Defining Riemannian Metrics
Everything built so far has concerned the differentiable structure of a manifold: how to
differentiate functions and maps, how vectors and covectors transform, how tensor fields are
pulled back and differentiated along flows. None of it measures anything. A smooth manifold by
itself has no notion of the length of a tangent vector, the angle between two directions, or the
distance between two points; a diffeomorphism may stretch and bend it arbitrarily, and nothing in
the smooth structure objects. To introduce geometry, in the metric sense of lengths and angles, we
must add one more piece of data. The model for what to add comes from linear algebra: on a single
vector space, the structure that produces lengths and angles is an inner product, and the symmetric
covariant \(2\)-tensors that carry such geometric content are exactly the inner products. The
construction of this section attaches one such tensor to every tangent space at once, smoothly,
placing an inner product on each \(T_pM\) that varies from point to point. It is one of the central
applications of the tensor machinery developed in the previous pages.
A choice of such a structure is the central object of this chapter. It is the smallest addition to
the smooth structure that turns a manifold into a space one can measure, and it is the foundation
on which the deeper theory rests. That deeper theory, the study of curvature, geodesics, and the
connections that relate nearby tangent spaces, is a subject of its own and lies beyond what we
develop here. Our aim is more contained: to define the structure precisely, to prove that every
manifold admits one, to extract from it the basic measurements of length and angle, and to see how
it transfers across maps. With these in hand we will reach, in the pages that follow, the result
that a connected manifold carrying such a structure becomes a metric space in the sense of analysis,
and we will see the first geometric obstructions that a single such structure cannot remove.
Definition: Riemannian Metric
Let \(M\) be a smooth manifold with or without boundary. A Riemannian metric on
\(M\) is a smooth
symmetric
covariant
\(2\)-tensor field
\(g\) on \(M\) that is positive definite at each point: for every \(p \in M\) and every nonzero
\(v \in T_pM\),
\[
g_p(v, v) > 0 .
\]
A Riemannian manifold is a pair \((M, g)\) consisting of a smooth manifold
\(M\) and a Riemannian metric \(g\) on \(M\); when the metric is understood, one says simply
that \(M\) is a Riemannian manifold. A Riemannian manifold with boundary is
defined in the same way, with \(M\) a smooth manifold with boundary.
Because \(g\) is symmetric and positive definite at each point, the value \(g_p\) is precisely an
inner product
on the tangent space \(T_pM\), a real inner product in the sense of the general axioms with the
scalar field taken to be \(\mathbb{R}\). For this reason the real number \(g_p(v, w)\) is often written
\(\langle v, w \rangle_g\), the subscript recording which metric supplies the pairing. A Riemannian
metric is thus a smoothly varying field of inner products, one on each tangent space.
A word on terminology is in order, because the word "metric" now carries two meanings. A Riemannian
metric is not a metric in the sense of metric spaces, where a metric is a distance function on a
set. The two are related, and a later section makes the relation precise by producing a genuine
distance function from a Riemannian metric. To keep them apart we reserve "metric" for the
Riemannian object and use "distance function" for the metric-space notion; the context will always
make clear which is meant.
In any smooth local coordinates \((x^i)\), a covariant \(2\)-tensor field is determined by its
component functions, and a Riemannian metric is written
\[
g = g_{ij}\, dx^i \otimes dx^j ,
\]
where \((g_{ij})\) is a matrix of smooth functions that is symmetric and positive definite at each
point. Symmetry of \(g\) lets us rewrite this in terms of the
symmetric product:
averaging the expression with its transpose and relabeling the summation indices in the second copy
gives
\[
\begin{align*}
g
&= g_{ij}\, dx^i \otimes dx^j \\\\
&= \tfrac{1}{2}\big(g_{ij}\, dx^i \otimes dx^j + g_{ji}\, dx^i \otimes dx^j\big) \\\\
&= \tfrac{1}{2}\big(g_{ij}\, dx^i \otimes dx^j + g_{ij}\, dx^j \otimes dx^i\big) \\\\
&= g_{ij}\, dx^i\, dx^j ,
\end{align*}
\]
where the second line uses \(g_{ij} = g_{ji}\), the third relabels \(i \leftrightarrow j\) in the
second term, and the last line writes the symmetric product \(dx^i\, dx^j = \tfrac{1}{2}(dx^i
\otimes dx^j + dx^j \otimes dx^i)\). This is the form in which a metric is most often displayed in
coordinates.
