The Length of a Curve
A Riemannian metric assigns a length to every tangent vector, and a curve is traced out by its
velocity vectors. Integrating the lengths of those velocity vectors along a curve measures how far
the curve travels, giving the notion of arc length on a manifold. This is the first measurement
that reaches beyond a single tangent space to the manifold as a whole, and from it the distance
between points will be built.
Throughout this discussion \((M, g)\) is a Riemannian manifold, taken without boundary, since the
theory of distance is most straightforward in that case. A curve here means a piecewise smooth
curve segment: a continuous map \(\gamma : [a, b] \to M\) for which there is a partition \(a = a_0 <
a_1 < \cdots < a_k = b\) such that \(\gamma\) restricted to each \([a_{i-1}, a_i]\) is smooth. Its
velocity
\(\gamma'(t)\) is a tangent vector at \(\gamma(t)\), defined at every \(t\) except possibly the
finitely many partition points, where left- and right-hand velocities may differ.
Definition: Length of a Curve
Let \((M, g)\) be a Riemannian manifold and let \(\gamma : [a, b] \to M\) be a piecewise smooth
curve segment. The length of \(\gamma\) is
\[
L_g(\gamma) = \int_a^b \big| \gamma'(t) \big|_g\, dt ,
\]
where \(\big| \gamma'(t) \big|_g = g_{\gamma(t)}\big(\gamma'(t), \gamma'(t)\big)^{1/2}\) is the
norm
of the velocity vector with respect to the metric.
The integrand \(\big| \gamma'(t) \big|_g\) is continuous wherever \(\gamma\) is smooth, so it is
continuous at all but the finitely many partition points, and at each of those it has well-defined
limits from the left and from the right. A bounded function continuous except at finitely many
points is
Riemann integrable,
so the integral is well defined; equivalently, the length is the sum of the lengths of the smooth
pieces, each an ordinary integral of a continuous function.
Length does not depend on how a curve is parametrized, only on its image traversed in a given
direction. This is what makes length a geometric quantity rather than an artifact of the
description, and it is the property that allows length to be compared and minimized across all
parametrizations.
Theorem: Parameter Independence of Length
Let \((M, g)\) be a Riemannian manifold and let \(\gamma : [a, b] \to M\) be a piecewise smooth
curve segment. If \(\tilde{\gamma} = \gamma \circ \varphi\) is a reparametrization of \(\gamma\)
by a diffeomorphism \(\varphi : [c, d] \to [a, b]\), then
\[
L_g(\tilde{\gamma}) = L_g(\gamma) .
\]
Proof:
Suppose first that \(\gamma\) and \(\varphi\) are smooth; the piecewise case follows by applying
the argument on each subinterval where \(\gamma\) is smooth. A diffeomorphism of intervals has
derivative of constant sign, so either \(\varphi' > 0\) everywhere or \(\varphi' < 0\)
everywhere. By the chain rule the velocity of the reparametrized curve is \(\tilde{\gamma}'(t) =
\varphi'(t)\, \gamma'(\varphi(t))\), so taking norms,
\[
\big| \tilde{\gamma}'(t) \big|_g = \big| \varphi'(t) \big|\, \big| \gamma'(\varphi(t)) \big|_g .
\]
Suppose \(\varphi' > 0\), so \(\varphi(c) = a\) and \(\varphi(d) = b\) and \(\big|\varphi'(t)\big|
= \varphi'(t)\). Then
\[
\begin{align*}
L_g(\tilde{\gamma})
&= \int_c^d \big| \tilde{\gamma}'(t) \big|_g\, dt
= \int_c^d \big| \gamma'(\varphi(t)) \big|_g\, \varphi'(t)\, dt \\\\
&= \int_a^b \big| \gamma'(s) \big|_g\, ds = L_g(\gamma) ,
\end{align*}
\]
where the third equality is the substitution \(s = \varphi(t)\). If instead \(\varphi' < 0\),
then \(\varphi(c) = b\) and \(\varphi(d) = a\), and two sign changes occur: one because
\(\big|\varphi'(t)\big| = -\varphi'(t)\), and one because the substitution reverses the limits of
integration. The two cancel, and the value of the integral is again \(L_g(\gamma)\).
The Riemannian Distance Function
With lengths of curves in hand, the distance between two points is defined as the length of the
shortest route between them. Since there is in general no single shortest curve, and since some
routes may be arbitrarily long, the right notion is the greatest lower bound of all the lengths:
the distance is what no path can undercut. Using curve segments as measuring tapes in this way turns
a manifold into a setting where points have numerical separations.
