Pointwise and Continuous Orientations
A single vector space has two orientations, and on
\(\mathbb{R}^n\)
one of them is singled out by universal agreement. A manifold is a space whose tangent spaces are
all isomorphic to \(\mathbb{R}^n\) but are not canonically identified with it, so each tangent space
must be oriented separately, and the orientations of nearby tangent spaces need not have anything to
do with one another.
The new phenomenon, absent in the linear setting, is that a consistent choice across the whole
manifold may be impossible. On a sphere one can decide unambiguously which rotational sense is
counterclockwise by viewing the surface from outside. On a Möbius band a frame transported once
around the band returns as its mirror image, so no global choice of sense exists. Capturing the
difference between these two cases is the purpose of this page.
Definition: Pointwise Orientation
Let \(M\) be a smooth \(n\)-manifold with or without boundary. A pointwise
orientation on \(M\) is a choice, for each \(p \in M\), of an
orientation
of the tangent space \(T_pM\).
By itself a pointwise orientation carries no information tying neighboring points together. On
\(\mathbb{R}^n\) one could assign the standard orientation at points with rational first coordinate
and its opposite elsewhere, a perfectly well-defined pointwise orientation with no relation to the
smooth structure.
To make the assignment respect the smooth structure, we require that it vary continuously, expressed
through frames. Recall that a
local frame
\((E_1, \dots, E_n)\) on an open set \(U\) is an ordered \(n\)-tuple of vector fields whose values
form a basis of \(T_pM\) at each \(p \in U\). Since vector fields are continuous, such a frame
furnishes a continuously varying basis.
Definition: Positively and Negatively Oriented Frames
Let \(M\) be a smooth \(n\)-manifold with or without boundary, endowed with a pointwise
orientation, and let \((E_1, \dots, E_n)\) be a local frame on an open set \(U\). The frame is
positively oriented if \(\bigl(E_1|_p, \dots, E_n|_p\bigr)\) is a positively
oriented basis of \(T_pM\) at every \(p \in U\), and negatively oriented if it
is negatively oriented at every such point.
Definition: Orientation of a Manifold
Let \(M\) be a smooth \(n\)-manifold with or without boundary. A pointwise orientation on
\(M\) is continuous if every point of \(M\) lies in the domain of a positively
oriented local frame. A continuous pointwise orientation is called an
orientation of \(M\). The manifold \(M\) is orientable if it
admits an orientation and nonorientable otherwise. An
oriented manifold is a pair \((M, \mathcal{O})\) consisting of an orientable
manifold together with a choice of orientation \(\mathcal{O}\). For each \(p \in M\), the
orientation of \(T_pM\) determined by \(\mathcal{O}\) is written \(\mathcal{O}_p\).
The continuity condition is what fails for the Möbius band and holds for the sphere. For a
zero-dimensional manifold the definition degenerates gracefully. A point has tangent space
\(\{0\}\), whose orientation is a choice of \(\pm 1\), so an orientation of a \(0\)-manifold is, by
convention, any choice of \(\pm 1\) at each point. Every \(0\)-manifold is therefore orientable.
The Orientation Form
Checking continuity frame by frame is workable but clumsy. The
covector description of orientation
on a vector space suggests a global substitute. A single
differential form
of top degree, defined on all of \(M\) and vanishing nowhere, assigns an orientation to every
tangent space at once. The next proposition shows that this device captures orientability exactly.
Proof.
Suppose \(\omega\) is a nowhere-vanishing \(n\)-form. At each \(p\) the covector \(\omega_p\) is
a nonzero element of \(\Lambda^n(T_p^*M)\) and so
determines an orientation
of \(T_pM\). This defines a pointwise orientation, and only its continuity is in question. When
\(n = 0\) both directions are immediate. A nowhere-vanishing \(0\)-form is a nowhere-zero
function whose sign at each point is the orientation there, and continuity is vacuous.
Conversely, the function returning the sign \(\mathcal{O}_p\) at each \(p\) is smooth, because a
\(0\)-manifold is discrete. Assume \(n \geq 1\) from here on. Given \(p\), choose any local
frame \((E_i)\) on a connected neighborhood \(U\) of \(p\), with dual
coframe
\((\varepsilon^i)\). Since the top exterior power \(\Lambda^n(T_p^*M)\) is one-dimensional with
basis \(\varepsilon^1 \wedge \dots \wedge \varepsilon^n\) at each point, the form is
\(\omega = f\,\varepsilon^1 \wedge \dots \wedge \varepsilon^n\) for a continuous function \(f\),
and evaluating on the frame gives
\[
\omega(E_1, \dots, E_n) = f .
