} r_{c} - 36 c_{2}^{2} c_{3} r_{a} r_{b} r_{c} + 36 c_{2}^{3} c_{3} r_{b}^{2} r_{c} + 24 c_{2}^{2} c_{3}^{2} r_{a} r_{c}^{2} - 48 c_{2}^{3} c_{3}^{2} r_{b} r_{c}^{2} + 24 c_{2}^{3} c_{3}^{3} r_{c}^{3})) / |
(r_{b}^{3} (- r_{a} + c_{2} r_{b})^{3}) | (r_{c}^{3} (- r_{b} + c_{3} r_{c})^{3} (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c})^{3}) | |
R^{a b c d} R_{a c} R_{b d} | 2 c_{2}^{3} / | (2 c_{3}^{3} (r_{a}^{3} - 4 c_{2} r_{a}^{2} r_{b} + 6 c_{2}^{2} r_{a} r_{b}^{2} - 4 c_{2}^{3} r_{b}^{3} + 5 c_{2} c_{3} r_{a}^{2} r_{c} - 17 c_{2}^{2} c_{3} r_{a} r_{b} r_{c} + 17 c_{2}^{3} c_{3} r_{b}^{2} r_{c} + 12 c_{2}^{2} c_{3}^{2} r_{a} r_{c}^{2} - 24 c_{2}^{3} c_{3}^{2} r_{b} r_{c}^{2} + 12 c_{2}^{3} c_{3}^{3} r_{c}^{3})) / |
(r_{b}^{3} (- r_{a} + c_{2} r_{b})^{3}) | (r_{c}^{3} (- r_{b} + c_{3} r_{c})^{3} (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c})^{3}) | |
R^{a b c d} R^{e}_{a} R_{b c d e} | 2 c_{2}^{3} / | (c_{3}^{3} (2 r_{a}^{3} - 7 c_{2} r_{a}^{2} r_{b} + 9 c_{2}^{2} r_{a} r_{b}^{2} - 6 c_{2}^{3} r_{b}^{3} + 8 c_{2} c_{3} r_{a}^{2} r_{c} - 20 c_{2}^{2} c_{3} r_{a} r_{b} r_{c} + 20 c_{2}^{3} c_{3} r_{b}^{2} r_{c} + 12 c_{2}^{2} c_{3}^{2} r_{a} r_{c}^{2} - 24 c_{2}^{3} c_{3}^{2} r_{b} r_{c}^{2} + 12 c_{2}^{3} c_{3}^{3} r_{c}^{3})) / |
(r_{b}^{3} (- r_{a} + c_{2} r_{b})^{3}) | (r_{c}^{3} (- r_{b} + c_{3} r_{c})^{3} (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c})^{3}) | |
R^{a b c d} R^{e f}_{a b} R_{c d e f} | 8 c_{2}^{3} / | (8 c_{3}^{3} (r_{a}^{3} - 3 c_{2} r_{a}^{2} r_{b} + 3 c_{2}^{2} r_{a} r_{b}^{2} - 2 c_{2}^{3} r_{b}^{3} + 3 c_{2} c_{3} r_{a}^{2} r_{c} - 6 c_{2}^{2} c_{3} r_{a} r_{b} r_{c} + 6 c_{2}^{3} c_{3} r_{b}^{2} r_{c} + 3 c_{2}^{2} c_{3}^{2} r_{a} r_{c}^{2} - 6 c_{2}^{3} c_{3}^{2} r_{b} r_{c}^{2} + 3 c_{2}^{3} c_{3}^{3} r_{c}^{3})) / |
(r_{b}^{3} (- r_{a} + c_{2} r_{b})^{3}) | (r_{c}^{3} (- r_{b} + c_{3} r_{c})^{3} (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c})^{3}) | |
R^{a b c d} R^{e}_{a}^{f}_{c} R_{b e d f} | 0 | (6 c_{2}^{2} c_{3}^{4}) / |
(r_{c}^{2} (- r_{b} + c_{3} r_{c})^{2} (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c})^{2}) | ||
R^{a b c d} R^{e}_{a}^{f}_{c} R_{b f d e} | - 2 c_{2}^{3} / | (- 2 c_{3}^{3} (r_{a}^{2} - c_{2} r_{a} r_{b} + c_{2}^{2} r_{b}^{2}) (r_{a} - 2 c_{2} r_{b} + 3 c_{2} c_{3} r_{c})) / |
