= Complete positively curved paraboloid
{title2=$g=(1+4r^2)dr^2+r^2d\theta^2,\quad K=4/(1+4r^2)^2$}
The graph $z=x^2+y^2$ in Euclidean three-space has the displayed induced <Riemannian metric> and positive <Gaussian curvature>. It is smooth at its vertex; the polar-coordinate singularity is only a coordinate artifact. The metric dominates the Euclidean plane metric, so <upward stability of Riemannian completeness> proves completeness. It has infinite diameter. In dimension two <Ricci curvature> equals Gaussian curvature times the metric, so this is a counterexample to compactness under merely pointwise positive Ricci curvature. Its curvature tends to zero at infinity.
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