In geodesic polar coordinates $(r, \theta_1, \dots, \theta_{n-1})$ on a Riemannian manifold $(M, g)$, the metric takes the form \begin{align*} g = dr \otimes dr + \sum_{\alpha, \beta} g_{\alpha\beta}(r, \theta)\, d\theta_\alpha \otimes d\theta_\beta. \end{align*} More explicitly: \begin{align*} g\!\left(\frac{\partial}{\partial r}, \frac{\partial}{\partial r}\right) = 1, \qquad g\!\left(\frac{\partial}{\partial r}, \frac{\partial}{\partial \theta_\alpha}\right) = 0 \quad \text{for all } \alpha. \end{align*}