An n-sphere of fixed radius r as embedded in Euclidean space of dimension n + 1 is the locus of all points satisfying
r^{2} = Σ x_{i}^{2}where the x_{i} 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
ds^{2} = r^{2} dΩ^{2},where
dΩ^{2} = dθ_{1}^{2} + sin(θ_{1})^{2} dθ_{2}^{2} + sin(θ_{1})^{2} sin(θ_{2})^{2} dθ_{3}^{2} + ... +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(θ_{i})^{2} dφ^{2}
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 θ_{1}, 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 θ_{i} = 0 and π. The coordinate chart is
0 < θ_{i} < πComputations are simplified by a transformation to coordinates0 <= φ < 2 π
c_{i} = cos (θ_{i})Degeneracy singularities are now at c_{i} = +- 1 in the new metric:
dΩ^{2} = dc_{1}^{2} / (1 - c_{1}^{2}) +We will use this chart often in the following to improve Mathematica efficiency.(1 - c_{1}^{2}) dc_{2}^{2} / (1 - c_{2}^{2}) +(1 - c_{1}^{2}) (1 - c_{2}^{2}) dc_{3}^{2} / (1 - c_{3}^{2}) + ... +
Π (1 - c_{i}^{2}) dφ^{2}
The nonzero Christoffel Symbols for n = 2, 3 and 4 are
n = 2 | n = 3 | n = 4 | |
Γ^{1}_{1 1} | c_{1} / (1 - c_{1}^{2}) | c_{1} / (1 - c_{1}^{2}) | c_{1} / (1 - c_{1}^{2}) |
Γ^{1}_{2 2} | c_{1} (1 - c_{1}^{2}) | c_{1} (1 - c_{1}^{2}) / (1 - c_{2}^{2}) | c_{1} (1 - c_{1}^{2}) / (1 - c_{2}^{2}) |
Γ^{1}_{3 3} | c_{1} (1 - c_{1}^{2}) (1 - c_{2}^{2}) | c_{1} (1 - c_{1}^{2}) (1 - c_{2}^{2}) / (1 - c_{3}^{2}) | |
Γ^{1}_{4 4} | c_{1} (1 - c_{1}^{2}) (1 - c_{2}^{2}) (1 - c_{3}^{2}) | ||
Γ^{2}_{1 2} | - c_{1} / (1 - c_{1}^{2}) | - c_{1} / (1 - c_{1}^{2}) | - c_{1} / (1 - c_{1}^{2}) |
Γ^{2}_{2 2} | c_{2} / (1 - c_{2}^{2}) | c_{2} / (1 - c_{2}^{2}) | |
Γ^{2}_{3 3} | c_{2} (1 - c_{2}^{2}) | c_{2} (1 - c_{2}^{2}) / (1 - c_{3}^{2}) | |
Γ^{2}_{4 4} | c_{2} (1 - c_{2}^{2}) (1 - c_{3}^{2}) | ||
Γ^{3}_{1 3} | - c_{1} / (1 - c_{1}^{2}) | - c_{1} / (1 - c_{1}^{2}) | |
Γ^{3}_{2 3} | - c_{2} / (1 - c_{2}^{2}) | - c_{2} / (1 - c_{2}^{2}) | |
Γ^{3}_{3 3} | c_{3} / (1 - c_{3}^{2}) | ||
Γ^{3}_{4 4} | c_{3} (1 - c_{3}^{2}) | ||
Γ^{4}_{1 4} | - c_{1} / (1 - c_{1}^{2}) | ||
Γ^{4}_{2 4} | - c_{2} / (1 - c_{2}^{2}) | ||
Γ^{4}_{3 4} | - c_{3} / (1 - c_{3}^{2}) |
We can see that c_{1} is a geodesic direction. For n = 2, these are the "great circles" crossing the poles. Similarly, for n > 2 the c_{1} and c_{2} directions define a geodesic hypersurface and for n > 3 c_{1}, c_{2} and c_{3} define a geodesic hypersurface. In general, there n - 2 geodesic hypersurfaces defined by {c_{1} ... c_{i}} for i from 2 to n - 1.
Note also that when c_{1} is zero, c_{2} is a geodesic direction: the great circle around the equator. In general, when c_{1} through c_{i} are zero, c_{i+1} is a geodesic direction.
The components of the Riemann Tensor are all positive definite and can be written as
R_{a b c d} = g_{a c} g_{b d} - g_{a d} g_{b c}while those of the Ricci Tensor are
R_{a b} = (n - 1) g_{a b} / r^{2}Manifolds 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 S^{n},
Λ = (n - 1) (n / 2 - 1) / r^{2}The invariants we have chosen to examine are
R | n (n - 1) / r^{2} |
R^{a b} R_{a b} | n (n - 1)^{2} / r^{4} |
R^{a b c d} R_{a b c d} | 2 n (n - 1) / r^{4} |
R^{a b} R^{c}_{a} R_{b c} | n (n - 1)^{3} / r^{6} |
R^{a b c d} R_{a c} R_{b d} | n (n - 1)^{3} / r^{6} |
R^{a b c d} R^{e}_{a} R_{b c d e} | n (n - 1)^{2} / r^{6} |
R^{a b c d} R^{e f}_{a b} R_{c d e f} | 4 n (n - 1) / r^{6} |
R^{a b c d} R^{e}_{a}^{f}_{c} R_{b e d f} | n (n - 1) (n - 2) / r^{6} |
R^{a b c d} R^{e}_{a}^{f}_{c} R_{b f d e} | n (n - 1) (n - 3) / r^{6} |
R^{a b; c} R_{a b; c} | 0 |
R^{a b; c} R_{a c; b} | 0 |
R^{a b}_{; a} R^{c}_{b; c} | 0 |
R^{a b c d; e} R_{a b c d; e} | 0 |
R^{a b c d}_{; a} R^{e}_{b c d; e} | 0 |
Euler class | 2 / Area (S^{n}) for even n, otherwise 0 |
ε^{a b c i ...} ε^{e f g}_{i ...} R_{b c e}^{h} R_{f g a h} | - n (2 n - 4) (2 n - 2) / r^{4} |
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 S^{n} is a factor in many computations. In general, it is
2 π^{(n + 1) / 2} r^{n} / 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 (S^{n}) 1 2 π r 2 4 π r^{2} 3 2 π^{2} r^{3} 4 8 π^{2} r^{4} / 3 5 π^{3} r^{5} 6 16 π^{3} r^{6} / 15 7 π^{4} r^{7} / 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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