Example (The Euclidean Metric):
The simplest Riemannian metric is the Euclidean metric \(\bar{g}\) on
\(\mathbb{R}^n\), given in standard coordinates by
\[
\bar{g} = \delta_{ij}\, dx^i\, dx^j = \big(dx^1\big)^2 + \cdots + \big(dx^n\big)^2 ,
\]
where \(\delta_{ij}\) is the Kronecker delta and \(\big(dx^i\big)^2\) abbreviates the symmetric
product \(dx^i\, dx^i\). Applied to tangent vectors \(v, w \in T_p\mathbb{R}^n\) with components
\(v^i\) and \(w^i\), it returns
\[
\bar{g}_p(v, w) = \delta_{ij}\, v^i w^j = \sum_{i=1}^{n} v^i w^i = v \cdot w ,
\]
the ordinary dot product. In other words \(\bar{g}\) is the \(2\)-tensor field whose value at
every point is the Euclidean inner product. The matrix \((\bar{g}_{ij}) = (\delta_{ij})\) is the
identity, which is symmetric and positive definite, so \(\bar{g}\) is indeed a Riemannian
metric. The explicit summation sign is written because the repeated upper index in \(v^i w^i\)
does not match the summation convention's pairing of one upper with one lower index; writing the
metric coefficients \(\delta_{ij}\) explicitly restores the convention.
Example (Product Metrics):
If \((M, g)\) and \((\widetilde{M}, \widetilde{g})\) are Riemannian manifolds, the product
manifold \(M \times \widetilde{M}\) carries a natural Riemannian metric \(g \oplus \widetilde{g}\),
called the
product metric, defined on the tangent space \(T_{(p,q)}(M \times
\widetilde{M}) \cong T_pM \oplus T_q\widetilde{M}\) by
\[
(g \oplus \widetilde{g})\big((v, \tilde{v}), (w, \tilde{w})\big) = g(v, w) + \widetilde{g}(\tilde{v}, \tilde{w}) .
\]
Given local coordinates \((x^1, \ldots, x^n)\) for \(M\) and \((y^1, \ldots, y^m)\) for
\(\widetilde{M}\), the pair \((x^1, \ldots, x^n, y^1, \ldots, y^m)\) is a coordinate system for
the product, and in these coordinates \(g \oplus \widetilde{g}\) is represented by the block
diagonal matrix
\[
\begin{pmatrix} g_{ij} & 0 \\ 0 & \widetilde{g}_{ij} \end{pmatrix} .
\]
A direct consequence is that the Euclidean metric on \(\mathbb{R}^{n+m}\) is the product of the
Euclidean metrics on \(\mathbb{R}^n\) and \(\mathbb{R}^m\), since the identity matrix of size
\(n + m\) is the block diagonal assembly of the two smaller identity matrices.
Existence on Every Manifold
Before measuring anything, one must know that there is something to measure with. It is not obvious
that an arbitrary smooth manifold carries any Riemannian metric at all: the definition demands a
field of inner products that is simultaneously positive definite at every point and smooth across
the whole manifold, and a manifold is in general assembled from coordinate patches with no
preferred way to reconcile the geometries they would each suggest. The resolution is that local
geometries can always be blended into a global one, and the device that performs the blending is a
partition of unity. The conclusion is unconditional.
Theorem: Existence of Riemannian Metrics
Every smooth manifold with or without boundary admits a Riemannian metric.
