An n-sphere of fixed radius r as embedded in Euclidean space of dimension n + 1 is the locus of all points satisfying
r2 = Σ xi2where the xi are coordinates in the Cartesian chart. It is important to realize that this definition is a convenient starting point, but that the sphere need not be considered as an embedding in order to be defined. In an intrinsic definition, the sphere is treated as a rotation of a (hyper-)hemisphere around a circle. The standard metric is
ds2 = r2 dΩ2,where
dΩ2 = dθ12 + sin(θ1)2 dθ22 + sin(θ1)2 sin(θ2)2 dθ32 + ... +and r is simply a parameter specifying the curvature of the sphere. We can use the following Mathematica code to generate this metric:Π sin(θi)2 dφ2
xu = Flatten[ {Table[ Symbol[ "t" <> ToString[i]], {i, dim - 1}], ph}];The first line creates a list whose elements are the variables t1, t2, etc., corresponding to θ1, etc. The second line creates a list whose elements are the diagonal entries in the final metric which is then created and filled in with the remaining line.gdiag = r^2 Flatten[ {1, Table[ Product[ Sin[x[[i - 1]]]^2, {i, 2, j}], {j, 2, dim}]}];
gll = DiagonalMatrix[ gdiag];
This metric has degeneracy singularities at θi = 0 and π. The coordinate chart is
0 < θi < πComputations are simplified by a transformation to coordinates0 <= φ < 2 π
ci = cos (θi)Degeneracy singularities are now at ci = +- 1 in the new metric:
dΩ2 = dc12 / (1 - c12) +We will use this chart often in the following to improve Mathematica efficiency.(1 - c12) dc22 / (1 - c22) +(1 - c12) (1 - c22) dc32 / (1 - c32) + ... +
Π (1 - ci2) dφ2
The nonzero Christoffel Symbols for n = 2, 3 and 4 are
| n = 2 | n = 3 | n = 4 | |
| Γ11 1 | c1 / (1 - c12) | c1 / (1 - c12) | c1 / (1 - c12) |
| Γ12 2 | c1 (1 - c12) | c1 (1 - c12) / (1 - c22) | c1 (1 - c12) / (1 - c22) |
| Γ13 3 | c1 (1 - c12) (1 - c22) | c1 (1 - c12) (1 - c22) / (1 - c32) | |
| Γ14 4 | c1 (1 - c12) (1 - c22) (1 - c32) | ||
| Γ21 2 | - c1 / (1 - c12) | - c1 / (1 - c12) | - c1 / (1 - c12) |
| Γ22 2 | c2 / (1 - c22) | c2 / (1 - c22) | |
| Γ23 3 | c2 (1 - c22) | c2 (1 - c22) / (1 - c32) | |
| Γ24 4 | c2 (1 - c22) (1 - c32) | ||
| Γ31 3 | - c1 / (1 - c12) | - c1 / (1 - c12) | |
| Γ32 3 | - c2 / (1 - c22) | - c2 / (1 - c22) | |
| Γ33 3 | c3 / (1 - c32) | ||
| Γ34 4 | c3 (1 - c32) | ||
| Γ41 4 | - c1 / (1 - c12) | ||
| Γ42 4 | - c2 / (1 - c22) | ||
| Γ43 4 | - c3 / (1 - c32) |
We can see that c1 is a geodesic direction. For n = 2, these are the "great circles" crossing the poles. Similarly, for n > 2 the c1 and c2 directions define a geodesic hypersurface and for n > 3 c1, c2 and c3 define a geodesic hypersurface. In general, there n - 2 geodesic hypersurfaces defined by {c1 ... ci} for i from 2 to n - 1.
Note also that when c1 is zero, c2 is a geodesic direction: the great circle around the equator. In general, when c1 through ci are zero, ci+1 is a geodesic direction.
The components of the Riemann Tensor are all positive definite and can be written as
Ra b c d = ga c gb d - ga d gb cwhile those of the Ricci Tensor are
Ra b = (n - 1) ga b / r2Manifolds which admit metrics whose Ricci Tensors are a constant multiple of the metric are called Einstein Manifolds. They are spaces of constant curvature and admit metrics which are vacuum solutions to Einstein's Equations with cosmological constant. For Sn,
Λ = (n - 1) (n / 2 - 1) / r2The invariants we have chosen to examine are
| R | n (n - 1) / r2 |
| Ra b Ra b | n (n - 1)2 / r4 |
| Ra b c d Ra b c d | 2 n (n - 1) / r4 |
| Ra b Rca Rb c | n (n - 1)3 / r6 |
| Ra b c d Ra c Rb d | n (n - 1)3 / r6 |
| Ra b c d Rea Rb c d e | n (n - 1)2 / r6 |
| Ra b c d Re fa b Rc d e f | 4 n (n - 1) / r6 |
| Ra b c d Reafc Rb e d f | n (n - 1) (n - 2) / r6 |
| Ra b c d Reafc Rb f d e | n (n - 1) (n - 3) / r6 |
| Ra b; c Ra b; c | 0 |
| Ra b; c Ra c; b | 0 |
| Ra b; a Rcb; c | 0 |
| Ra b c d; e Ra b c d; e | 0 |
| Ra b c d; a Reb c d; e | 0 |
| Euler class | 2 / Area (Sn) for even n, otherwise 0 |
| εa b c i ... εe f gi ... Rb c eh Rf g a h | - n (2 n - 4) (2 n - 2) / r4 |
Note that all of them are positive except the last, consistent with the knowledge that a sphere is a space of positive curvature, and that the invariants involving covariant derivatives are zero, consistent with the knowledge that a sphere is a space of constant curvature. The overwhelming similarity of these invariants is indicative of the geometrical simplicity of the sphere: they only depend on the dimension and the parameter r.
The surface area of Sn is a factor in many computations. In general, it is
2 π(n + 1) / 2 rn / Γ((n + 1) / 2)where Γ is the Gamma Function. This gives us, for example,
These values can be computed using the Mathematica Gamma function, or directly is an integration over the volume element:
n Area (Sn) 1 2 π r 2 4 π r2 3 2 π2 r3 4 8 π2 r4 / 3 5 π3 r5 6 16 π3 r6 / 15 7 π4 r7 / 3
rdg = simpler[ Sqrt[ Simplify[ Det[gll]]]];The area function is an example of a recursive function; it assumes the standard coordinate chart and that the metric does not depend on φ. Its argument is the coordinate list, which it needs in order to control the recursion.area[ x_List] := If [ Length[x] == 1, 2 Pi * rdg, Integrate[ area[ Rest[x]], {x[[1]], 0, Pi}]];
Simplify[ area[xu]]
The next section explores tori in arbitrary dimensions.
©2013, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.
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