\]
Because \(\omega\) is nowhere zero, \(f\) is nowhere zero on \(U\). Since \(U\) is connected and
\(f\) is continuous, \(f\) has constant sign there. The frame is therefore positively oriented
on all of \(U\) when \(f \gt 0\), and negatively oriented when \(f \lt 0\). In the latter case,
replacing \(E_1\) by \(-E_1\) yields a positively oriented frame on \(U\). Either way \(p\) lies
in the domain of a positively oriented frame, so the pointwise orientation is continuous. It is
the unique orientation making \(\omega\) positively oriented, since the orientation of each
\(T_pM\) is forced by \(\omega_p\).
Conversely, suppose \(M\) is oriented. Cover \(M\) by smooth charts with connected domains. On
such a domain \(U_\alpha\), each point has a neighborhood carrying a positively oriented local
frame \((E_j)\), and there the determinant of the matrix \(\bigl(dx^i(E_j)\bigr)\) is continuous
and nowhere zero. Its sign records whether the coordinate frame is positively oriented, so it is
locally constant and hence constant on the connected set \(U_\alpha\). Accordingly exactly one
of the two smooth \(n\)-forms \(\pm\, dx^1 \wedge \dots \wedge dx^n\), which we call
\(\omega_\alpha\), is positively oriented at every point of \(U_\alpha\).
Let \(\{\psi_\alpha\}\) be a
partition of unity
subordinate to this cover and set \(\omega = \sum_\alpha \psi_\alpha \omega_\alpha\). Local
finiteness makes the sum finite near each point, so \(\omega\) is a smooth \(n\)-form.
At a point \(p\), every \(\omega_\alpha\) with \(\psi_\alpha(p) \gt 0\) is a positive multiple
of any fixed positively oriented \(n\)-covector at \(p\). Indeed, the fiber
\(\Lambda^n(T_p^*M)\) is one-dimensional, and all the \(\omega_\alpha(p)\) lie on the same open
half-line determined by the orientation. A sum of nonnegative multiples of vectors on one open
half-line of a line, not all zero, again lies on that half-line. Since
\(\sum_\alpha \psi_\alpha(p) = 1\), the coefficients are not all zero, so \(\omega_p\) is a
positive multiple of the chosen covector. Thus \(\omega\) is positively oriented, hence nowhere
vanishing, at every point.
A nowhere-vanishing \(n\)-form is accordingly called an orientation form, and the
orientation it determines is the one for which the form is positively oriented. If
\(\omega\) and \(\widetilde{\omega}\) are two positively oriented forms for the same orientation,
then \(\widetilde{\omega} = f\omega\) for a strictly positive smooth function \(f\), since at each
point the two are positive multiples of one another. For a \(0\)-manifold the same statement reads
as follows. A nowhere-vanishing \(0\)-form is a nowhere-zero real-valued function, and it assigns
the orientation \(+1\) where it is positive and \(-1\) where it is negative.
The orientation also singles out a class of charts. A smooth
coordinate chart
on an oriented manifold is positively oriented if its coordinate frame
\(\bigl(\partial / \partial x^i\bigr)\) is positively oriented, and negatively
oriented if that frame is negatively oriented. On a chart with connected domain the
coordinate frame is one or the other throughout, by the constant-sign argument just used, so every
connected chart is of exactly one type.
Oriented Atlases and Their Consequences
There is a third way to record an orientation, purely in terms of an atlas, and it is the one most
convenient for computation. Two positively oriented charts overlap in a way constrained by the
sign of a Jacobian, because the
change-of-coordinates law for a top-degree form
introduces exactly the determinant of the transition map. This motivates the following notion.
Definition: Consistently Oriented Atlas
A smooth
atlas
\(\{(U_\alpha, \varphi_\alpha)\}\) for a smooth manifold, or a smooth atlas of charts into
\(\mathbb{R}^n\) or \(\mathbb{H}^n\) for a
smooth manifold with boundary,
is consistently oriented if for each \(\alpha, \beta\) the
transition map
\(\varphi_\beta \circ \varphi_\alpha^{-1}\) has positive Jacobian determinant everywhere on
\(\varphi_\alpha(U_\alpha \cap U_\beta)\).
