Riemannian Submanifolds and Induced Metrics
A metric was shown to pull back along any immersion, and a basic immersion is the
inclusion of a submanifold into the manifold that contains it. Through that inclusion every
submanifold of a Riemannian manifold inherits a metric of its own, with no additional choice
required. This inherited metric is the setting in which curved surfaces acquire their geometry from
the space surrounding them, and it is the source of nearly every concrete Riemannian manifold one
meets in practice.
Let \((M, g)\) be a Riemannian manifold with or without boundary, and let \(S \subseteq M\) be a
submanifold, either
immersed
or
embedded,
with inclusion map \(\iota : S \hookrightarrow M\). Since the inclusion of a submanifold is a smooth
immersion, the pullback criterion of the previous page guarantees that \(\iota^*g\) is a Riemannian
metric on \(S\). It is this metric, and no other, that a submanifold carries by default.
Definition: Riemannian Submanifold and Induced Metric
Let \((M, g)\) be a Riemannian manifold with or without boundary, and let \(S \subseteq M\) be a
submanifold with inclusion \(\iota : S \hookrightarrow M\). The
pullback metric
\(\iota^*g\) is called the induced metric on \(S\), and \(S\) equipped with it
is a Riemannian submanifold of \(M\) (with or without boundary).
The induced metric has a transparent meaning. Under the usual identification of the
tangent space to a submanifold
\(T_pS\) with a subspace of the ambient tangent space \(T_pM\), the differential \(d\iota_p\) is the
inclusion of that subspace. Writing out the pullback on a pair of vectors \(v, w \in T_pS\),
\[
(\iota^*g)_p(v, w) = g_p\big(d\iota_p(v),\, d\iota_p(w)\big) = g_p(v, w) ,
\]
so the induced metric is simply the restriction of \(g\) to pairs of vectors tangent to \(S\).
Lengths and angles measured on \(S\) are exactly the lengths and angles those vectors already have
as elements of the ambient tangent spaces; the submanifold borrows its geometry verbatim from its
surroundings.
In practice the induced metric is computed not through the inclusion directly but through a local
parametrization. Recall that a smooth local parametrization of \(S\) is an injective smooth
immersion \(X : U \to M\) from an open subset \(U \subseteq \mathbb{R}^k\) onto an open subset of
\(S\), serving as the inverse of a coordinate chart. Because \(\iota \circ X = X\) as a map into
\(M\), the coordinate representation of the induced metric is the pullback \(X^*g = X^*(\iota^*g)\),
which by the pullback computation amounts to substituting the component functions of \(X\) into the
expression for \(g\). The examples that follow all proceed this way.
The Round Metric on the Sphere
The sphere is a basic example of a curved Riemannian manifold, and the metric it carries as a
submanifold of Euclidean space is the one against which spherical geometry is done. Because the
construction is nothing more than the induced metric of the previous section applied to the unit
sphere, it requires no new machinery, only a name.
Definition: The Round Metric
Let \(\iota : \mathbb{S}^n \hookrightarrow \mathbb{R}^{n+1}\) be the inclusion of the unit
sphere.
The metric induced on \(\mathbb{S}^n\) by the Euclidean metric \(\bar{g}\),
\[
\mathring{g} = \iota^*\bar{g} ,
\]
is called the round metric (or the standard metric) on the
sphere.
To compute the round metric in coordinates we use a local parametrization, following the procedure
set out in the previous section. The most uniform parametrizations express a piece of the sphere as
the graph of a function, so it is worth recording the induced metric in graph coordinates once and
for all.
Example (Induced Metric in Graph Coordinates):
Let \(U \subseteq \mathbb{R}^n\) be open and let \(S \subseteq \mathbb{R}^{n+1}\) be the graph of
a smooth function \(f : U \to \mathbb{R}\), parametrized by the map
\[
X(u^1, \ldots, u^n) = \big(u^1, \ldots, u^n,\, f(u^1, \ldots, u^n)\big) .
\]
Substituting these coordinate functions for \(x^1, \ldots, x^{n+1}\) in the Euclidean metric
\(\bar{g} = (dx^1)^2 + \cdots + (dx^{n+1})^2\) gives the induced metric in graph coordinates:
\[
X^*\bar{g} = (du^1)^2 + \cdots + (du^n)^2 + df^2 ,
\]
where \(df = \tfrac{\partial f}{\partial u^i}\, du^i\) is the differential of \(f\) and \(df^2\)
denotes its symmetric product with itself. Each of the first \(n\) coordinate functions of \(X\)
is simply \(u^i\), contributing \((du^i)^2\), and the last contributes \(df^2\); there are no
cross terms among the first \(n\), so the expression is as stated.
