I am reading an article about the geometric foundation of Hamiltonian Markov Chain Monte Carlo (see https://projecteuclid.org/journals/bernoulli/volume-23/issue-4A/The-geometric-foundations-of-Hamiltonian-Monte-Carlo/10.3150/16-BEJ810.full) and I stumbled upon some lines which to my knowledge don't make sense. I would appreciate any clarifications on my potential misunderstanding.
From the paper:
A smooth fiber bundle, $\omega : Z \mapsto Q$, combines an $(n + k)$-dimensional total space, Z, an $n$-dimensional base space, $Q$, and a smooth projection, $\omega$ , that injectively maps, or submerses, the total space into the base space.
Question: I think it should be a surjective map not an injective. Is that correct?
Next part:
We will refer to a positively-oriented fiber bundle as a fiber bundle in which both the total space and the base space are positively-oriented and the projection operator is orientation-preserving
Question: How can the projection map be orientation preserving if the dimensionality of both spaces differ? I think orientation preserving maps are only possible for local diffeomorphisms.
Next part:
... while horizontal vector fields, X˜i, push forward to the tangent space of the base space
Again: pushforwards are - to the best of my knowledge - only defined for diffeomorphisms.
And finally:
Formally, volume forms are defined as positive, top-rank differential forms, $M(Q) = \{ \mu \in \Omega^n(Q)| \mu_q > 0, \forall \in Q \}$
What is the meaning of $\mu_q>0$. If I read it pointwise it doesn't make sense: How can a n-form be pointwise positive? If it is positive for a particular set of tangent vectors $v_1, \dots v_n$, then it will be negative, when using $-v_1, v_2, \dots v_n$.