# Sn

We begin our analyses with the sphere because it is both topologically and geometrically the simplest of manifolds, and because its natural metric is such an important ingredient in so many other metrics.

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 xi2
where 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 2,
where
2 = 12 + sin(θ1)2 22 + sin(θ1)2 sin(θ2)2 32 + ... +
Π sin(θi)2 2
and r is simply a parameter specifying the curvature of the sphere. We can use the following Mathematica code to generate this metric:
xu = Flatten[ {Table[ Symbol[ "t" <> ToString[i]], {i, dim - 1}], ph}];

gdiag = r^2 Flatten[ {1, Table[ Product[ Sin[x[[i - 1]]]^2, {i, 2, j}], {j, 2, dim}]}];

gll = DiagonalMatrix[ gdiag];

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.

This metric has degeneracy singularities at θi = 0 and π. The coordinate chart is

0 < θi < π

0 <= φ < 2 π

Computations are simplified by a transformation to coordinates
ci = cos (θi)
Degeneracy singularities are now at ci = +- 1 in the new metric:
2 = dc12 / (1 - c12) +
(1 - c12) dc22 / (1 - c22) +

(1 - c12) (1 - c22) dc32 / (1 - c32) + ... +

Π (1 - ci2) 2

We will use this chart often in the following to improve Mathematica efficiency.

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 c
while those of the Ricci Tensor are
Ra b = (n - 1) ga b / r2
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 Sn,
Λ = (n - 1) (n / 2 - 1) / r2
The 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,
 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
These values can be computed using the Mathematica Gamma function, or directly is an integration over the volume element:
rdg = simpler[ Sqrt[ Simplify[ Det[gll]]]];

area[ x_List] := If [ Length[x] == 1, 2 Pi * rdg, Integrate[ area[ Rest[x]], {x[], 0, Pi}]];

Simplify[ area[xu]]

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.

The next section explores tori in arbitrary dimensions.