An Einstein Metric on $S^2\times S^3$¶
The historical Egison demo bearing the name “Riemann curvature tensor of $S^2\times S^3$” uses a non-diagonal five-dimensional metric, not the elementary block product of two round metrics. Its goal is the Einstein identity
$$ \operatorname{Ric}_{ij}=4g_{ij},\qquad \mathcal R=20. $$
This notebook preserves that metric and makes its symbolic workload explicit. Small metric and connection components are suitable interactive outputs; the full Einstein residual is defined for deliberate evaluation.
Coordinates and abbreviations¶
In the coordinate order $x=(\phi,\theta,\psi,y,\alpha)$, introduce
$$ p=1-y,\quad u=a-y^2,\quad v=a-3y^2+2y^3,\quad q=a-2y+y^2. $$
These factors expose the repeated structure of the source metric and make its coordinate domain restrictions ($p u v\ne0$) visible.
declare symbol φ, θ, ψ, y, α, a: MathValue
def x : Vector MathValue := [| φ, θ, ψ, y, α |]
def p : MathValue := `(1 - y)
def u : MathValue := `(a - y^2)
def v : MathValue := `(a - 3 * y^2 + 2 * y^3)
def q : MathValue := `(a - 2 * y + y^2)
(p, u, v, q)
Local frame and induced metric¶
Put $f=q/(6u)$. The metric has the orthonormal-frame decomposition
$$ \begin{split} ds^2={}&\frac p6(d\theta^2+\sin^2\theta\,d\phi^2) +\frac{p}{2v}dy^2 +\frac{v}{9u}(d\psi-\cos\theta\,d\phi)^2\\ &+\frac{2u}{p} \bigl(d\alpha+f(d\psi-\cos\theta\,d\phi)\bigr)^2. \end{split} $$
Each row of e_i_j below is the coordinate tangent vector $e_i$
expressed in this orthonormal frame. As in the round-sphere notebooks,
Egison constructs every metric component from tangent-vector inner
products, $g_{ij}=e_i\mathbin{\cdot}e_j$. The coupled $(1,3,5)$ block is
produced by the two appearances of $d\psi-\cos\theta\,d\phi$.
def f : MathValue := q / (6 * u)
def e_i_j : Matrix MathValue :=
[| [| sqrt (p / 6) * sin θ
, 0
, -(sqrt (v / u) / 3) * cos θ
, 0
, -sqrt (2 * u / p) * f * cos θ |]
, [| 0, sqrt (p / 6), 0, 0, 0 |]
, [| 0, 0, sqrt (v / u) / 3, 0, sqrt (2 * u / p) * f |]
, [| 0, 0, 0, sqrt (p / (2 * v)), 0 |]
, [| 0, 0, 0, 0, sqrt (2 * u / p) |]
|]_i_j
def g_i_j : Matrix MathValue :=
generateTensor (\[i, j] -> V.* e_i_# e_j_#) [5, 5]
g_2_2
g_4_4
(g_1_3, g_3_1, g_1_5, g_5_1)
Connection and curvature pipeline¶
The first-kind symbols require only derivatives of the displayed metric. Raising an index introduces the inverse of the coupled block. The Riemann and Ricci definitions are nevertheless identical to the lower-dimensional examples:
$$ R^i{}_{jkl}=\partial_k\Gamma^i{}_{jl}-\partial_l\Gamma^i{}_{jk} +\Gamma^m{}_{jl}\Gamma^i{}_{mk}-\Gamma^m{}_{jk}\Gamma^i{}_{ml}, \qquad \operatorname{Ric}_{ij}=R^m{}_{imj}. $$
def g~i~j : Matrix MathValue := M.inverse g_#_#
def Γ_i_j_k : Tensor MathValue :=
(1 / 2) * (∂/∂ g_i_k x~j + ∂/∂ g_i_j x~k - ∂/∂ g_j_k x~i)
def Γ~i_j_k : Tensor MathValue := withSymbols [m]
g~i~m . Γ_m_j_k
def R~i_j_k_l : Tensor MathValue := withSymbols [m]
∂/∂ Γ~i_j_l x~k - ∂/∂ Γ~i_j_k x~l
+ Γ~m_j_l . Γ~i_m_k - Γ~m_j_k . Γ~i_m_l
def Ric_i_j : Matrix MathValue := withSymbols [m]
sum (contract R~m_i_m_j)
def einsteinResidual_i_j : Matrix MathValue := withSymbols [i, j]
expandAll' (Ric_i_j -' 4 *' g_i_j)
Γ_2_2_4
Γ_2_4_2
Interpretation and evaluation strategy¶
The two sampled first-kind symbols come from the simple $g_{22}=p/6$
entry and show the expected symmetry $\Gamma_{ijk}=\Gamma_{ikj}$. The
complete test is einsteinResidual_#_#; mathematically it is the zero
matrix, so the scalar curvature is $5\cdot4=20$.
Expanding all 25 residual entries involves the inverse coupled block and
many rational simplifications. It is intentionally not an automatic
output cell: evaluate individual entries such as
einsteinResidual_2_2 when investigating the identity interactively.