FLRW metric with spherical coordinates - General Relativity, Astroparticles and Cosmology

The goal is to compute

\mathrm{d} s^2=-c^2\mathrm{d} t^2 + a^2(t) \left(\mathrm{d} \vec \sigma^2 + k\frac{(\vec \sigma \cdot \mathrm{d} \vec \sigma)^2}{1 - k \sigma^2} \right)

(1)

with

\begin{align} \sigma_1 & = \sigma \sin \theta \cos \phi \\ \sigma_2 & = \sigma \sin \theta \sin \phi \\ \sigma_3 & = \sigma \cos \theta \end{align}

(2)

from sympy import *

sigma, theta, phi, k = symbols("sigma theta phi k")
dsigma, dtheta, dphi = symbols(r"d\sigma d\theta d\phi")

sigma1 = sigma * sin(theta) * cos(phi)
sigma2 = sigma * sin(theta) * sin(phi)
sigma3 = sigma * cos(theta)
Sigma = Matrix([sigma1, sigma2, sigma3])
Sigma

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Sigma2 = simplify(Sigma.T * Sigma)[0]
Sigma2

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dsigma1 = diff(sigma1, sigma)*dsigma + diff(sigma1, theta)*dtheta + diff(sigma1, phi)*dphi
dsigma2 = diff(sigma2, sigma)*dsigma + diff(sigma2, theta)*dtheta + diff(sigma2, phi)*dphi
dsigma3 = diff(sigma3, sigma)*dsigma + diff(sigma3, theta)*dtheta + diff(sigma3, phi)*dphi
dSigma = Matrix([dsigma1, dsigma2, dsigma3])
dSigma

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dSigma2 = simplify(dSigma.T * dSigma)[0]
dSigma2

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SigmadSigma2 = simplify((Sigma.T * dSigma)**2)[0]
SigmadSigma2

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dSigma2 + k * SigmadSigma2 / (1 - k * Sigma2)

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This is the expected result.