Proof:
Let \(M\) be a smooth manifold, and cover it by smooth coordinate charts \((U_\alpha,
\varphi_\alpha)\), where each \(\varphi_\alpha : U_\alpha \to \widehat{U}_\alpha \subseteq
\mathbb{R}^n\) is a diffeomorphism onto an open subset of \(\mathbb{R}^n\). On each coordinate
domain, transport the Euclidean metric back to \(U_\alpha\) by setting
\[
g_\alpha = \varphi_\alpha^*\, \bar{g} ,
\]
the pullback of the Euclidean metric \(\bar{g}\) under \(\varphi_\alpha\). Because
\(\varphi_\alpha\) is a diffeomorphism, its
differential
is a linear isomorphism at each point,
so \(g_\alpha\) is a smooth symmetric \(2\)-tensor field on \(U_\alpha\) that is positive
definite: for nonzero \(v \in T_pU_\alpha\),
\[
(g_\alpha)_p(v, v) = \bar{g}\big(d(\varphi_\alpha)_p(v),\, d(\varphi_\alpha)_p(v)\big)
= \big| d(\varphi_\alpha)_p(v) \big|^2 > 0 ,
\]
the last inequality because \(d(\varphi_\alpha)_p\) is injective and so sends the nonzero \(v\)
to a nonzero vector. In coordinates \(g_\alpha\) is simply \(\delta_{ij}\, dx^i\, dx^j\). Each
\(U_\alpha\) thus carries a Riemannian metric of its own; what remains is to assemble these into
a single metric on all of \(M\).
Let \((\psi_\alpha)\) be a smooth
partition of unity
subordinate to the cover \((U_\alpha)\), which exists by the
existence of partitions of unity,
and define
\[
g = \sum_\alpha \psi_\alpha\, g_\alpha ,
\]
each term interpreted as zero outside \(\operatorname{supp}\psi_\alpha \subseteq U_\alpha\), so
that it extends smoothly by zero to all of \(M\). The defining property of a partition of unity
is that its supports are locally finite, so in a neighborhood of any point only finitely many
terms are nonzero; the sum is therefore a finite sum near each point and defines a smooth
\(2\)-tensor field on \(M\). It is symmetric, being a sum of symmetric tensors. Only positive
definiteness remains to be checked.
Fix \(p \in M\) and a nonzero \(v \in T_pM\). Evaluating the field on the pair \((v, v)\) gives
\[
g_p(v, v) = \sum_\alpha \psi_\alpha(p)\, (g_\alpha)_p(v, v) .
\]
Every term is nonnegative: \(\psi_\alpha(p) \ge 0\) by the partition-of-unity conditions, and
\((g_\alpha)_p(v, v) > 0\) wherever \(\psi_\alpha(p) \neq 0\), since there \(p \in U_\alpha\) and
\(g_\alpha\) is positive definite. The functions \(\psi_\alpha\) sum to \(1\) at \(p\), so at
least one of them is strictly positive there; for that index the corresponding term is strictly
positive, and the whole sum is therefore strictly positive. Hence \(g_p(v, v) > 0\), and \(g\)
is a Riemannian metric on \(M\). The construction is identical on a manifold with boundary, using
charts into the half-space and the partition of unity that exists in that setting as well.
The construction makes a structural point that survives long after the proof. The metric produced
depends on every arbitrary choice along the way: which charts cover the manifold, how the partition
of unity is built, how the local pieces overlap. Different choices yield genuinely different metrics,
and there is nothing canonical about any of them. The same manifold can be given metrics whose
geometries differ as widely as one likes, and a manifold therefore does not come with a metric the
way it comes with its smooth structure; a metric is extra data, freely chosen. This abundance is the
source of a question that recurs whenever geometry is brought to bear on a concrete problem: among
the supply of metrics a space admits, is there one the situation itself singles out? When a
manifold arises as a family of probability distributions, imposing natural invariance requirements
singles out a metric essentially uniquely, and that metric is the subject toward which a later part
of this development is aimed.
Length, Angle, and Orthogonality
A Riemannian metric was introduced to make geometric measurement possible, and the measurements it
supports are exactly those that an inner product supplies on a single vector space, now available on
every tangent space at once. The length of a tangent vector, the angle between two of them, and the
relation of being perpendicular all transfer verbatim from linear algebra, with the fixed Euclidean
inner product replaced by the metric's pairing \(\langle \cdot, \cdot \rangle_g\) at the point in
question.
Definition: Length, Angle, and Orthogonality
Let \((M, g)\) be a Riemannian manifold and \(p \in M\).
