I have a question on the notion of overtwisted discs. They are defined in many similar ways, for example by Hofer (p. 517, equation (3)) as a disc $D\subset \Sigma$ in a contact manifold $(\Sigma,\xi)$ so that $$T_x \partial D \subset\xi_x, \text{ and } T_x D\not\subset\xi_x \text{ for all }x\in\partial D,$$ or e.g. in this short script (page 3, definition 3) as a disc $D\subset\Sigma$, so that $$T_x D = \xi_x $$ at all $x\in \partial D$. However, it is also noted under this definition that this means that
The interior of $D$ must be transervsal to $\xi$ everywhere, and $\partial D$ must be tangent to $\xi$.
Now, I see how $\partial D$ being tangent to $\xi$ is the same as the first property in Hofer's definition, but I am wondering about everything else.
- Why doesn't Hofer just write $T_x D = \xi_x$?
- Couldn't $T_x D\not\subset\xi_x$ also be satisfied if $T_x D$ was transverse to $\xi_x$ at each $x\in\partial D$?
- Is the option mentioned in the second question not possible because of $T_x \partial D\subset\xi_x$? Why?
I feel very stupid asking this stuff, but it's my first time reading about contact geometry.