Proposition: Orientation Determined by an Atlas
Let \(M\) be a smooth positive-dimensional manifold with or without boundary. Any consistently
oriented smooth atlas for \(M\) determines a unique orientation for which each chart of the
atlas is positively oriented. Conversely, if \(M\) is oriented and either
\(\partial M = \varnothing\) or \(\dim M \gt 1\), the collection of all positively oriented
smooth charts is a consistently oriented atlas.
Proof.
Given a consistently oriented atlas, each chart determines a pointwise orientation on its domain
by declaring its coordinate frame positively oriented. Where two charts overlap, the transition
matrix between their coordinate frames is the Jacobian of the transition map, whose determinant
is positive by hypothesis. The two charts therefore induce the same orientation at each common
point. The pointwise orientation so defined is continuous, since each point lies in the domain
of an oriented coordinate frame, and it is the unique orientation making every chart positively
oriented.
Conversely, assume \(M\) is oriented. Each point lies in the domain of a smooth chart with
connected domain, obtained by restricting any chart to the preimage of a small ball around the
image of the point (intersected with \(\mathbb{H}^n\) for a boundary chart), chosen inside the
image of the chart. As noted after the definition of positively oriented charts, such a chart is
either positively or negatively oriented. If it is negatively oriented, one may replace the first
coordinate \(x^1\) by \(-x^1\) to obtain a positively oriented chart. The resulting charts cover
\(M\), so the collection of all positively oriented smooth charts is an atlas. Any two of them
have coordinate frames inducing the same orientation, so the determinant of their Jacobian is
positive, and this atlas is consistently oriented.
The replacement step deserves a comment when boundary charts are present. In a boundary chart
the last coordinate \(x^n\) is constrained to be nonnegative, while \(x^1\) is free. Negating
\(x^1\) therefore reverses the chart's orientation while keeping it a valid boundary chart,
provided \(x^1\) is not itself the constrained coordinate. This holds whenever \(\dim M \gt 1\),
and fails only for \(\dim M = 1\), where \(x^1 = x^n\) is the sole coordinate and cannot be
negated without leaving the half-space. Hence the stated hypothesis.
Except in dimension \(0\) and in dimension \(1\) with nonempty boundary, where the proposition just
proved does not supply a consistently oriented atlas for every orientation, the three descriptions
assembled so far are interchangeable presentations of the same structure. They are a continuous
choice of pointwise orientation, a nowhere-vanishing orientation form, and a consistently oriented
atlas. Each is preferred in a different situation: frames for pointwise reasoning, the form for
integration, the atlas for coordinate computation. Three standard consequences follow, recorded
here without full proofs since each is a direct application of one of these descriptions.
Proposition: Products, Components, and Codimension Zero
Let all manifolds below be smooth, with or without boundary.
(a) Products. If \(M_1, \dots, M_k\) are oriented manifolds without boundary,
then \(M_1 \times \dots \times M_k\) carries a unique product orientation for
which, whenever each \(\omega_i\) is a positively oriented orientation form for \(M_i\), the
form \(\pi_1^* \omega_1 \wedge \dots \wedge \pi_k^* \omega_k\) is positively oriented, where
\(\pi_i\) denotes the projection onto the \(i\)-th factor.
(b) Components. A nonempty connected orientable manifold has exactly two
orientations, and two orientations that agree at a single point are equal.
(c) Codimension zero. If \(M\) is oriented and \(D \subseteq M\) is a smooth
codimension-\(0\)
submanifold with or without boundary,
the orientation of \(M\) restricts to an orientation of \(D\). If \(\omega\) is an orientation form for \(M\), its pullback to \(D\)
under the inclusion is an orientation form for \(D\).
Proof Sketch.
For (a), the displayed wedge product is nowhere vanishing, because at each point of the
product, evaluating it on a basis assembled from bases of the individual tangent spaces
\(T_{p_i}M_i\) yields the product of the nonzero values of the factors \(\omega_i\) on those
bases. The wedge product is therefore an orientation form. Any two such forms differ by a
positive function, so the orientation class is independent of the chosen \(\omega_i\). The
factors are taken to have empty boundary. A nonempty product in which two factors have
nonempty boundary contains points at which two boundary coordinates vanish together, as at the
corner \((0, 0)\) of \([0, 1] \times [0, 1]\). Near such a point the product charts take
values in a product of two half-spaces, which is not an open subset of a half-space, so these
charts are not boundary charts and the product construction does not make such a product a
smooth manifold with boundary.