(r_{b}^{3} (- r_{a} + c_{2} r_{b})^{3}) | (r_{c}^{3} (- r_{b} + c_{3} r_{c})^{3} (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c})^{3}) | |
R^{a b; c} R_{a b; c} | 2 (1 - c_{2}^{2}) r_{a}^{2} / | |
(r_{b}^{4} (- r_{a} + c_{2} r_{b})^{4}) | ||
R^{a b; c} R_{a c; b} | (1 - c_{2}^{2}) r_{a}^{2} / | |
(r_{b}^{4} (- r_{a} + c_{2} r_{b})^{4})) | ||
R^{a b}_{; a} R^{c}_{b; c} | (1 - c_{2}^{2}) r_{a}^{2} / | |
(r_{b}^{4} (- r_{a} + c_{2} r_{b})^{4})) | ||
R^{a b c d; e} R_{a b c d; e} | 4 (1 - c_{2}^{2}) r_{a}^{2} / | |
(r_{b}^{4} (- r_{a} + c_{2} r_{b})^{4}) | ||
R^{a b c d}_{; a} R^{e}_{b c d; e} | 2 (1 - c_{2}^{2}) r_{a}^{2} / | |
(r_{b}^{4} (- r_{a} + c_{2} r_{b})^{4}) | ||
ε^{a b c i ...} ε^{e f g}_{i ...} R_{b c e}^{h} R_{f g a h} | (- 8 c_{2} c_{3}^{2} (- (r_{a} r_{b}) + c_{2} r_{b}^{2} + 2 c_{3} r_{a} r_{c} - 4 c_{2} c_{3} r_{b} r_{c} + 3 c_{2} c_{3}^{2} r_{c}^{2})) / | |
(r_{c}^{2} (- r_{b} + c_{3} r_{c})^{2} (r_{a} - c_{2} r_{b} + c_{2} c_{3} r_{c})^{2}) |
Euler_{n = 4} = (3 c_{2} c_{3}^{2} c_{4}^{3}) / (4 π^{2} r_{d} (- r_{c} + c_{4} r_{d}) (r_{b} - c_{3} r_{c} + c_{3} c_{4} r_{d}) (- r_{a} + c_{2} r_{b} - c_{2} c_{3} r_{c} + c_{2} c_{3} c_{4} r_{d}))
Euler_{n = 6} = (15 c_{2} c_{3}^{2} c_{4}^{3} c_{5}^{4} c_{6}^{5}) / (8 π^{3} r_{f} (- r_{e} + c_{6} r_{f}) (r_{d} - c_{5} r_{e} + c_{5} c_{6} r_{f}) (- r_{c} + c_{4} r_{d} - c_{4} c_{5} r_{e} + c_{4} c_{5} c_{6} r_{f}) (r_{b} - c_{3} r_{c} + c_{3} c_{4} r_{d} - c_{3} c_{4} c_{5} r_{e} + c_{3} c_{4} c_{5} c_{6} r_{f}) (- r_{a} + c_{2} r_{b} - c_{2} c_{3} r_{c} + c_{2} c_{3} c_{4} r_{d} - c_{2} c_{3} c_{4} c_{5} r_{e} + c_{2} c_{3} c_{4} c_{5} c_{6} r_{f}))
Note the simlilarity among the invariants for T^{2}, and the corresponding lack of simliarity for T^{3}. In general, the complexity of the results make them rather useless for larger dimension. For instance, for the Kretschmann Invariant, there are 31 terms for n = 4, 163 terms for n = 5, 882 terms for n = 6 and 4896 terms for n = 7. It is interesting to note that for all invariants we computed, the denominator is equal to a power of the numerator of the determinant of the metric. Hence all are regular. The zeroes obviously have no invariant meaning since they can be removed by simple translation of the angular coordinates.
The next section explores cosmological solutions to Einstein's Equations.
©2005, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.
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