= Solution
The two coordinate tangent vectors of the embedding are
$$
e_\varphi=(-\rho\sin\varphi,\rho\cos\varphi,0),\qquad e_{\widehat z}=(0,0,1).
$$
Their Euclidean inner products give the <pullback of a Riemannian metric>, equivalently the <induced metric>:
$$
\boxed{\phi^*g=\rho^2d\varphi^2+d\widehat z^2,\qquad
\gamma_{ij}=\begin{pmatrix}\rho^2&0\\0&1\end{pmatrix}.}
$$
Locally set $u=\rho\varphi$. The line element becomes $du^2+d\widehat z^2$, so \b[the induced <Riemann curvature tensor> vanishes identically]. This is the <intrinsic flatness of a circular cylinder>. Its bending in the ambient space is <extrinsic curvature>, not intrinsic curvature; periodicity of the angular coordinate does not change the local flatness.
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