Definition: Riemannian Distance
Let \((M, g)\) be a connected Riemannian manifold. For points \(p, q \in M\), the
Riemannian distance from \(p\) to \(q\) is the
infimum
\[
d_g(p, q) = \inf\, \big\{ L_g(\gamma) : \gamma \text{ is a piecewise smooth curve segment from } p \text{ to } q \big\} ,
\]
the greatest lower bound of the lengths of all piecewise smooth curve segments joining \(p\) to
\(q\).
For this to be meaningful, the set whose infimum is taken must be nonempty: there must exist at
least one piecewise smooth curve segment from \(p\) to \(q\). A connected manifold is
path-connected,
so any two of its points are joined by a continuous path; on a smooth manifold such a path can be
replaced by a piecewise smooth curve segment with the same endpoints, by smoothing it within
coordinate charts. The set of competing lengths is therefore nonempty, and being a set of
nonnegative real numbers it has a well-defined infimum. The distance \(d_g(p, q)\) is thus defined
for every pair of points in a connected Riemannian manifold.
On Euclidean space the Riemannian distance recovers the elementary notion of distance, confirming
that the construction generalizes rather than replaces it.
Example (Euclidean Distance):
Consider \(\mathbb{R}^n\) with the Euclidean metric \(\bar{g}\). A straight line segment from
\(x\) to \(y\) has length \(|x - y|\), so \(d_{\bar{g}}(x, y) \le |x - y|\). In the other
direction, any piecewise smooth curve segment \(\gamma\) from \(x\) to \(y\) satisfies
\[
L_{\bar{g}}(\gamma) = \int_a^b \big| \gamma'(t) \big|\, dt \ge \left| \int_a^b \gamma'(t)\, dt \right| = \big| \gamma(b) - \gamma(a) \big| = |x - y| ,
\]
the inequality being the triangle inequality for integrals. Taking the infimum over all such
\(\gamma\) gives \(d_{\bar{g}}(x, y) \ge |x - y|\). The two bounds combine to
\[
d_{\bar{g}}(x, y) = |x - y| ,
\]
so the Riemannian distance on \(\mathbb{R}^n\) is exactly the Euclidean distance, with straight
line segments achieving the infimum.
The example also shows that on \(\mathbb{R}^n\) the infimum is attained, by the straight segment.
On a general Riemannian manifold a length-minimizing curve need not exist between two given points,
which is why the distance is defined as an infimum rather than a minimum. Whether the infimum is
realized, and by what curves, is the question that leads to the theory of geodesics; the distance
function defined here is what those curves are required to minimize.
Riemannian Manifolds as Metric Spaces
The distance function ought to make a manifold into a metric space, with all the structure that
entails, and its metric topology ought to agree with the topology the manifold already carries. Both
statements are true, and the bridge between them is a comparison: in any single coordinate chart, the
Riemannian metric is squeezed between constant multiples of the Euclidean metric over any compact
region. That local comparison is what transfers Euclidean facts about distance to the manifold.
Lemma: Local Comparison with the Euclidean Metric
Let \(g\) be a Riemannian metric on an open subset \(U \subseteq \mathbb{R}^n\), and let
\(\bar{g}\) denote the Euclidean metric. For every compact subset \(K \subseteq U\) there exist
positive constants \(c, C\) such that for all \(x \in K\) and all \(v \in T_x\mathbb{R}^n\),
\[
c\, |v|_{\bar{g}} \le |v|_g \le C\, |v|_{\bar{g}} .
\]
Proof:
Identifying \(T\mathbb{R}^n\) with \(\mathbb{R}^n \times \mathbb{R}^n\), consider the set
\[
L = \big\{ (x, v) : x \in K,\ |v|_{\bar{g}} = 1 \big\} = K \times \mathbb{S}^{n-1} ,
\]
a product of a compact set with the unit sphere, hence compact. The function \((x, v) \mapsto
|v|_g\) is continuous and strictly positive on \(L\), since \(|v|_g = 0\) would force \(v = 0\),
contradicting \(|v|_{\bar{g}} = 1\). By the
extreme value theorem
it attains a positive minimum \(c\) and a maximum \(C\) on \(L\), so \(c \le |v|_g \le C\)
whenever \(x \in K\) and \(|v|_{\bar{g}} = 1\).
