In my class of Differential Geometry, the teacher defined geodesics as follows:
A regular curve on a regular surface, denoted as $\gamma:I\subset\Bbb{R}\to S$, ($S$ is the surface) is a geodesic if, $\forall t\in I$, the vector $\gamma"(t)$ is a normal vector to $S$ at the point $\gamma(t)$.
With this definition, I must prove for an exposition project that the möbius band can have geodesics.
The problem is that saying that a vector is normal to a surface implies orientation, and the möbius band is un-orientable.
So this is my question: How can i define geodesics on an un-orientable surface like the möbius band, and with that, how do i calculate them?
Can't the möbius band have any geodesics at all, because of its un-orientability?
If you can provide me a reference, i would apreciate it.
Best Answer
Normality is not dependent on orientation.
In $\mathbb R^3$, if I give you a plane $P$ and a vector $V$ based at a point of $P$, I can tell you whether $V$ is normal to $P$ without mentioning any orientation of $P$: $V$ is normal to $P$ if and only if $V \cdot W = 0$ for all vectors $W$ parallel to $P$. All I've used to formulate this definition is the (standard) inner product on the vector space $\mathbb R^3$.
You can now apply this principle at the point $\gamma(t)$, using the tangent plane $P = T_{\gamma(t)} S$ and the vector $V = \gamma''(t)$.