Consider a harmonic map $\Phi : \Sigma \to S^n \times \mathbb{R}$, where $\Sigma$ is a surface, and the metric on $S^n \times \mathbb{R}$ is given by the product metric. Choose local spherical (polar) coordinates $\omega_1,..,\omega_n$ on $S^n$ (coming from embedding in $\mathbb{R}^{n + 1}$) in the usual way. I am trying to see what local coordinate equations would be satisfied by the coordinates $r$ and $\omega_i$. Thanks for your help.
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