= Induced metric
{title2=$g=F^*h$}
{wiki}
An <immersion> $F:M\to N$ into a <Riemannian manifold> $(N,h)$ inherits the <Riemannian metric> $g=F^*h$: measure two <tangent vectors> after mapping them into the surrounding <tangent space>. This is a <pullback of a Riemannian metric>, with
$$
g_p(u,v)=h_{F(p)}(dF_pu,dF_pv).
$$
The injective derivative of an <immersion> makes this <inner product> <positive-definite>. For a parametrized <surface> in <Euclidean space>, its coefficients are simply the <scalar products> of the coordinate tangent vectors: the <first fundamental form>.
For the <circular cylinder> $F(\theta,z)=(R\cos\theta,R\sin\theta,z)$, the tangent vectors $F_\theta$ and $F_z$ are <orthogonal>, with squared <norms> $R^2$ and $1$. Consequently
$$
ds^2=R^2\,d\theta^2+dz^2.
$$
The unrolled coordinate $x=R\theta$ gives $ds^2=dx^2+dz^2$. This calculation explains <intrinsic flatness of a circular cylinder> and makes <Riemannian distance on a circular cylinder> a planar problem with repeated endpoint lifts.
Preserving a <Riemannian metric> locally preserves the <arc length> of every mapped <curve>, but it need not preserve distances measured by unrestricted paths in the surrounding space. The <great-circle distance> between two sphere points, for example, exceeds their straight chord <length> unless they coincide. https://www.math.ucla.edu/~petersen/RG.pdf[Peter Petersen, Riemannian Geometry, §1.1: induced metrics and Riemannian immersions]{ref}.
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