(i) The length or norm of a tangent vector
\(v \in T_pM\) is
\[
|v|_g = \langle v, v \rangle_g^{1/2} = g_p(v, v)^{1/2} .
\]
(ii) The angle between two nonzero tangent vectors \(v, w \in
T_pM\) is the unique \(\theta \in [0, \pi]\) satisfying
\[
\cos\theta = \frac{\langle v, w \rangle_g}{|v|_g\, |w|_g} .
\]
(iii) Two tangent vectors \(v, w \in T_pM\) are orthogonal if
\(\langle v, w \rangle_g = 0\). This holds when one or both vectors are zero, or when the angle
between them is \(\pi/2\).
The length is well defined because \(g_p(v, v) \ge 0\), with equality only when \(v = 0\), so the
square root is real and vanishes precisely on the zero vector. The angle requires slightly more: for
the prescription to determine a genuine angle, the ratio defining \(\cos\theta\) must lie in the
interval \([-1, 1]\), so that exactly one \(\theta \in [0, \pi]\) has the required cosine. Since
\(g_p\) is an inner product on \(T_pM\), the
Cauchy-Schwarz inequality
applies to it directly, giving
\[
\big| \langle v, w \rangle_g \big| \le |v|_g\, |w|_g
\]
for all \(v, w \in T_pM\). For nonzero \(v\) and \(w\), dividing by the positive product \(|v|_g\,
|w|_g\) places the ratio in the definition of the angle in \([-1, 1]\), so a unique \(\theta \in [0,
\pi]\) realizes it and the angle is well defined. With lengths and angles in hand, the metric
delivers on every plane of measurement that an inner product offers on a vector space, now varying
smoothly across the manifold.
Orthonormal Frames
On a vector space with an inner product, the most convenient bases are the orthonormal ones, in
which the inner product reduces to the identity matrix and computations become transparent. The
same convenience is available locally on a Riemannian manifold, through frames of vector fields
that are orthonormal with respect to the metric at every point. Such frames are the working tool
for much of the local theory, and they always exist near any point, though, as we will see, only
rarely as the coordinate frame of a chart.
Recall that a smooth local
frame
for \(M\) over an open set \(U\) is an ordered tuple of smooth vector fields whose values form a
basis of the tangent space at each point of \(U\). The metric singles out those frames for which
that basis is orthonormal.
Definition: Orthonormal Frame on a Riemannian Manifold
Let \((M, g)\) be an \(n\)-dimensional Riemannian manifold with or without boundary, and let
\(U \subseteq M\) be open. A smooth local frame \((E_1, \ldots, E_n)\) for \(M\) over \(U\) is an
orthonormal frame if at every point \(p \in U\) the vectors \(E_1|_p, \ldots,
E_n|_p\) form an orthonormal basis of \(T_pM\) with respect to \(g_p\); equivalently, if
\[
\langle E_i, E_j \rangle_g = \delta_{ij}
\]
as functions on \(U\), for all \(i, j\).
When \(M = \mathbb{R}^n\) carries the Euclidean metric, this reduces to the notion of an orthonormal
frame already familiar in the ambient setting, where orthonormality is measured by the dot product.
The metric version replaces that fixed product with the pointwise pairing \(\langle \cdot, \cdot
\rangle_g\), and the construction that produces such frames is the same Gram-Schmidt process, applied
fiber by fiber.
Theorem: Existence of Local Orthonormal Frames
Let \((M, g)\) be a Riemannian manifold with or without boundary. If \((X_1, \ldots, X_n)\) is a
smooth local frame for \(M\) over an open set \(U\), then there is a smooth orthonormal frame
\((E_1, \ldots, E_n)\) over \(U\) satisfying
\[
\operatorname{span}\big(E_1|_p, \ldots, E_j|_p\big)
= \operatorname{span}\big(X_1|_p, \ldots, X_j|_p\big)
\]
for each \(j = 1, \ldots, n\) and each \(p \in U\). Consequently, every point of \(M\) has a
neighborhood on which a smooth orthonormal frame exists.
