7 Orientation

Definition 7.1 (Orientation of vector space) If \(V\) is a vector space of dimension \(n\), the set of all bases of \(V\) is divided in two equivalence classes by the following equivalence relation: \[ \begin{pmatrix} a_1 & \cdots & a_n \end{pmatrix} \sim \begin{pmatrix} b_1 & \cdots & b_n \end{pmatrix} \] if the change of basis matrix \(A\) \[ \begin{pmatrix} a_1 & \cdots & a_n \end{pmatrix} A = \begin{pmatrix} b_1 & \cdots & b_n \end{pmatrix} \] has a positive determinant.

Definition 7.2 (Pointwise orientation) Suppose \(M\) is a manifold, a pointwise orientation \(\mu\) assigns each point \(p \in M\) an orientation of its tangent space \(T_pM\). Since orientations are equivalent classes of bases, the pointwise orientation can be represented as a set of vector field \(X_1, \dots, X_n\) on \(M\) \[ \mu = [X_1, \dots, X_n] \] such that \(\mu(p)\) is the orientation specified by the basis \(X_1(p), \dots, X_n(p)\).

Definition 7.3 (Continuous orientation) A pointwise orientation \(\mu\) of a manifold is said to be continuous at \(p \in M\) of dimension \(n\) if it can be represented by a collection of continuous vector fields \(X_i\) defined at a neighborhood of \(p\) \[ \mu = [X_1, \dots, X_n] \] where \(X_1, \dots, X_n\) makes up a basis for the tangent space of any point in the neighborhood of \(p\).

A manifold is orientable with orientation \(\mu\) if \(\mu\) is continuous everywhere.

7.1 Orientation by differential forms

Theorem 7.1 A pointwise orientation \([X_1, \dots, X_n]\) on a manifold \(M\) is continuous at \(p\) if and only if everywhere there is a local coordinate \((U, x^1, \dots, x^n)\) such that \[ dx^1 \wedge \dots \wedge dx^n (X_1, \dots, X_n) > 0 \] on \(U\).

Theorem 7.2 A manifold \(M\) of dimension \(n\) is orientable if and only if there exists a nowhere vanishing smooth \(n\)-form.

Proof. Apply partition of unity on the previous theorem.

7.2 Orientation by atlas

Definition 7.4 (Oriented altas) An atlas \(\{(\phi)\}\) is oriented if the Jacobian of the change of coordinates \[ \psi \circ \phi^{-1} \] is everywhere positive.

Theorem 7.3 An atlas \(\{(x^1, \dots, x^n)\}\) is oriented if and only if there exists a orientation \(n\)-form \(\omega\) such that \[ \omega\left ( \frac{\partial}{\partial x^1}, \dots, \frac{\partial}{\partial x^n} \right ) > 0 \] written in local coordinates.