In mathematics, a complete manifold (or geodesically complete manifold) M is a (pseudo-) Riemannian manifold for which, starting at any point p of M, there are straight paths extending infinitely in all directions.

Formally, a manifold ${\displaystyle M}$ is (geodesically) complete if for any maximal geodesic ${\displaystyle \ell :I\to M}$, it holds that ${\displaystyle I=(-\infty ,\infty )}$. A geodesic is maximal if its domain cannot be extended.

Equivalently, ${\displaystyle M}$ is (geodesically) complete if for all points ${\displaystyle p\in M}$, the exponential map at ${\displaystyle p}$ is defined on ${\displaystyle T_{p}M}$, the entire tangent space at ${\displaystyle p}$.

The Hopf–Rinow theorem gives alternative characterizations of completeness. Let ${\displaystyle (M,g)}$ be a connected Riemannian manifold and let ${\displaystyle d_{g}:M\times M\to [0,\infty )}$ be its Riemannian distance function.

The Hopf–Rinow theorem states that ${\displaystyle (M,g)}$ is (geodesically) complete if and only if it satisfies one of the following equivalent conditions:

Euclidean space ${\displaystyle \mathbb {R} ^{n}}$, the sphere ${\displaystyle \mathbb {S} ^{n}}$, and the tori ${\displaystyle \mathbb {T} ^{n}}$ (with their natural Riemannian metrics) are all complete manifolds.

All compact Riemannian manifolds and all homogeneous manifolds are geodesically complete. All symmetric spaces are geodesically complete.

A simple example of a non-complete manifold is given by the punctured plane ${\displaystyle \mathbb {R} ^{2}\smallsetminus \lbrace 0\rbrace }$ (with its induced metric). Geodesics going to the origin cannot be defined on the entire real line. By the Hopf–Rinow theorem, we can alternatively observe that it is not a complete metric space: any sequence in the plane converging to the origin is a non-converging Cauchy sequence in the punctured plane.