Proof:
Apply the
Gram-Schmidt process
to the given frame at each point, but with the Euclidean inner product replaced throughout by
the metric pairing \(\langle \cdot, \cdot \rangle_g\). Concretely, define vector fields
\(V_1, \ldots, V_n\) and \(E_1, \ldots, E_n\) on \(U\) by the recursion
\[
\begin{align*}
V_1 &= X_1, &\quad E_1 &= \frac{V_1}{|V_1|_g}, \\\\
V_j &= X_j - \sum_{i=1}^{j-1} \langle X_j, E_i \rangle_g\, E_i, &\quad E_j &= \frac{V_j}{|V_j|_g}
\qquad (j = 2, \ldots, n).
\end{align*}
\]
Each step is well posed. Inductively, \(\operatorname{span}(E_1|_p, \ldots, E_{j-1}|_p) =
\operatorname{span}(X_1|_p, \ldots, X_{j-1}|_p)\), and the vectors \(X_1|_p, \ldots, X_j|_p\) are
linearly independent, so \(X_j|_p\) lies outside that span; hence \(V_j|_p\), which differs from
\(X_j|_p\) by a combination of the earlier \(E_i|_p\), is nonzero. Thus \(|V_j|_g\)
is nowhere zero on \(U\) and the division is legitimate. The resulting fields are smooth: the
coefficients \(\langle X_j, E_i \rangle_g\) are smooth functions because \(g\), \(X_j\), and the
already-constructed \(E_i\) are smooth, and the norm \(|V_j|_g = \langle V_j, V_j
\rangle_g^{1/2}\) is a smooth, nowhere-vanishing function whose reciprocal is therefore smooth.
A direct induction shows the frame is orthonormal. Each \(E_j\) is a unit vector by
construction, and for \(i < j\) the defining sum subtracts off precisely the components of
\(X_j\) along \(E_1, \ldots, E_{j-1}\), so
\[
\langle V_j, E_k \rangle_g
= \langle X_j, E_k \rangle_g - \sum_{i=1}^{j-1} \langle X_j, E_i \rangle_g\, \langle E_i, E_k \rangle_g
= \langle X_j, E_k \rangle_g - \langle X_j, E_k \rangle_g = 0
\]
for each \(k < j\), using the inductive hypothesis \(\langle E_i, E_k \rangle_g = \delta_{ik}\).
Dividing by \(|V_j|_g\) preserves this, so \(\langle E_j, E_k \rangle_g = 0\) for \(k < j\), and
together with \(|E_j|_g = 1\) this gives \(\langle E_i, E_j \rangle_g = \delta_{ij}\) throughout.
Finally, each \(E_j\) is a nonzero multiple of \(V_j\), itself \(X_j\) plus a combination of the
earlier \(E_i\), so the spans coincide stagewise as claimed. The last assertion follows by
starting from any smooth coordinate frame on a chart around a given point.
One feature of these frames carries a warning that the next stage of the theory will make precise.
Producing an orthonormal frame does not produce coordinates whose coordinate frame is orthonormal.
The fields \(E_i\) constructed above are seldom of the form \(\partial/\partial x^i\) for any chart;
the coordinate frame of a chart is orthonormal only in special circumstances, and demanding that
such a chart exist near every point turns out to be a strong restriction on the metric rather than a
general fact. The question of which metrics admit such charts is exactly the question of whether the
metric is, in a sense to be defined, geometrically trivial, and it is the entry point to the study
of curvature.
Pullback Metrics and Isometries
A metric on one manifold can be carried to another along a smooth map, by the same pullback
operation that transports any covariant tensor field. The result is always a smooth symmetric
\(2\)-tensor field, but it need not be a metric: positive definiteness can fail. Determining exactly
when it survives identifies a familiar class of maps and supplies the mechanism by which curved
surfaces inherit geometry from the space around them.
Let \(M\) and \(N\) be smooth manifolds, let \(g\) be a Riemannian metric on \(N\), and let \(F : M
\to N\) be smooth. The
pullback
\(F^*g\) is a smooth covariant \(2\)-tensor field on \(M\), symmetric because \(g\) is. Its value on
a pair of tangent vectors is, by definition of the pullback,
\[
(F^*g)_p(v, w) = g_{F(p)}\big(dF_p(v),\, dF_p(w)\big) ,
\]
where \(dF_p\) is the
differential
of \(F\) at \(p\). Whether \(F^*g\) is a Riemannian metric depends entirely on whether this form is
positive definite, and the answer is governed by a single condition on \(F\).
