Boundaryless domain for geometric completeness

Let $M$ be a smooth Hausdorff second-countable manifold without boundary, in the sense of Smooth manifolds and their smooth charts. Let $g$ be a positive definite smooth Riemannian metric as in Riemannian metrics, symmetric cotangent-bundle connections, and covariant Hessians. Use the Levi-Civita connection of Levi civita connection, whose existence and uniqueness for the supplied smooth Riemannian metric are proved by Fundamental theorem of riemannian geometry. The coordinate formula is supplied by Christoffel formula for the levi civita connection: its coefficients are ${\mathrm{\Gamma }}_{ij}^k=\frac{1}{2}{g}^{kl}({\mathrm{\partial }}_i{g}_{jl}+{\mathrm{\partial }}_j{g}_{il}-{\mathrm{\partial }}_l{g}_{ij})$. A parametrized affine geodesic is a smooth curve $\gamma :J\to M$ on an interval with nonempty interior satisfying ${\mathrm{\nabla }}_{\dot{\gamma }}\dot{\gamma }=0$, equivalently ${\ddot{\gamma }}^k+{\mathrm{\Gamma }}_{ij}^k(\gamma ){\dot{\gamma }}^i{\dot{\gamma }}^j=0$, where covariant differentiation along the curve is defined by Covariant derivative along a curve; the coordinate equation is the expansion of that derivative in a coordinate basis. Its maximal domain is an interval on which it cannot be extended with the same affine parameter as a solution. Two-sided geodesic completeness means that every maximal affine geodesic is defined on all of $\mathbb{R}$.

Independently, for a real metric $d$ in Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric, a sequence $(x_n)$ is Cauchy when for every $\epsilon >0$ there is $N$ with $d(x_m,x_n)<\epsilon$ whenever $m,n\ge N$. It converges to $x$ when for every $\epsilon >0$ eventually $d(x_n,x)<\epsilon$. Metric completeness means that every Cauchy sequence converges in the given metric space. In a Riemannian application the metric must be the distance induced by $g$ on a connected component: the infimum of lengths $\int \sqrt{g(\dot{\alpha },\dot{\alpha })}dt$ over piecewise-$C^1$ connecting curves. Its definition and finite metric property are supplied by Riemannian distance on a connected manifold and Riemannian distance is a metric. Specify that induced metric, rather than an arbitrary compatible metric. Distinct components are treated separately; no infinite distance is inserted into a real metric.

These are different completeness conditions. Their equivalence is a Hopf–Rinow theorem requiring its own hypotheses and proof; it is not part of this definition. In particular, a Lorentzian metric is not a positive definite distance, and this convention supplies no Lorentzian Hopf–Rinow implication.

The boundaryless restriction matters: the Euclidean interval $[0,1]$ is metrically complete, while the curve $\gamma (t)=1\mathrm{/}2+t$ stays inside it only for $-1\mathrm{/}2\le t\le 1\mathrm{/}2$ and cannot be continued there as a two-sided affine geodesic with velocity $1$. Completeness of this interval follows because a real Cauchy sequence converges in $\mathbb{R}$ by The reals are complete and its limit remains between $0$ and $1$. This boundary example explains the domain restriction; it is not an instance of the boundaryless equivalence.