Denote $$ U^2= \{ (x,y,z)\in\mathbb R^2: x^2+y^2+z^2=1 \} $$ $g$ is the induced metric. Under the polar coordinates, $\theta$ is polar angle, $\varphi$ is azimuthal angle, the metric is $$ g=\begin{pmatrix} 1 &0 \\ 0 &\sin^2\theta \end{pmatrix} $$ As the 3.4 section of Petersen's Riemannian Gemetry (the picture below), since $$ (\sin^2\theta)'|_{\theta=0}=0 ~~~~~~ (\sin^2\theta)'|_{\theta=\pi}=0 $$ the $(U^2, g)$ is not smooth. But, obviously, the induced metric of unit sphere is smooth. Where is my mistake ?
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1If I'm reading Petersen's notation correctly, $\sin^{2}\theta$ is $\varphi(t)^{2}$, i.e., $\varphi(t) = \sin t$. :) – Andrew D. Hwang Aug 28 '23 at 14:02
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1@AndrewD.Hwang I think you are right. Thanks. – Enhao Lan Aug 28 '23 at 14:57