Hopf–Rinow theorem
The Hopf–Rinow theorem is a cornerstone result in Riemannian geometry that characterizes the completeness of connected Riemannian manifolds through several equivalent conditions. It asserts that for a connected Riemannian manifold (M, g), the following properties are equivalent: (M, d) is a complete metric space, where d is the distance function induced by the Riemannian metric g; the manifold is geodesically complete, meaning every geodesic can be extended to all real time parameters; the exponential map at some point p \in M is defined on the entire tangent space T_p M; and every closed and bounded subset of M is compact, embodying the Heine-Borel property.[1] Additionally, under any of these conditions, any two points in M can be joined by a geodesic of length exactly equal to their distance d(p, q).[2] Originally established by Heinz Hopf and his student Willi Rinow in 1931 for surfaces, the theorem extends naturally to higher-dimensional manifolds with proofs that follow similar lines.[3] The result bridges metric, geodesic, and topological aspects of Riemannian manifolds, highlighting how completeness ensures the existence of minimizing geodesics between points, analogous to straight lines in Euclidean space.[1] This equivalence is pivotal for understanding global properties of manifolds, such as compactness, and has implications in areas like general relativity and optimization on curved spaces.[2] Key corollaries include the fact that complete Riemannian manifolds behave like Euclidean spaces in terms of bounded sets being compact, and that the injectivity radius at points can be analyzed via the exponential map.[1] Examples illustrate the theorem's sharpness: the open unit ball in \mathbb{R}^n with the Euclidean metric is geodesically incomplete despite being connected, as Cauchy sequences may converge outside the ball, violating metric completeness.[2] In contrast, compact manifolds without boundary are always complete. The theorem's proof typically involves showing that geodesic incompleteness implies a missing endpoint in the metric space, using compactness arguments on minimizing sequences of curves.[1]Prerequisites
Riemannian geometry basics
A Riemannian manifold is a smooth manifold M equipped with a Riemannian metric, which is a smooth assignment of a positive-definite inner product to each tangent space T_p M at every point p \in M. This metric is typically denoted by a tensor field g, where g_p: T_p M \times T_p M \to \mathbb{R} satisfies g_p(v, v) > 0 for all nonzero v \in T_p M, and varies smoothly across the manifold.[4][5] The inner product induced by g on each tangent space enables the measurement of lengths and angles intrinsically on the manifold. Specifically, the length of a tangent vector v \in T_p M is given by \|v\|_p = \sqrt{g_p(v, v)}, and the angle between two vectors v, w \in T_p M is \theta = \cos^{-1} \left( \frac{g_p(v, w)}{\|v\|_p \|w\|_p} \right). This structure allows for local geometric computations analogous to those in Euclidean space, but adapted to the curved nature of the manifold.[4] From the Riemannian metric, one defines a distance function d: M \times M \to [0, \infty) between points p, q \in M as the infimum of the lengths of all smooth curves \gamma: [a, b] \to M connecting them, where the length L(\gamma) = \int_a^b \sqrt{g_{\gamma(t)}(\dot{\gamma}(t), \dot{\gamma}(t))} \, dt. Thus, d(p, q) = \inf \{ L(\gamma) \mid \gamma(a) = p, \gamma(b) = q \}, which induces a metric topology on M compatible with its smooth structure.[6] Classic examples of Riemannian manifolds include Euclidean space \mathbb{R}^n with the standard flat metric g = \delta_{ij}, the sphere S^n endowed with the round metric of constant positive curvature, and hyperbolic space \mathbb{H}^n with its metric of constant negative curvature, each illustrating how the metric tensor shapes local and global geometry.[7]Geodesics and completeness
In a Riemannian manifold (M, g), the Riemannian metric g induces a notion of length for smooth curves, allowing the definition of geodesics as curves that locally minimize this length. A smooth curve \gamma: I \to M, where I \subseteq \mathbb{R} is an interval, is a geodesic if it satisfies the geodesic equation \nabla_{\gamma'(t)} \gamma'(t) = 0 for all t \in I, meaning the covariant derivative of its velocity field along itself vanishes.[8] This condition ensures that the acceleration of the curve is zero with respect to the Levi-Civita connection, analogous to straight lines in Euclidean space. The exponential map at a point p \in M, denoted \exp_p: T_p M \to M, provides a way to parameterize geodesics emanating from p. For a tangent vector v \in T_p M, \exp_p(v) is defined as the endpoint \gamma(1), where \gamma: [0,1] \to M is the unique maximal geodesic segment satisfying \gamma(0) = p and \gamma'(0) = v, provided such a segment exists within its domain of definition.[8] The image of \exp_p traces points reachable by geodesics of unit speed from p. The metric g further induces a distance function d: M \times M \to [0, \infty) on the manifold, defined as the infimum of lengths of smooth curves connecting points. A Riemannian manifold (M, g) is metrically complete if the metric space (M, d) is complete, meaning every Cauchy sequence in d converges to a point in M.[8] Geodesic completeness concerns the extendability of geodesics themselves. A Riemannian manifold is geodesically complete if every geodesic \gamma: I \to M admits a reparameterization to a geodesic defined on the entire real line \mathbb{R}, or equivalently, if the domain of \exp_p is the entire tangent space T_p M for every p \in M.[8] In the context of Riemannian manifolds, properness relates to topological compactness properties induced by the metric. A Riemannian manifold M is proper if it is a proper metric space, meaning every closed and bounded subset of (M, d) is compact.[9]The Theorem
Formal statement
The Hopf–Rinow theorem provides a characterization of completeness for connected Riemannian manifolds. Let (M, g) be a connected smooth Riemannian manifold equipped with the induced distance function d: M \times M \to [0, \infty), where d(p, q) denotes the infimum of lengths of piecewise smooth curves connecting p and q. The exponential map \exp_p: T_p M \to M at a point p \in M$ is defined on a domain D_p \subseteq T_p M$ consisting of all initial velocities for which the corresponding geodesic is defined on some maximal interval.[3][1] The following conditions are equivalent:(1) (M, d) is a complete metric space, meaning every Cauchy sequence in M converges to a point in M;
(2) M is geodesically complete, meaning that for every p \in M and v \in T_p M, the geodesic \gamma_v(t) with \gamma_v(0) = p and \dot{\gamma}_v(0) = v is defined for all t \in \mathbb{R} (equivalently, D_p = T_p M for every p);
(3) every closed and bounded subset of M is compact (the Heine–Borel property holds for the metric d).[3][1] This equivalence holds for finite-dimensional smooth connected Riemannian manifolds.[3][1]