Take the 2-minute tour ×  Let $M$ and $N$ be smooth manifolds
and $f: M \to N$ a smooth map. Define the pullback bundle
$\pi_f^*:f^*(T^*N) \to M$ as usual by $ f^*(T^*N) =
\{(x,j^1_{f(x)}g) \in M \times T^*N \}$ with projection
$\pi^*_f(x,j^1_{f(x)}g)=x$ then the pullback of $f$ is a smooth
morphism $$f^*: f^*(T^*N) \to T^*M$$ and moreover a vector
bundle morphism over the identity defined by $f^*(x,j^1_{f(x)}g)
= j^1_{x}(g \circ f)$. Now the questions are:  1.) Suppose $f$
is an embedding. Is $f^*$ a surjective submersion?  2.) Suppose
$f$ is a surjective submersion. Is $f^*$ an embedding?  3.)
Suppose $f$ is a diffeomorphism. Is $f^*$ a diffeomorphism?
(Regarded $f^*$ just a smooth map, forgetting the additional
bundle structure)  4.) Suppose $f$ is a diffeomorphism. Is $f^*$
a vector bundle isomorphism?  share|improve this question  1
Answer 1  up vote 1 down vote accepted  Let me adopt slightly
different notation, because otherwise the symbol $f^\ast$ will
be overloaded. Fix $f: M \to N$, and let $E$ be