For a nonzero \(v \in T_x\mathbb{R}^n\) with \(x \in K\), set \(\lambda = |v|_{\bar{g}}\). Then
\(\lambda^{-1} v\) is a unit vector, so \(c \le |\lambda^{-1} v|_g \le C\). Both norms are
homogeneous of degree one, so multiplying through by \(\lambda\) gives
\[
c\, |v|_{\bar{g}} = c\lambda \le |v|_g = \lambda\, \big| \lambda^{-1} v \big|_g \le C\lambda = C\, |v|_{\bar{g}} .
\]
The inequalities hold trivially when \(v = 0\), so they hold for all \(v\).
The comparison lemma is the analytic core of the next theorem, which collects everything the
distance function provides. It is the point at which the manifold, first met as a topological space
patched from coordinate charts, acquires a notion of distance compatible with that original
topology.
Theorem: Riemannian Manifolds as Metric Spaces
Let \((M, g)\) be a connected Riemannian manifold. With the Riemannian distance function
\(d_g\), the set \(M\) is a
metric space,
and the
metric topology
it determines coincides with the original manifold topology.
Proof:
We first verify the metric space axioms. Lengths are nonnegative, so \(d_g(p, q) \ge 0\). A
constant curve has velocity zero and hence length zero, giving \(d_g(p, p) = 0\). Any curve
segment from \(p\) to \(q\) can be
reparametrized
to run from \(q\) to \(p\) with the same length, so \(d_g(p, q) = d_g(q, p)\). For the triangle
inequality, let \(\gamma_1\) run from \(p\) to \(q\) and \(\gamma_2\) from \(q\) to \(r\); the
concatenation that follows \(\gamma_1\) and then \(\gamma_2\), reparametrized to a single
domain, is a piecewise smooth curve segment from \(p\) to \(r\) whose length is \(L_g(\gamma_1) +
L_g(\gamma_2)\). Hence \(d_g(p, r) \le L_g(\gamma_1) + L_g(\gamma_2)\); taking the infimum over
\(\gamma_1\) and \(\gamma_2\) separately yields \(d_g(p, r) \le d_g(p, q) + d_g(q, r)\). It is
precisely the use of piecewise smooth curves that makes this concatenation admissible.
It remains to show that \(d_g(p, q) > 0\) when \(p \ne q\). Let \(U\) be a smooth coordinate
domain containing \(p\) but not \(q\), identified with an open subset of \(\mathbb{R}^n\), and
let \(\bar{g}\) be the Euclidean metric in these coordinates. Choose a
regular coordinate ball
\(V\) of radius \(\varepsilon\) centered at \(p\) with \(\overline{V} \subseteq U\); the comparison
lemma applied to the compact set \(\overline{V}\) supplies constants \(c, C\) with \(c\,
|v|_{\bar{g}} \le |v|_g \le C\, |v|_{\bar{g}}\) there. For any piecewise smooth curve segment
\(\gamma\) lying entirely in \(V\), integrating the left inequality gives \(L_g(\gamma) \ge
c\, L_{\bar{g}}(\gamma)\).
Now let \(\gamma : [a, b] \to M\) be any piecewise smooth curve segment from \(p\) to \(q\).
Since \(q \notin \overline{V}\), the curve must leave \(V\); let \(t_0\) be the infimum of times
\(t\) with \(\gamma(t) \notin V\), so \(\gamma(t_0)\) lies on the boundary sphere of \(V\), at
Euclidean distance \(\varepsilon\) from \(p\), while \(\gamma\) stays in \(V\) up to \(t_0\).
Restricting to \([a, t_0]\) and using the Euclidean computation from the previous section,
\[
L_g(\gamma) \ge L_g\big(\gamma|_{[a, t_0]}\big) \ge c\, L_{\bar{g}}\big(\gamma|_{[a, t_0]}\big) \ge c\, d_{\bar{g}}\big(p, \gamma(t_0)\big) = c\varepsilon .
\]
Taking the infimum over all \(\gamma\) gives \(d_g(p, q) \ge c\varepsilon > 0\). The metric space
axioms are established.
Finally we show the metric topology equals the manifold topology, by proving each contains the
other's open sets. Suppose \(U\) is open in the manifold topology and \(p \in U\). Choosing a
regular coordinate ball \(V\) of radius \(\varepsilon\) around \(p\) with \(\overline{V}
\subseteq U\), the estimate just proved shows \(d_g(p, q) \ge c\varepsilon\) whenever \(q \notin
V\). Contrapositively, \(d_g(p, q) < c\varepsilon\) forces \(q \in V \subseteq U\), so the metric
ball of radius \(c\varepsilon\) about \(p\) lies in \(U\). Thus \(U\) is open in the metric
topology.