For (b), the difference between two orientations on a connected manifold is a locally constant
sign, hence constant by
connectedness.
Two orientations agreeing at one point agree everywhere, and the two global sign choices give
the only two orientations.
For (c), a codimension-\(0\) submanifold inherits the ambient coordinate frames at each of its
points, on which the restriction of an orientation form is again nowhere vanishing, so the
inclusion pulls an orientation form back to an orientation form.
Pullbacks, Parallelizability, and Lie Groups
At each point, a diffeomorphism between oriented manifolds either respects the two orientations or
reverses them, and in positive dimension its differential there detects which. This is the manifold
counterpart of a linear isomorphism carrying one basis class to the other, and it lets an
orientation be transported across a diffeomorphism.
Definition: Orientation-Preserving and Reversing Maps
Let \((M, \mathcal{O}_M)\) and \((N, \mathcal{O}_N)\) be oriented smooth manifolds of the same
dimension, with or without boundary, and let \(F : M \to N\) be a
local diffeomorphism.
At a point \(p\), the differential \(dF_p : T_pM \to T_{F(p)}N\) is a linear isomorphism, so it
carries the orientation \((\mathcal{O}_M)_p\) to one of the two orientations of \(T_{F(p)}N\).
The map \(F\) is orientation-preserving if this image is
\((\mathcal{O}_N)_{F(p)}\) for every \(p\), and orientation-reversing if it is
the opposite orientation for every \(p\). When \(M\) and \(N\) are \(0\)-manifolds the
differential carries no information. Instead, \(F\) is orientation-preserving if \(p\) and
\(F(p)\) carry the same sign \(\pm 1\) for every \(p\), and orientation-reversing if they carry
opposite signs.
Not every local diffeomorphism is one or the other, since it may preserve orientation on one
component and reverse it on another. On a connected domain, however, the two exhaust the
possibilities, by the constant-sign argument used throughout. The pullback of an orientation form
makes the same distinction analytically and supplies orientations on the domain of any local
diffeomorphism.
Proposition: Pullback of an Orientation
Suppose \(M\) and \(N\) are smooth \(n\)-manifolds with or without boundary and \(F : M \to N\)
is a local diffeomorphism. If \(N\) is oriented with orientation form \(\omega\), then the
pullback
\(F^*\omega\) is an orientation form on \(M\), and the orientation it determines depends only on
the orientation of \(N\), not on the choice of \(\omega\). This orientation is the unique one
making \(F\) orientation-preserving, and it is called the orientation pulled back
by \(F\).
Proof.
Since \(F\) is a local diffeomorphism, each differential \(dF_p\) is an isomorphism, so the
pullback covector \((F^*\omega)_p = (dF_p)^* \omega_{F(p)}\) is nonzero whenever
\(\omega_{F(p)}\) is. Thus \(F^*\omega\) is nowhere vanishing and is an orientation form on
\(M\). For an ordered basis \((v_1, \dots, v_n)\) of \(T_pM\),
\[
(F^*\omega)_p(v_1, \dots, v_n) = \omega_{F(p)}\bigl(dF_p(v_1), \dots, dF_p(v_n)\bigr),
\]
so the basis is positively oriented for \(F^*\omega\) exactly when its image under \(dF_p\) is
positively oriented for \(\omega\). Hence the orientation determined by \(F^*\omega\) is the one
for which \(dF_p\) carries positively oriented bases to positively oriented bases, that is, the
unique orientation making \(F\) orientation-preserving. Replacing \(\omega\) by \(f\omega\) for
a positive smooth function \(f\) on \(N\) replaces \(F^*\omega\) by \((f \circ F)\, F^*\omega\),
again a positive multiple, and so leaves the orientation unchanged.
A recurring way to certify orientability bypasses forms and atlases entirely. It is enough to
produce a single global frame. A manifold admitting one is
parallelizable,
and parallelizability is a strictly stronger condition than orientability.
