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A reference for this material is Chapter 15 of John M. Lee. Introduction to smooth
manifolds. Second edition. Grad. Texts in Math., 218. Springer, New York, 2013.
xvi+708pp. ISBN: 978-1-4419-9981-8.
Orientation Let \(M\) be a smooth manifold. An atlas \(\mathcal {A} \) is said to be oriented if for each \( (U, \varphi ) , (V, \psi ) \in \mathcal {A}\),
the change of coordinates map
\[ \psi \circ \varphi ^{-1} : \varphi (U \cap V ) \to \psi (U \cap V ) \]
is orientation preserving. That is, the matrix \( d_p (\psi \circ \varphi ^{-1} ) \) has
positive determinant for all \(p \in \varphi (U \cap U ) \).
An orientation is a maximal oriented atlas.
A smooth manifold is called oriented if it is equipped with an orientation.
A smooth manifold is called orientable if it admits an orientation.
Orientation compatible In a manifold \(M\) equipped with an orientation \(\mathcal {A}\), we say a chart \((U, \varphi )\)
is orientation compatible if for all \((V, \psi ) \in \mathcal {A}\), the change of coordinates map between \(\varphi \) and \(\psi \) is
orientation preserving.
Orientation form Let \(M\) be a smooth oriented \(n\)-dimensional manifold and \(\omega \in \Omega ^n (M)\). We say that \(\omega \)determines the orientation if for any chart \((U , \varphi )\) compatible with the orientation,
