Analysis in Metric Spaces
This article discusses several ideas which are foundational in the study of curves in metric spaces. In short, curves in a metric space can be given well-defined properties such as ‘length’ or ‘speed’, which generalize the classical notions from Euclidean space. These properties then allow one to define minimum-length curves (or geodesics), which then give a natural way to interpolate between points in a metric space. Beyond being useful in their own right, these ideas are widespread in optimal transport.
Curves in Metric Spaces
Let \((X,d)\) be a metric space. A curve in \(X\) is a function \(\gamma: [a,b] \to X\), where \([a,b] \subset \mathbb{R}\). The independent variable \(t \in [a,b]\) is typically considered to represent time. Notice that a curve is therefore time-dependent by definition – that is, it is not merely a set of points in \(X\).
A curve \(\gamma\) is said to be continuous is it satisfies the usual definition: for all \(\epsilon > 0\) and for all \(t,\tau \in [a,b]\), there exists \(\delta = \delta(t,\epsilon) > 0\) such that \(|t - \tau| < \delta\) implies \(d(\gamma(t),\gamma(\tau)) < \epsilon\). Similarly, \(\gamma\) is said to be uniformly continuous if \(\delta\) can be chosen independently of \(t\).
However, we will mainly be interested in a stronger notion of continuity in this article: absolute continuity. In the case \(X = \mathbb{R}\), a function \(f: [a,b] \to \mathbb{R}\) is said to be absolutely continuous if there exists \(g \in L^1([a,b])\) such that \[ f(s) - f(r) ~=~ \int_r^s g(t) \, dt \] for all \(r,s \in [a,b]\). In other words, \(f\) has a derivative almost everywhere (i.e. \(g\)) and is equal to the integral of such. This is the class of curves for which notions such as ‘length’ or ‘speed’ are well-behaved.
In a general metric space (\(X \neq \mathbb{R}\)), there is no way to define integrals or derivatives of curves, and so we must relax the definition above. In this setting, a curve \(\gamma: [a,b] \to X\) is said to be absolutely continuous if there exists \(g \in L^1([a,b])\) such that \[ d(\gamma(r),\gamma(s)) ~\leq~ \int_r^s g(t) \, dt \] for all \(a \leq r \leq s \leq b\). We emphasize again that this is a stronger notion than continuity or uniform continuity, and remark that the original definition can indeed be recovered when \(X = \mathbb{R}\). The space of absolutely continuous curves from \([a,b]\) to \(X\) will be denoted \(\text{AC}([a,b],X)\).
Length and Speed
While there is no way to define the derivative of a curve \(\gamma: [a,b] \to X\), there is a well-defined notion of speed: the metric derivative. For \(\gamma \in \text{AC}([a,b],X)\), its metric derivative is defined as \[ |\gamma'|(t) ~:=~ \lim_{h \to 0} \frac{d(\gamma(t),\gamma(t+h))}{|h|} . \] The metric derivative exists for a.e. \(t \in (a,b)\), and is the minimal \(g\) that can be chosen in the definition of absolute continuity above [Theorem 9.2, Ambrosio et al. (2021)].
The length of \(\gamma\) is then defined as \[ \ell(\gamma) ~:=~ \int_a^b |\gamma'|(t) \, dt . \] Observe that the above definitions imply that for any curve \(\gamma \in \text{AC}([a,b],X)\), it holds that \(\ell(\gamma) \geq d(\gamma(a),\gamma(b))\), i.e., that the length of \(\gamma\) is greater than the distance between its endpoints. This should be intuitively obvious.
Reparameterizations
As remarked before, curves in a metric space are time-dependent by definition. However, it is often desirable to standardize this time-dependence as much as possible. For example, by reparameterizing the time variable, any curve \(\gamma \in \text{AC}([a,b],X)\) can be transformed into a curve \(\tilde{\gamma} \in \text{AC}([0,1],X)\) with constant speed (which is then equal to its length).