Example (The Round Metric in Graph Coordinates):
The upper hemisphere of \(\mathbb{S}^2\) is the graph of \(f(u, v) = \sqrt{1 - u^2 - v^2}\) over
the open unit disk, parametrized by
\[
X(u, v) = \big(u,\, v,\, \sqrt{1 - u^2 - v^2}\big) .
\]
Its differential is
\[
df = \frac{-u\, du - v\, dv}{\sqrt{1 - u^2 - v^2}} ,
\]
so by the graph-coordinate formula the round metric is
\[
\begin{align*}
\mathring{g}
&= du^2 + dv^2 + df^2 \\\\
&= du^2 + dv^2 + \left( \frac{u\, du + v\, dv}{\sqrt{1 - u^2 - v^2}} \right)^2 \\\\
&= \frac{(1 - v^2)\, du^2 + 2uv\, du\, dv + (1 - u^2)\, dv^2}{1 - u^2 - v^2} ,
\end{align*}
\]
valid on the unit disk where \(u^2 + v^2 < 1\). The denominator vanishes at the equator, where
these coordinates break down, which reflects the fact that no single chart covers the whole
sphere; the other hemispheres and the equatorial bands are described by analogous graphs over
the remaining coordinate planes.
Surfaces of Revolution
A second family of induced metrics, broad enough to include the torus, the sphere, and the cylinder,
comes from revolving a curve about an axis. These surfaces of revolution are the testing ground on
which we will shortly decide which curved surfaces are flat and which are not, so it is worth
computing their induced metrics in a single uniform formula.
Let \(C\) be an embedded \(1\)-dimensional submanifold of the half-plane \(H = \{(r, z) : r > 0\}\),
and let \(S_C \subseteq \mathbb{R}^3\) be the surface of revolution swept out by rotating \(C\) about
the \(z\)-axis. If \(\gamma(t) = \big(a(t), b(t)\big)\) is a smooth local parametrization of \(C\),
with \(a(t) > 0\) the distance from the axis and \(b(t)\) the height, then \(S_C\) is parametrized
locally by
\[
X(t, \theta) = \big(a(t)\cos\theta,\, a(t)\sin\theta,\, b(t)\big) ,
\]
valid where \((t, \theta)\) ranges over a sufficiently small open set so that \(X\) is an injective
immersion. Substituting these coordinate functions into the Euclidean metric \(\bar{g} = dx^2 + dy^2
+ dz^2\) gives the induced metric
\[
\begin{align*}
X^*\bar{g}
&= d\big(a(t)\cos\theta\big)^2 + d\big(a(t)\sin\theta\big)^2 + d\big(b(t)\big)^2 \\\\
&= \big(a'(t)\cos\theta\, dt - a(t)\sin\theta\, d\theta\big)^2
+ \big(a'(t)\sin\theta\, dt + a(t)\cos\theta\, d\theta\big)^2
+ \big(b'(t)\, dt\big)^2 \\\\
&= \big(a'(t)^2 + b'(t)^2\big)\, dt^2 + a(t)^2\, d\theta^2 ,
\end{align*}
\]
where the cross terms in \(dt\, d\theta\) cancel and the angular coefficients combine through
\(\cos^2\theta + \sin^2\theta = 1\). In particular, if \(\gamma\) is a unit-speed
curve, meaning \(|\gamma'(t)|^2 = a'(t)^2 + b'(t)^2 = 1\), this simplifies to
\[
X^*\bar{g} = dt^2 + a(t)^2\, d\theta^2 .
\]
Three familiar surfaces arise as special cases, each from a unit-speed generating curve.
Example (Torus, Sphere, and Cylinder):
(a) The torus. The circle \((r - 2)^2 + z^2 = 1\) in the half-plane, generating
a torus of revolution, has the unit-speed parametrization \(\gamma(t) = (2 + \cos t,\, \sin
t)\). Here \(a(t) = 2 + \cos t\), so the induced metric is
\[
dt^2 + (2 + \cos t)^2\, d\theta^2 .
