Let $M$ be a smooth $n$-manifold (which is not assumed to be orientable), and write $o(TM)\to M$ for its orientation bundle. Equivalently, it is the top exterior bundle $\Lambda^n(TM)\to M$. In any case, it is a real line bundle and is flat, where the locally constant transition functions are given by the sign of the Jacobian matrix of the transition functions of $TM\to M$. Let $\nabla^{o(TM)}$ be the corresponding flat connection.

My questions are:

- Is the flat connection on $o(TM)\to M$ unique (in some/any sense)? I saw in somewhere the following sentence "Let $\nabla^{o(TM)}$ be the natural flat connection on $o(TM)\to M$". I don't understand what natural means there.
- Is the flat connection $\nabla^{o(TM)}$ induced by some other connection? Let say I put a Riemannian metric on $M$, and denote by $\nabla^{TM}$ its Levi-Civita connection. What is the difference $\nabla^{o(TM)}-\nabla^{\Lambda^n(TM)}$?

Any reference related to the orientation bundle and its flat connection will be appreciated.