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How do we explicitly compute the curvature form $\Omega$ of the Levi-Civita connection $\nabla^{L.C.}$ for the $n$-sphere $S^n$?

Thus, how do we calculate the trace over the matrix logarithm $\log((\sigma_2 \otimes I_{n/2})^T\cdot\Omega)$ for $\sigma_2$ the second Pauli matrix, and $I_{n/2}$ the identity matrix of dimension $n/2$?

Thanks in advance! Any help would be much appreciated.

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