Proposition: Parallelizable Manifolds Are Orientable
Let \(M\) be a parallelizable smooth \(n\)-manifold with or without boundary. Then \(M\) is
orientable, and if \(n \geq 1\), a global frame \((E_1, \dots, E_n)\) for \(M\) determines the
orientation in which that frame is positively oriented at every point.
Proof.
If \(n = 0\), then \(M\) is orientable because every \(0\)-manifold is. Assume \(n \geq 1\). A
global frame is in particular a positively oriented local frame on all of \(M\) once its own
class is declared positive, so the pointwise orientation it defines is continuous everywhere.
Equivalently, its dual coframe \((\varepsilon^i)\) gives the nowhere-vanishing orientation form
\(\varepsilon^1 \wedge \dots \wedge \varepsilon^n\). The converse fails. Orientability does not
return a global frame, as the even-dimensional spheres of positive dimension illustrate. Those
spheres are orientable, yet not parallelizable, since they admit no nowhere-vanishing vector
field at all. That these spheres admit no nowhere-vanishing vector field is the hairy ball
theorem, which we state without proof.
The most important parallelizable manifolds for what follows are the Lie groups, where, in positive
dimension, a global frame is produced by left-translating a basis of the tangent space at the
identity. Orienting this frame positively gives an orientation that every left translation
preserves, determined by the orientation of that tangent space given by the basis, and this
construction is our entry point for invariant integration.
Proposition: Lie Groups Are Orientable
Let \(G\) be a Lie group
with identity \(e\), and call an orientation of \(G\) left-invariant if, for
every \(g \in G\), the left translation \(L_g : G \to G\), \(h \mapsto gh\), which is a
diffeomorphism with inverse \(L_{g^{-1}}\), is orientation-preserving. Then \(G\) is orientable
and has precisely two left-invariant orientations, and the assignment
\(\mathcal{O} \mapsto \mathcal{O}_e\) is a bijection from them onto the two orientations of the
tangent space \(T_eG\). When \(G\) has positive dimension, a basis of \(T_eG\) extends by left
translation to a left-invariant global frame,
and the orientation in which this frame is positively oriented is the left-invariant
orientation corresponding to the orientation of \(T_eG\) determined by that basis.
Proof.
If \(G\) has dimension \(0\), it is orientable because every \(0\)-manifold is. Suppose now
that \(G\) has positive dimension \(n\), and let \((v_1, \dots, v_n)\) be a basis of \(T_eG\).
The evaluation isomorphism
supplies left-invariant vector fields \(X_i\) with \((X_i)_e = v_i\). Since
\((X_i)_g = d(L_g)_e(v_i)\) and \(d(L_g)_e\) is an isomorphism, \((X_1, \dots, X_n)\) is a
left-invariant global frame, so \(G\) is
parallelizable
and hence orientable by the previous proposition, with the orientation in which this frame is
positively oriented. The value of this orientation at \(e\) is the class of
\((v_1, \dots, v_n)\), since the frame evaluated at \(e\) is that basis. Because each left
translation carries the frame to itself, its differential sends the positively oriented frame
at one point to the positively oriented frame at the image point, so left translation preserves
the orientation, which is therefore left-invariant. The construction depends on a choice of
basis at the identity, but any two choices related by a positive determinant give the same
orientation.
Finally, a left-invariant orientation is determined by its value at the identity alone. In
positive dimension, left-invariance forces the orientation at each \(g\) to be the image of
\(\mathcal{O}_e\) under the differential of \(L_g\) at \(e\). In dimension \(0\), it forces
each \(g\) to carry the same sign as \(e\), since \(L_g\) carries \(e\) to \(g\). Either way
the orientation is recovered from \(\mathcal{O}_e\) throughout \(G\). Conversely, each of the
two orientations of \(T_eG\) is the value at \(e\) of a left-invariant orientation, given in
positive dimension by the frame construction above applied to a basis in that class, and in
dimension \(0\) by the constant sign. Hence \(G\) has precisely two left-invariant
orientations, in bijection with the orientations of \(T_eG\).
With a left-invariant orientation of a Lie group fixed, a left-invariant orientation form for it is
obtained by left-translating a nonzero top-degree covector at \(e\) that is positive for the
orientation of \(T_eG\), and it is unique up to a positive constant factor. No metric is needed.
Such a form is the geometric object whose
integral is invariant under left translations, and it is the foundation on which averaging over a
compact group is built.