= Intrinsic flatness of a circular cylinder
{title2=$ds^2=\rho^2d\varphi^2+dz^2$}
A <circular cylinder> of radius $\rho$ has <induced metric> locally equal to $du^2+dz^2$ with $u=\rho\varphi$. Its <Riemann curvature tensor> therefore vanishes, despite its nonzero <extrinsic curvature>. With outward unit normal and convention $K_{ij}=e_i^ae_j^b\nabla_an_b$, its trace is $1/\rho$ and its <principal curvatures> are $1/\rho$ and zero. Its averaged <mean curvature> is $1/(2\rho)$. The angular identification changes global topology, not local flatness.
Back to article page
