Orientable surfaces:

A smooth surface \(S\) embedded in \(\mathbb{R}^3\) is orientable if there exists a unit vector field \(\vec{N}(P)\) defined on \(S\) that varies continuously as \(P\) ranges over \(S\) and that is normal to \(S\) for any \(P \in S\). Any such vector field \(\vec{N}(P)\) determines an orientation of \(S\).

Given a particular choice of orienting unit normal field \(\vec{N}\), the surface must have two sides since \(\vec{N}(P)\) can have only one value at each point \(P\).

The Möbius strip is not orientable:

On the Möbius strip \(M\), a point \(P\) may be moved continuously so that it is always normal to the surface and arrives at its starting point in the opposite direction. It follows that for any choice of unit normal field \(\vec{N}\) on \(M\), for any \(P\) in \(M\) there are two possible values for \(\vec{N}(P)\) so \(M\) is not orientable.

A technical definition of Oriented manifolds via Jacobians, Charts and Atlases:

A smooth orientation of an n-dimensional smooth manifold \(M\) is the choice of a maximal smooth oriented atlas. A smooth atlas \(\{(U_i,\phi_i): i \in I \}\) is oriented if the Jacobian determinant of all coordinate transformations between distinct charts:

\begin{equation} \phi_i \circ \phi_j^{-1}: \phi_j(U_i \cap U_j) \rightarrow \phi_i(U_i \cap U_j) \end{equation}

is positive so \(\phi_i \circ \phi_j^{-1}\) is a diffeomorphism.

A smooth oriented atlas \(\{(U_{\alpha},\phi_{\alpha}):\alpha \in I \}\) is maximal if it contains every chart \((U_{\beta},\phi_{\beta})\) that defines a diffeomorphism \(\phi_{\beta} \circ \phi_{\alpha}^{-1}\) for any \(\phi_{\alpha}\) in the atlas. In other words, it can’t be enlarged by adding another smooth chart.

References:

  1. John Lee. Introduction to Smooth Manifolds. Springer. 2012.
  2. M. Lübke. Introduction to manifolds. 2018.