Theorem: Pullback Metric Criterion
Let \(F : M \to N\) be a smooth map and let \(g\) be a Riemannian metric on \(N\). Then \(F^*g\)
is a Riemannian metric on \(M\) if and only if \(F\) is a smooth
immersion.
When this holds, \(F^*g\) is called the pullback metric determined by \(F\).
Proof:
The field \(F^*g\) is a smooth symmetric \(2\)-tensor for any smooth \(F\), so the only question
is positive definiteness, which we examine point by point. Fix \(p \in M\) and a nonzero \(v \in
T_pM\). From the formula above,
\[
(F^*g)_p(v, v) = g_{F(p)}\big(dF_p(v),\, dF_p(v)\big) = \big| dF_p(v) \big|_g^2 .
\]
Since \(g_{F(p)}\) is positive definite, this is nonnegative, and it is strictly positive
exactly when \(dF_p(v) \neq 0\). Thus \((F^*g)_p\) is positive definite if and only if \(dF_p\)
sends every nonzero vector to a nonzero vector, that is, if and only if \(dF_p\) is injective.
Requiring this at every \(p\) is precisely the condition that \(F\) be a smooth immersion, and so
\(F^*g\) is positive definite at every point, hence a Riemannian metric, if and only if \(F\) is
an immersion.
When \(F\) is itself a diffeomorphism, the pullback not only produces a metric on \(M\) but matches
the geometries of the two manifolds exactly. Maps with this property are the structure-preserving
maps of Riemannian geometry, and they are what it means for two such manifolds to be
indistinguishable as geometric objects.
Definition: Isometry and Local Isometry
Let \((M, g)\) and \((\widetilde{M}, \widetilde{g})\) be Riemannian manifolds. A smooth map \(F
: M \to \widetilde{M}\) is a (Riemannian) isometry if it is a
diffeomorphism
satisfying \(F^*\widetilde{g} = g\). If such a map exists, \((M, g)\) and \((\widetilde{M},
\widetilde{g})\) are said to be isometric. More generally, \(F\) is a
local isometry if every point of \(M\) has a neighborhood \(U\) on which
\(F|_U\) is an isometry onto an open subset of \(\widetilde{M}\). This holds precisely when \(F\)
is a
local diffeomorphism
satisfying \(F^*\widetilde{g} = g\): the pullback condition is pointwise, so once \(F\) restricts
to a diffeomorphism on a neighborhood of each point, \(F^*\widetilde{g} = g\) there is exactly the
statement that each such restriction is an isometry. Two manifolds are locally
isometric if each point of one has a neighborhood isometric to an open subset of the
other.
The condition \(F^*\widetilde{g} = g\) says exactly that \(F\) preserves the inner products of
tangent vectors: \(\widetilde{g}\big(dF_p(v), dF_p(w)\big) = g_p(v, w)\) for all \(v, w \in T_pM\).
Lengths, angles, and ultimately distances are therefore unchanged by an isometry, and the study of
the properties left invariant by all isometries, local and global, is the substance of Riemannian
geometry.
Pullback metrics are easy to compute once a coordinate expression for the map is known: by the
coordinate formula for tensor pullbacks,
the computation amounts to substituting the component functions of \(F\) for the coordinates of
\(N\) in the expression for \(g\). Two examples illustrate the procedure.
Example (The Helicoid):
Consider the smooth map \(F : \mathbb{R}^2 \to \mathbb{R}^3\) given by
\[
F(u, v) = (u \cos v,\, u \sin v,\, v) .