This is done as follows. Let \(\gamma \in \text{AC}([a,b],X)\) be a curve with time variable \(t\), metric derivative \(|\gamma'|(t)\), and length \(\ell(\gamma)\). Define the function \[ \sigma(t) ~:=~ \frac{1}{\ell(\gamma)} \int_a^t |\gamma'|(s) \, ds . \] The function \(\sigma(t)\) represents the fraction of \(\gamma\) which has been traversed by time \(t\). Since \(|\gamma'|\) is integrable and nonnegative, \(\sigma\) is an absolutely continuous and monotone nondecreasing function, and thus admits a right inverse \(\sigma^{-R}\). We then define the new time variable \(\tau := \sigma(t)\) and the reparameterized curve \(\tilde{\gamma}\) by \[ \tilde{\gamma}(\tau) ~:=~ \gamma(\sigma^{-R}(\tau)) ~=~ (\gamma \circ \sigma^{-R})(\tau) . \] It can then be verified that \(\tilde{\gamma}\) is an absolutely continuous curve on the interval \([0,1]\) with \(\tilde{\gamma}(0) = \gamma(a)\), \(\tilde{\gamma}(1) = \gamma(b)\), and \(|\tilde{\gamma}'| \equiv \ell(\tilde{\gamma}) = \ell(\gamma) = \text{constant}\) [Proposition 9.6, Ambrosio et al. (2021)].
It is then often conventient to work in the space of constant-speed absolutely continuous curves on \([0,1]\) rather than in the spaces \(\text{AC}([a,b],X)\).
Geodesics
As remarked before, for any curve \(\gamma \in \text{AC}([a,b],X)\), it holds that \(\ell(\gamma) \geq d(\gamma(a),\gamma(b))\). Any curve which achieves equality here (i.e. for which \(\ell(\gamma) = d(\gamma(a),\gamma(b))\)) is termed a geodesic. Geodesics are thus curves of minimal length between their endpoints, and in this way, generalize the notion of ‘straight line’ from Euclidean space to a general metric space. (In Euclidean space, ‘straight’ typically evokes a notion of angle, however, there is no notion of angle in a general metric space. Therefore, this generealization relies on the variational characterization of straight lines as the shortest paths between their endpoints.)
Just as straight lines are a natural way to interpolate between points in a Euclidean space, geodesics are a natural way to interpolate between points in a metric space. We are thus led to the following question: Given two points \(a,b\) in a metric space \((X,d)\), does a geodesic between them always exist, and if so, is it unique? The answer is generically ‘no’, and this can fail in many ways:
If the space is not path-connected, then curves connecting \(a\) and \(b\) may not exist at all.
Curves connecting \(a\) and \(b\) may exist, but there may not be one of minimum length (e.g. \(\mathbb{R}^2 \backslash \{ 0 \}\) with \(a = -b\)).
A minimal curve connecting \(a\) and \(b\) may exist, but may have length strictly larger than \(d(a,b)\) (e.g. the unit circle \(\mathbb{S}^1\) embedded in \(\mathbb{R}^2\) with the inherited metric).
Even when geodesics do exist, they are not always unique (e.g. the unit sphere \(\mathbb{R}^2\) with \(a,b\) antipodal points).
We are thus led to make the following definitions. A metric space \((X,d)\) is said to be a length space if for all \(x,y \in X\), \[ d(x,y) ~=~ \inf \{ \ell(\gamma) ~:~ \gamma \in \text{AC}([a,b],X) ,~ \gamma(a) = x ,~ \gamma(b)=y \}. \] It is said to be a geodesic space if there exists such a \(\gamma\) achieving the infimum (i.e. a geodesic). It is said to be a uniquely geodesic space if such a \(\gamma\) is unique.
There are a large number of results in differential and metric geometry establishing necessary and sufficient conditions for a space to have these properties.