Orientation.

Let $V$ be a real vector space of dimension $n$. Then define $\det(V) = \Lambda^n(V)$. This is a real, one dimensional vector space. So the set $\det(V) = \{0\}$ is disconnected. An orientation of the vector space $V$ is a choice of one of these connected components. If $X$ is an invertible linear map from $V$ to $V$ then it induces a linear map from $\det(V) \to \det(V)$ which is therefore multiplication by a complex number. This number is just $\det(X)$ the determinant of $X$ If $M$ is a manifold of dimension $n$ then the same applies to $M$; $\det(T_xM)- \{0\}$ is a disconnected set. We define

Definition 5.2   A manifold is orientable if there is a non-vanishing $n$-form on $M$. Otherwise it is called non-orientable.

If $\eta$ and $\zeta$ are two non-vanishing $n$ forms then $\eta = f \zeta$ for some function $f$ which is either strictly negative or strictly positive. Hence the set of non-vanishing $n$ forms divides into two sets. We have

Definition 5.3 (Orientation)   An orientation on $M$ is a choice of one of these two sets.

An orientation defines an orientation on each tangent space $T_xM$. We call an $n$ form positive if it coincides with the chosen orientation negative otherwise. We say a chart $(U, \psi)$ is positive or oriented if $d\psi^1\wedge \dots \wedge d\psi^n$ is positive. Note that if a chart is not positive we can make it so by changing the sign of one co-ordinate function so oriented charts exits. If we chose two oriented charts then we have that

$$\chi \circ \psi^{-1}_{\vert\psi(U\cap V)}$$

is an oriented diffeomorphism. The converse is also true.

Proposition 5.6   Assume we have a covering of $M$ by co-ordinate charts $\{(U_\alpha , \psi_\alpha )\}_{\alpha \in I}$ such that for any two $(U_\alpha, \psi_\alpha)$ and $(U_\b , \psi_\b )$ the diffeomorphism

$$\psi_\b\circ {\psi_\alpha ^{-1}}_{I\psi_\alpha (U_\alpha \cap U_\b )}$$

is orientation preserving. Then there is an orientation of $M$ which makes each all these charts oriented.

Proof. Notice that the fact that

$$\psi_\b\circ {\psi_\alpha ^{-1}}_{\vert\psi_\alpha (U_\alpha \cap U_\b )}$$

is an oriented diffeomorphism means that if $x\in U_\alpha \cap U_\b$ then

$$d\psi_\alpha ^1\wedge \dots d\psi_\alpha ^n (x)$$

is a positive multiple of

$$d\psi_\b ^1\wedge \dots d\psi_\b ^n (x)$$

Hence if $\phi_\alpha$ is a partition of unity then

$$\rho = \sum \phi_\alpha d\psi_\alpha ^1\wedge \dots d\psi_\alpha ^n (x)$$

is a non-vanishing $n$ form. Clearly this defines the required orientation. $\qedsymbol$