\]
Its differential has matrix with columns \(\partial F/\partial u = (\cos v, \sin v, 0)\) and
\(\partial F/\partial v = (-u\sin v, u\cos v, 1)\), which are linearly independent at every
point, so \(F\) is an immersion; its image is a surface
called a helicoid, an endlessly winding spiral ramp. To find the pullback of
the Euclidean metric \(\bar{g} = dx^2 + dy^2 + dz^2\), substitute \(x = u\cos v\), \(y = u\sin
v\), \(z = v\):
\[
\begin{align*}
F^*\bar{g}
&= d(u\cos v)^2 + d(u\sin v)^2 + d(v)^2 \\\\
&= (\cos v\, du - u\sin v\, dv)^2 + (\sin v\, du + u\cos v\, dv)^2 + dv^2 \\\\
&= \cos^2 v\, du^2 - 2u\sin v\cos v\, du\, dv + u^2\sin^2 v\, dv^2 \\\\
&\qquad + \sin^2 v\, du^2 + 2u\sin v\cos v\, du\, dv + u^2\cos^2 v\, dv^2 + dv^2 \\\\
&= du^2 + (u^2 + 1)\, dv^2 .
\end{align*}
\]
The cross terms cancel and the squared coefficients combine through \(\cos^2 v + \sin^2 v = 1\),
leaving a metric on \(\mathbb{R}^2\) that differs from the Euclidean one by the factor \(u^2 + 1\)
on the \(dv^2\) term. Here, by the convention for symmetric products, \(du^2\) abbreviates the
symmetric product \(du\, du\), not the differential of \(u^2\).
Example (Polar Coordinates):
The same substitution mechanism expresses a fixed metric in new coordinates, viewing the change
of coordinates as the identity map written in two coordinate systems. To express the Euclidean
metric \(\bar{g} = dx^2 + dy^2\) on \(\mathbb{R}^2\) in polar coordinates, substitute \(x = r\cos
\theta\), \(y = r\sin\theta\):
\[
\begin{align*}
\bar{g}
&= d(r\cos\theta)^2 + d(r\sin\theta)^2 \\\\
&= (\cos\theta\, dr - r\sin\theta\, d\theta)^2 + (\sin\theta\, dr + r\cos\theta\, d\theta)^2 \\\\
&= (\cos^2\theta + \sin^2\theta)\, dr^2 + (r^2\sin^2\theta + r^2\cos^2\theta)\, d\theta^2 \\\\
&\qquad + (-2r\cos\theta\sin\theta + 2r\sin\theta\cos\theta)\, dr\, d\theta \\\\
&= dr^2 + r^2\, d\theta^2 .
\end{align*}
\]
This is the polar form of the Euclidean metric, valid where \((r, \theta)\) are genuine
coordinates. The coefficient \(r^2\) on \(d\theta^2\) records that a fixed change \(d\theta\) in
angle corresponds to an arc whose length grows with the radius.
Flatness and Local Isometry
With isometries defined, we can name the most basic geometric property a Riemannian metric can
have: agreement, locally, with the flat geometry of Euclidean space. The orthonormal frames of an
earlier section raised the question implicitly. Such frames always exist near a point, yet the
coordinate frame of a chart is rarely orthonormal; the metrics for which a chart with orthonormal
coordinate frame does exist near every point are exactly those that are locally indistinguishable
from Euclidean space, and they form the trivial case against which all curvature is measured.
Definition: Flat Metric
A Riemannian manifold \((M, g)\) is flat, and \(g\) is a flat
metric, if \((M, g)\) is locally isometric to \((\mathbb{R}^n, \bar{g})\) with its
Euclidean metric: every point of \(M\) has a neighborhood isometric to an open subset of
Euclidean space.
Flatness is defined through isometries, but it can be detected by criteria that are far easier to
check in practice, phrased in terms of coordinates and frames. The following theorem assembles
them; it is the tool used to decide whether a given metric is flat.
Theorem: Characterization of Flat Metrics
For a Riemannian manifold \((M, g)\), the following are equivalent.
(a) \(g\) is flat.
(b) Each point of \(M\) lies in the domain of a smooth coordinate chart in
which \(g\) has the coordinate representation \(g = \delta_{ij}\, dx^i\, dx^j\).
(c) Each point of \(M\) lies in the domain of a smooth coordinate chart whose
coordinate frame is orthonormal.