Conversely, suppose \(W\) is open in the metric topology and \(p \in W\). Take a regular
coordinate ball \(V\) of radius \(r\) around \(p\) with comparison constants \(c, C\) on
\(\overline{V}\), and choose \(\varepsilon < r\) small enough that the metric ball of radius
\(C\varepsilon\) about \(p\) lies in \(W\). Let \(V_\varepsilon\) be the set of points in \(V\)
whose Euclidean distance from \(p\) is less than \(\varepsilon\). For \(q \in V_\varepsilon\),
the straight coordinate segment \(\gamma\) from \(p\) to \(q\) stays in \(V\), so integrating the
right inequality of the comparison lemma,
\[
d_g(p, q) \le L_g(\gamma) \le C\, L_{\bar{g}}(\gamma) = C\, |p - q| < C\varepsilon .
\]
Hence \(V_\varepsilon\) is contained in the metric ball of radius \(C\varepsilon\), therefore in
\(W\). Since \(V_\varepsilon\) is a neighborhood of \(p\) in the manifold topology, \(W\) is open
in the manifold topology as well. The two topologies coincide.
Metrizability and Completeness
The metric space theorem was stated for connected manifolds, but its reach extends to all of them.
Every smooth manifold, connected or not, admits a distance function inducing its topology. This
closes a circle opened at the very beginning of the theory: a manifold is introduced as a
topological space satisfying certain axioms, and it now turns out that such a space always carries a
compatible metric.
Corollary: Smooth Manifolds Are Metrizable
Every smooth manifold with or without boundary is metrizable; that is, it admits a distance
function whose induced metric topology is the given manifold topology.
Proof:
Suppose first that \(M\) is a smooth manifold without boundary, and equip it with a
Riemannian metric
\(g\). If \(M\) is connected, the preceding theorem shows directly that \(d_g\) induces the
manifold topology, so \(M\) is metrizable. In general, let \(\{M_i\}\) be the
connected components
of \(M\), each of which is itself a connected smooth manifold and so carries a Riemannian
distance \(d_g\) inducing its topology. Choose a point \(p_i \in M_i\) for each \(i\). Define a
function on \(M\) by setting, for \(x \in M_i\) and \(y \in M_j\),
\[
d(x, y) =
\begin{cases}
d_g(x, y), & i = j, \\\\
d_g(x, p_i) + 1 + d_g(p_j, y), & i \ne j .
\end{cases}
\]
One pictures a bridge of length \(1\) joining each pair of chosen basepoints, so that crossing
from one component to another costs the trip to the basepoint, the bridge, and the trip onward.
This is a distance function: symmetry and positivity are clear, and the triangle inequality holds
because any cross-component step contributes at least the bridge cost \(1\), so a route that
changes components cannot be shorter than going through the basepoints, while a route staying
within one component is governed by the triangle inequality for \(d_g\). It induces the given
topology on \(M\): within a single component it agrees with \(d_g\), and distinct components,
being open as well as closed, are separated by the construction.
Finally, if \(M\) has nonempty boundary, it embeds into its double, a smooth manifold without
boundary obtained by gluing two copies of \(M\) along their common boundary. The double is
metrizable by the case just settled, and a subspace of a metrizable space is metrizable, so
\(M\) is metrizable as well.
With this corollary the development comes full circle. A
topological manifold
was defined at the outset as a space that is locally Euclidean, Hausdorff, and second countable,
with no metric assumed. The route through smooth structures, tangent vectors, tensor fields, and
Riemannian metrics has now returned a distance function on that same space, recovering its topology
exactly. The manifold and the metric space are, in the end, two descriptions of one object.
Because a connected Riemannian manifold is a metric space, the vocabulary of metric spaces applies
to it without change. A connected Riemannian manifold \((M, g)\) is called complete,
and \(g\) a complete Riemannian metric, if the metric space \((M, d_g)\) is
complete,
meaning every Cauchy sequence in \(M\) converges to a point of \(M\). A subset \(B \subseteq M\) is
bounded if there is a constant \(K\) with \(d_g(x, y) \le K\) for all \(x, y \in B\).
These notions, together with the distance function itself, are the foundation on which the geometry
of geodesics, curvature, and length-minimizing paths is built.