\]
(b) The sphere. The unit sphere, minus its north and south poles, is the
surface of revolution generated by the semicircle \(\gamma(t) = (\sin t,\, \cos t)\) for \(0 < t
< \pi\), which is unit-speed. With \(a(t) = \sin t\), the induced metric is
\[
dt^2 + \sin^2 t\, d\theta^2 ,
\]
the round metric on \(\mathbb{S}^2\) written in the coordinates \((t, \theta)\), where \(t\) is
the angle from the north pole and \(\theta\) the longitude.
(c) The cylinder. The unit cylinder \(x^2 + y^2 = 1\) is the surface of
revolution generated by the vertical line \(\gamma(t) = (1, t)\), which is unit-speed with
\(a(t) = 1\) constant. The induced metric is
\[
dt^2 + d\theta^2 ,
\]
which is the Euclidean metric in the coordinates \((t, \theta)\).
The last example is worth examining further. The induced metric on the cylinder is, in the coordinates
\((t, \theta)\), exactly the Euclidean metric on the \((t, \theta)\)-plane. A suitable restriction
of the parametrization therefore gives, for each point of the cylinder, a Riemannian isometry
between an open subset of \((\mathbb{R}^2, \bar{g})\) and a neighborhood of that point in the
cylinder. The induced metric on the cylinder is consequently flat: a two-dimensional being living in
the cylinder could not distinguish its surroundings from the Euclidean plane by any local
measurement of lengths and angles, even though the cylinder is visibly curved when viewed from the
surrounding space. This shows that whether a metric is flat can have an answer that the ambient
picture does not suggest, and it is the cylinder, not the sphere, that turns out to be the flat one.
The Sphere Is Not Flat
The flatness criterion established earlier turns the geometric question "is this surface flat?" into
a checkable condition on coordinate frames. We now apply it to surfaces of revolution and reach the
conclusion that motivated the whole discussion: the round sphere admits no flattening, unlike
the cylinder. The result is an exact characterization of which generating curves
produce a flat surface.
Theorem: Flatness of Surfaces of Revolution
Let \(C\) be a connected embedded \(1\)-dimensional submanifold of the half-plane \(H = \{(r, z)
: r > 0\}\), and let \(S_C\) be the surface of revolution generated by \(C\). The induced metric
on \(S_C\) is
flat
if and only if \(C\) is part of a straight line.
Proof:
Suppose first that \(C\) is part of a straight line, so it has a parametrization \(\gamma(t) =
(Pt + K,\, Qt + L)\) for constants \(P, Q, K, L\) with \(P\) and \(Q\) not both zero. Rescaling
\(t\) makes \(\gamma\) unit-speed, so \(P^2 + Q^2 = 1\). There are three cases. If \(Q = 0\),
then \(C\) is a horizontal segment and \(S_C\) is an open subset of the plane \(z = L\), which is
flat. If \(P = 0\), then \(C\) is a vertical segment, \(a(t) = K\) is constant, and \(S_C\) is
part of the cylinder \(x^2 + y^2 = K^2\), shown flat in the previous section by the same
computation that flattened the unit cylinder. If neither \(P\) nor \(Q\) is zero, then \(S_C\)
is part of a cone, and the induced metric is \(dt^2 + (Pt + K)^2\, d\theta^2\). In a
neighborhood of any point the change of coordinates
\[
(u, v) = \big((t + K/P)\cos P\theta,\ (t + K/P)\sin P\theta\big)
\]
pulls the Euclidean metric \(du^2 + dv^2\) back to \(dt^2 + (Pt + K)^2\, d\theta^2\), exhibiting
a local isometry to the plane; the cone is flat, as one sees physically by slitting a paper cone
along one side and unrolling it onto the plane. In every case the metric is flat.
Conversely, suppose the induced metric on \(S_C\) is flat. Let \(\gamma(t) = \big(a(t),
b(t)\big)\) be a unit-speed local parametrization of \(C\), so that \(a'(t)^2 + b'(t)^2 = 1\); by
the previous section the induced metric is \(dt^2 + a(t)^2\, d\theta^2\). The frame
\[
E_1 = \frac{\partial}{\partial t} , \qquad E_2 = \frac{1}{a}\, \frac{\partial}{\partial \theta}
\]
is orthonormal for this metric. Because the metric is flat, the
flatness criterion
provides, near each point, a commuting orthonormal frame \((\widetilde{E}_1, \widetilde{E}_2)\).