(d) Each point of \(M\) lies in the domain of a commuting orthonormal frame.
Proof:
We prove the implications in the cycle (a) \(\Rightarrow\) (b) \(\Rightarrow\) (c)
\(\Rightarrow\) (d) \(\Rightarrow\) (a).
(a) \(\Rightarrow\) (b). Suppose \(g\) is flat, and let \(p \in M\). By
definition there is a neighborhood \(U\) of \(p\) and an isometry \(\varphi : U \to V\) onto an
open subset \(V \subseteq \mathbb{R}^n\), meaning \(\varphi\) is a diffeomorphism with
\(\varphi^*\bar{g} = g\). Such a \(\varphi\) is a smooth coordinate chart on \(U\). In the
standard coordinates of \(\mathbb{R}^n\) the Euclidean metric is \(\bar{g} = \delta_{ij}\, dx^i\,
dx^j\), and pulling back replaces those coordinates by the coordinate functions of \(\varphi\),
so in this chart \(g = \varphi^*\bar{g} = \delta_{ij}\, dx^i\, dx^j\). This is the
representation required by (b).
(b) \(\Rightarrow\) (c). In a chart where \(g = \delta_{ij}\, dx^i\, dx^j\), the
metric evaluated on the coordinate frame \((\partial/\partial x^i)\) gives \(\langle
\partial/\partial x^i,\, \partial/\partial x^j \rangle_g = g_{ij} = \delta_{ij}\), so the
coordinate frame is orthonormal. This is (c).
(c) \(\Rightarrow\) (d). The
coordinate vector fields
\((\partial/\partial x^i)\) of any chart commute, satisfying \([\partial/\partial x^i,\,
\partial/\partial x^j] = 0\) for all \(i, j\), since acting on a smooth function they reproduce
the equality of mixed second partial derivatives. If in addition this frame is orthonormal, as
(c) provides, then it is a commuting orthonormal frame containing the given point in its domain,
which is (d).
(d) \(\Rightarrow\) (a). Let \((E_1, \ldots, E_n)\) be a commuting orthonormal
frame on a neighborhood \(W\) of a point \(p\). Because the fields commute and are linearly
independent, the
canonical form for commuting vector fields
provides a smooth chart \((U, (s^1, \ldots, s^n))\) centered at \(p\) in which \(E_i =
\partial/\partial s^i\) for each \(i\). In this chart the metric components are
\[
g_{ij} = \langle \partial/\partial s^i,\, \partial/\partial s^j \rangle_g
= \langle E_i,\, E_j \rangle_g = \delta_{ij} ,
\]
using orthonormality of the frame, so \(g = \delta_{ij}\, ds^i\, ds^j\) on \(U\). The chart
\(\sigma = (s^1, \ldots, s^n) : U \to \sigma(U) \subseteq \mathbb{R}^n\) is a diffeomorphism
onto an open subset, and the identity \(g = \delta_{ij}\, ds^i\, ds^j\) says exactly that
\(\sigma^*\bar{g} = g\), since \(\bar{g} = \delta_{ij}\, dx^i\, dx^j\) pulls back to
\(\delta_{ij}\, ds^i\, ds^j\) under \(\sigma\). Hence \(\sigma\) is an isometry of \(U\) onto an
open subset of \((\mathbb{R}^n, \bar{g})\). Every point thus has a neighborhood isometric to an
open subset of Euclidean space, so \(g\) is flat. This closes the cycle.
The criterion makes flatness a property one can test, but it should not suggest that flatness is
common. In dimension one it is automatic: every Riemannian metric on a curve can be straightened to
the Euclidean one by an arc-length parameter, so there is no one-dimensional obstruction. Beginning
in dimension two the situation changes, and most metrics are not flat. We are not yet in a position
to exhibit the obstruction, because the quantity that measures the failure of flatness, the
curvature, lies beyond the constructions developed here. What the present criterion does supply is
the means to recognize the obstruction once a concrete metric is in hand: a surface for which no
chart can make the coordinate frame orthonormal cannot be flat. The round sphere is the standard
example, and showing that it admits no such chart, and is therefore not flat, is the goal toward
which the next stage of examples is directed.