Why is $$e_\mu=\partial_\mu$$always said to be the unit vector ?
Doesn't the size of the vector $\partial_\mu$ kindoff depend on the underlying manifold ?
2026-03-25 18:43:45.1774464225
Differential geometry unit vector
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In a Riemannian manifold with metric $g$ the length-squared of $\partial_{\mu}$ would be $g(\partial_{\mu},\partial_{\mu})$. Why should this be one?
It is possible to choose an orthonormal frame by rescaling a coordinate frame at a point. However, it is generally not possible to orthonormalize a coordinate frame over some open set in the manifold (unless of course, the thing is flat...)