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For a smooth surface obtained by translating the plane graph $z=F(y)$ along the $x$ direction, the induced metric is $d{s}^2=d{x}^2+[1+F^{\prime }(y{)}^2]d{y}^2$. The transverse arc length coordinate $u(y)={\int }^y\sqrt{1+F^{\prime }(s{)}^2}ds$ makes it $d{s}^2=d{x}^2+d{u}^2$. This is a local isometry to the Euclidean plane, extending the intrinsic flatness of a circular cylinder to arbitrary smooth generating graphs.

Its geodesics therefore lift to straight lines in $(x,u)$ coordinates. Those that are graphs over $x$ satisfy $u=ax+b$, while the additional family $x=\text{constant}$ cannot be written as $y(x)$. The same graph formula follows from the Beltrami identity for the arc-length integrand $\sqrt{1+[1+F^{\prime }(y{)}^2]y^{\prime 2}}$. These straight lines are locally minimizing; on cylinders with a closed generating curve, global identifications can create several different winding paths between two points.

An immersion $F:M\to N$ into a Riemannian manifold $(N,h)$ inherits the Riemannian metric $g=F^{\ast }h$: measure two tangent vectors after mapping them into the surrounding tangent space. This is a pullback of a Riemannian metric, withThe 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 $1$. ConsequentlyThe unrolled coordinate $x=R\theta$ gives $d{s}^2=d{x}^2+d{z}^2$. 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. ^[ref].

For local surface coordinates $u^1,u^2$ and metric coefficients $g_{ij}$, affinely parametrized geodesics satisfyThese are the geodesic equation and Christoffel symbols. They follow by applying the Euler-Lagrange equation to the energy $\frac{1}{2}{g}_{ij}{\dot{u}}^i{\dot{u}}^j$. Equivalently, in an embedded surface a constant-speed geodesic has acceleration normal to the surface.

Parameterize a circular cylinder by $(a\cos \theta ,a\sin \theta ,z)$, where $a>0$. Its induced metric is $a^{2}d{\theta }^2+d{z}^2$, with constant coefficients, so all Christoffel symbols vanish in this chart. ThusThese geodesics on a circular cylinder are straight generators when $c=0$, circles around the cylinder when $d=0$, and helices when $cd\ne 0$. Constant curves arise if both vanish. For unit-speed nonconstant curves, $a^2{c}^2+d^2=1$. This description is global after allowing the angle to run through arbitrary multiples of $2\pi$.

The unrolled coordinate $u=a\theta$ exhibits the intrinsic flatness of a circular cylinder: every displayed family is a straight line in $(u,z)$. A circle is therefore a geodesic despite bending in ambient three-dimensional space; its acceleration is normal to the cylinder.

The two coordinate tangent vectors of the embedding areTheir Euclidean inner products give the pullback of a Riemannian metric, equivalently the induced metric:Locally set $u=\rho \phi$. The line element becomes $d{u}^2+d{\hat{z}}^2$, so 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.

On the lateral surface of an infinite circular cylinder of radius $R$, the induced metric is $R^{2}d{\theta }^2+d{z}^2$. Lifting a curve to the unrolled plane $(x,z)=(R\theta ,z)$ preserves its arc length. If the endpoints differ by angular coordinate $\Delta \theta$ and height $\Delta z$, the possible lifted endpoint displacements are $(R(\Delta \theta +2\pi k),\Delta z)$ for integers $k$. Each straight segment gives a geodesic, and the shortest of their lengths isChoose the representative of $\Delta \theta$ with smallest absolute value modulo $2\pi$; if its absolute value is $\pi$, the two angular directions tie. This minimum is attained because the candidate lengths tend to infinity as $|k|\to \infty$. The formula concerns travel on the lateral surface; it does not allow a shortcut through the cylinder's interior or across end caps.

For $R=1$, take $p=(1,0,0)$ and $q=(0,1,3)$ in surrounding Euclidean space. The angular difference is $\pi /2$ and height difference is $3$. The $k=0$ route has length $\sqrt{9+{\pi }^2/4}\approx 3.386$. The $k=-1$ route goes the other way around and has length $\sqrt{9+9{\pi }^2/4}\approx 5.586$. Both are among the geodesics on a circular cylinder, but only the first is a minimizing geodesic between these endpoints.

This is a global consequence of intrinsic flatness of a circular cylinder: locally everything is Euclidean, yet the periodic angular coordinate leaves several competing routes. A shortest-route problem on the curved surface becomes a family of ordinary planar distance calculations.

Riemannian geometry studies smooth manifolds equipped with a Riemannian metric: an inner product at each tangent space, varying smoothly with position. This local measuring rule determines angles, arc length, Riemannian distance and geodesics. An inhabitant of a surface can discover its geometry by measuring within it, without knowing how it sits in a larger Euclidean space.

A circular cylinder is a particularly useful example. Its surface bends in space, yet unrolling it preserves the induced metric: a small patch has the geometry of a flat sheet. By contrast, a round sphere has positive Gaussian curvature. The distinction between intrinsic flatness of a circular cylinder and extrinsic curvature separates the geometry an inhabitant measures from the way an outside observer sees the surface bend.

Geodesics supply the analogue of straight lines. On a round sphere they follow great circles; on a circular cylinder they become straight lines when lifted to an unrolled sheet. Their arc length lets us compare candidate routes, as in great-circle distance and Riemannian distance on a circular cylinder. The same measuring rule also defines differential operators such as the Laplace-Beltrami operator, connecting geometry with diffusion and spectral geometry.

Bernhard Riemann's 1854 inaugural lecture proposed studying spaces of arbitrary dimension through their infinitesimal measuring rules, including a positive quadratic expression for the squared line element. This moved the subject beyond a particular surrounding space: the measuring rule itself became part of the geometry. ^[ref].