Replacing \(\widetilde{E}_2\) by \(-\widetilde{E}_2\) if necessary, which leaves the bracket
\([\widetilde{E}_1, \widetilde{E}_2]\) zero, we may assume \((\widetilde{E}_1, \widetilde{E}_2)\)
agrees in orientation with \((E_1, E_2)\). Any orthonormal frame so agreeing in orientation is
obtained from \((E_1, E_2)\) by a pointwise rotation, so we may write
\[
\widetilde{E}_1 = u E_1 + v E_2 , \qquad \widetilde{E}_2 = -v E_1 + u E_2 ,
\]
with \(u, v\) smooth functions of \((t, \theta)\) satisfying \(u^2 + v^2 = 1\).
Imposing \([\widetilde{E}_1, \widetilde{E}_2] = 0\) and expanding through the
coordinate formula for the Lie bracket
produces, after simplification, the two scalar equations
\[
u v_t - v u_t = 0 , \qquad v u_\theta - u v_\theta = a' ,
\]
where subscripts denote partial derivatives and \(a' = a_t\) since \(a\) depends only on \(t\).
The simplification uses \(u^2 + v^2 = 1\), which upon differentiation gives \(u u_t + v v_t = 0\)
and \(u u_\theta + v v_\theta = 0\). The first equation, paired with \(u u_t + v v_t = 0\),
forces \(u_t = v_t = 0\) wherever the frame is defined, so \(u\) and \(v\) are independent of
\(t\). The second equation then has a left side independent of \(t\), so \(a'\) is independent of
\(t\) as well; that is, \(a''(t) = 0\).
Therefore \(a(t) = \alpha t + \beta\) is an affine function of \(t\). The unit-speed condition
\(a'^2 + b'^2 = 1\) gives \(b'(t)^2 = 1 - \alpha^2\), a constant, so \(b'(t)\) is constant and
\(b(t) = \delta t + \varepsilon\) is affine as well. Hence \(\gamma(t) = (\alpha t + \beta,\,
\delta t + \varepsilon)\) traces a straight line, and since \(C\) is connected, all of \(C\) lies
on a single line. This completes the proof.
The sphere now follows at once. We saw that the unit sphere, away from its poles, is the surface of
revolution generated by a semicircle, and a semicircle is not part of any straight line.
Corollary: The Round Sphere Is Not Flat
The round metric on \(\mathbb{S}^2\) is not flat.
Proof:
The round metric on \(\mathbb{S}^2\) is the induced metric of the surface of revolution
generated by the semicircle \(\gamma(t) = (\sin t,\, \cos t)\), as computed in the previous
section. This generating curve is an arc of the unit circle, which is not contained in any
straight line. By the preceding theorem the induced metric is therefore not flat. Since
flatness is a local property and the poles are isolated points, the conclusion holds for the
round metric on all of \(\mathbb{S}^2\).
This is the obstruction promised when orthonormal frames were first introduced. No chart on the
sphere can have an orthonormal coordinate frame, because such a chart would exhibit the round metric
as locally Euclidean, which the theorem forbids. The failure is not a defect of any particular
coordinate system but an intrinsic feature of spherical geometry, and the quantity that measures it
precisely, the curvature, is the gateway to the deeper theory that begins where this development
ends. It is also the reason that constructions designed for flat domains do not transfer unchanged
to data that lives on a sphere: the sphere is not, even locally, a piece of the plane in disguise.
The Normal Bundle
A Riemannian submanifold carries not only its own tangent spaces but, at each point, the directions
of the ambient manifold that are perpendicular to it. Collecting these perpendicular directions
produces the normal bundle, a counterpart to the tangent bundle that records how the submanifold sits
inside its surroundings. The metric is what makes the construction possible, since perpendicularity
is a metric notion, and the same metric guarantees that the normal directions fit together smoothly.
The ambient manifold supplies, over each point of the submanifold, a tangent space larger than the
submanifold's own. Recall that for a submanifold \(S \subseteq M\) the restriction \(TM|_S\) is the
ambient tangent bundle,
whose fiber at \(p\) is the full ambient tangent space \(T_pM\), inside which the submanifold
tangent space \(T_pS\) sits as a subspace. The metric singles out a complement.
