§ 4. Differential Manifolds
265
Conversely, suppose that this condition holds; among all charts of X,
only consider those leading to the chosen coherent orientation of normals;
67
the previous arguments then show that the relation J θ > 0 holds for any
two of these charts, and the general definition of orientability given above is
recovered.
In the general case, which we now return to, consider two atlases (U i , ϕ i )
and (V p , ψ p ) with positive Jacobians. For x ∈ X, choose indices i and p
such that x ∈ U i ∩ V p ; let ε(x) = ±1 be the sign at the point x of the
Jacobian of the chart change ϕ i (x) → ψ p (x). It only depends on x, since if
x ∈ U j ∩ V q for another couple of indices, the Jacobians of the chart changes
(U i , ϕ i ) −→ (U j , ϕ j ) and (V p , ψ p ) −→ (V q , ψ q ) are positive by assumption.
The result is then an immediate consequence of the multiplication formula
for Jacobians. Having said this, observe that as the Jacobian of every chart
change (U i , ϕ i ) −→ (V p , ψ p ) is a continuous function on the open subset
U i ∩ V p , its sign remains constant in the neighbourhood of every x ∈ U i ∩ V p ;
hence ε(x) is a continuous function on X. The two atlases considered will be
said to define the same orientation (resp. opposite orientations) if ε(x) = +1
(resp. −1) for all x ∈ X. If X is connected, there is clearly no other possibility;
in other words, there are at most two ways of orienting a connected manifold.
Whether X is connected or not, an equivalence relation can be defined on
the set of all charts for X by setting two charts to be equivalent if and only
if the Jacobian of chart change is everywhere positive. This makes it possible
to divide the charts into equivalence classes, each of these classes being an
atlas of X with the following “ maximality ” property: given an atlas, if the
Jacobians of change of coordinates between a chart for X and those in the
atlas are all positive, then the chart belongs to it. Orienting a manifold then
consists in choosing one of these classes; the charts which belong to it will
then be said to be compatible with the orientation of X.
The simplest manifolds being Cartesian spaces, the orientation problem
already arises for them and, in the same way, for their open subsets; in this
case, there is a purely algebraic definition of orientation.
Indeed, let (a i ) be a basis for the Cartesian space E; this immediately
gives global chart for E (or for an open subset of E) by associating to each
x ∈ E its coordinates with respect to this basis; it could be called the linear
chart for E associated to the chosen basis. This chart being by itself an
atlas of E, the signs of all the Jacobians of its changes of charts are easy
to compute. . . Hence, E is orientable, and if this chart is used to orient E,
then the choice of a basis for E defines an orientation of E. Now let (b i ) be
another basis for E; there is another linear chart for E associated to it and
the latter can be obtained from the former by formulas known by everyone.
Hence the orientations defined by the two bases considered are identical or
67 If a chart (U, ϕ) does not satisfy this condition, it suffices to compose it with the
diffeomorphism (s, t) −→ (t, s) to obtain in the same open subset of X a chart
compatible with the orientation of the normals.
265
Conversely, suppose that this condition holds; among all charts of X,
only consider those leading to the chosen coherent orientation of normals;
67
the previous arguments then show that the relation J θ > 0 holds for any
two of these charts, and the general definition of orientability given above is
recovered.
In the general case, which we now return to, consider two atlases (U i , ϕ i )
and (V p , ψ p ) with positive Jacobians. For x ∈ X, choose indices i and p
such that x ∈ U i ∩ V p ; let ε(x) = ±1 be the sign at the point x of the
Jacobian of the chart change ϕ i (x) → ψ p (x). It only depends on x, since if
x ∈ U j ∩ V q for another couple of indices, the Jacobians of the chart changes
(U i , ϕ i ) −→ (U j , ϕ j ) and (V p , ψ p ) −→ (V q , ψ q ) are positive by assumption.
The result is then an immediate consequence of the multiplication formula
for Jacobians. Having said this, observe that as the Jacobian of every chart
change (U i , ϕ i ) −→ (V p , ψ p ) is a continuous function on the open subset
U i ∩ V p , its sign remains constant in the neighbourhood of every x ∈ U i ∩ V p ;
hence ε(x) is a continuous function on X. The two atlases considered will be
said to define the same orientation (resp. opposite orientations) if ε(x) = +1
(resp. −1) for all x ∈ X. If X is connected, there is clearly no other possibility;
in other words, there are at most two ways of orienting a connected manifold.
Whether X is connected or not, an equivalence relation can be defined on
the set of all charts for X by setting two charts to be equivalent if and only
if the Jacobian of chart change is everywhere positive. This makes it possible
to divide the charts into equivalence classes, each of these classes being an
atlas of X with the following “ maximality ” property: given an atlas, if the
Jacobians of change of coordinates between a chart for X and those in the
atlas are all positive, then the chart belongs to it. Orienting a manifold then
consists in choosing one of these classes; the charts which belong to it will
then be said to be compatible with the orientation of X.
The simplest manifolds being Cartesian spaces, the orientation problem
already arises for them and, in the same way, for their open subsets; in this
case, there is a purely algebraic definition of orientation.
Indeed, let (a i ) be a basis for the Cartesian space E; this immediately
gives global chart for E (or for an open subset of E) by associating to each
x ∈ E its coordinates with respect to this basis; it could be called the linear
chart for E associated to the chosen basis. This chart being by itself an
atlas of E, the signs of all the Jacobians of its changes of charts are easy
to compute. . . Hence, E is orientable, and if this chart is used to orient E,
then the choice of a basis for E defines an orientation of E. Now let (b i ) be
another basis for E; there is another linear chart for E associated to it and
the latter can be obtained from the former by formulas known by everyone.
Hence the orientations defined by the two bases considered are identical or
67 If a chart (U, ϕ) does not satisfy this condition, it suffices to compose it with the
diffeomorphism (s, t) −→ (t, s) to obtain in the same open subset of X a chart
compatible with the orientation of the normals.