Complete manifold

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

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

Hopf–Rinow theorem

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

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

• The metric space $(M,d_g)$ is complete (every $d_g$-Cauchy sequence converges),
• All closed and bounded subsets of $M$ are compact.

Examples and non-examples

Euclidean space ${ℝ}^n$, the sphere ${𝕊}^n$, and the tori ${𝕋}^n$ (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.

Non-examples

The punctured plane ${ℝ}^2\mathrm{\setminus }\{(0,0)\}$ is not geodesically complete because the maximal geodesic with initial conditions $p=(1,1)$, $v=(1,1)$ does not have domain $ℝ$.

A simple example of a non-complete manifold is given by the punctured plane ${ℝ}^2\setminus \{0\}$ (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.

There exist non-geodesically complete compact pseudo-Riemannian (but not Riemannian) manifolds. An example of this is the Clifton–Pohl torus.

In the theory of general relativity, which describes gravity in terms of a pseudo-Riemannian geometry, many important examples of geodesically incomplete spaces arise, e.g. non-rotating uncharged black-holes or cosmologies with a Big Bang. The fact that such incompleteness is fairly generic in general relativity is shown in the Penrose–Hawking singularity theorems.

Extendibility

If $M$ is geodesically complete, then it is not isometric to an open proper submanifold of any other Riemannian manifold. The converse does not hold.

References

Sources

• do Carmo, Manfredo Perdigão (1992), Riemannian geometry, Mathematics: theory and applications, Boston: Birkhäuser, pp. xvi+300, ISBN 0-8176-3490-8
• Lee, John (2018). Introduction to Riemannian Manifolds. Graduate Texts in Mathematics. Springer International Publishing AG.
• O'Neill, Barrett (1983). Semi-Riemannian Geometry. Academic Press. Chapter 3. ISBN 0-12-526740-1.