Definition: Normal Space and Normal Bundle
Let \((M, g)\) be an \(n\)-dimensional Riemannian manifold with or without boundary, and let \(S
\subseteq M\) be a \(k\)-dimensional Riemannian submanifold. A vector \(v \in T_pM\) is
normal to \(S\) at \(p \in S\) if it is orthogonal, with respect to \(g_p\), to
every vector in \(T_pS\). The normal space to \(S\) at \(p\) is the orthogonal
complement
\[
N_pS = (T_pS)^{\perp} \subseteq T_pM ,
\]
an \((n - k)\)-dimensional subspace, and the normal bundle of \(S\) is the
union
\[
NS = \bigsqcup_{p \in S} N_pS \subseteq TM\big|_S
\]
of all normal spaces, with the projection \(\pi : NS \to S\) sending each normal vector to its
base point.
When the ambient manifold is \(\mathbb{R}^n\) with its Euclidean metric, this reduces to the normal
bundle of a submanifold of Euclidean space already studied through the ambient dot product. The
Riemannian definition replaces that fixed product with \(g\), and the structural conclusion is the
same: the normal spaces assemble into a smooth subbundle.
Theorem: The Normal Bundle of a Riemannian Submanifold
Let \((M, g)\) be a Riemannian \(n\)-manifold with or without boundary, and let \(S \subseteq
M\) be an immersed \(k\)-dimensional Riemannian submanifold. The normal bundle \(NS\) is a smooth
rank-\((n - k)\)
subbundle
of the ambient tangent bundle \(TM|_S\). For each \(p \in S\), there is a smooth frame for \(NS\)
on a neighborhood of \(p\) that is orthonormal with respect to \(g\). When \(M = \mathbb{R}^n\)
with the Euclidean metric, this specializes to the statement that the
Euclidean normal bundle
is a subbundle.
Proof:
Fix \(p \in S\). On a neighborhood of \(p\) in \(S\) choose a smooth local frame \((X_1, \ldots,
X_k)\) for \(TS\). Regarded as sections of the ambient tangent bundle \(TM|_S\), these are
\(k\) smooth local sections that are linearly independent at each point, so by the
completion of local frames
they extend, on a possibly smaller neighborhood of \(p\), to a smooth local frame \((X_1,
\ldots, X_k, Y_{k+1}, \ldots, Y_n)\) for \(TM|_S\). This step takes place entirely within the
vector bundle \(TM|_S\) over \(S\), so it requires nothing of \(S\) beyond its being a
submanifold; the distinction between immersed and embedded does not enter. Applying the
Gram-Schmidt process for Riemannian frames
to this ambient frame, with the metric \(g\), yields a smooth orthonormal frame \((E_1, \ldots,
E_n)\) for \(TM|_S\) such that for each \(j\),
\[
\operatorname{span}\big(E_1|_q, \ldots, E_j|_q\big) = \operatorname{span}\big(X_1|_q, \ldots, X_j|_q\big) \quad (j \le k) .
\]
In particular \((E_1, \ldots, E_k)\) spans \(T_qS\) at each point \(q\), so it is an orthonormal
frame for \(TS\), while the remaining fields \((E_{k+1}, \ldots, E_n)\), being orthonormal and
orthogonal to all of \(E_1, \ldots, E_k\), take values in the orthogonal complement \(N_qS\) at
each point.
Because \(N_qS\) has dimension \(n - k\) and \((E_{k+1}, \ldots, E_n)\) consists of \(n - k\)
linearly independent vectors lying in it, these fields form a basis of \(N_qS\) at each \(q\).
Thus \((E_{k+1}, \ldots, E_n)\) is a smooth orthonormal frame for \(NS\) on a neighborhood of
\(p\), exhibiting the family of subspaces \(q \mapsto N_qS\) as locally spanned by smooth
sections of \(TM|_S\). By the
local frame criterion for subbundles,
\(NS\) is therefore a smooth rank-\((n - k)\) subbundle of \(TM|_S\). The orthonormal frame
just constructed is the asserted local orthonormal frame for \(NS\).
The tangent and normal bundles together account for the whole ambient tangent bundle along the
submanifold: at each point the orthonormal frame splits \(T_pM\) into \(T_pS\) and \(N_pS\), and
every ambient tangent vector decomposes uniquely into a part tangent to \(S\) and a part normal to
it. This decomposition is the starting point for relating the geometry of a submanifold to that of
its ambient space, the direction in which the study of curvature continues beyond the present
development.