Orientation of submanifold

Hello,

I try to solve this problem:

Let M, N be orientable Manifolds (of dimension m,n) and f:M->N a smooth submersion. =>$\displaystyle G:=f^{-1}(p)$ is orientable.

Ok i have a proof, but it can't be correct, since the same "proof" would also work for any embedded submanifold.:

We know that $\displaystyle G$ is an embedded submanifold. And therefore we have an canonical atlas for G. If $\displaystyle (U_i,f_i)$ is an atlas for M, then$\displaystyle (U_i \cap S, \pi \circ f_i) $is an atlas for G, whereas $\displaystyle \pi $is the projection on the first (m-n) coordinates.

Now we have to show that det $\displaystyle d(\pi \circ f_i) \circ(\pi \circ f_j)^{-1})>0$

And this follows, since the f_i are orientable.(?) This can't be. (because not every submanifold is orient.)

But why is it >0?? We know also that the projection is in some nbh. equal to the map:

$\displaystyle g_j\circ f \circ (f_i)^{-1}$ whereas $\displaystyle g_j$ is a coordinate map of N and $\displaystyle f_i$ a coordinate maß of M.

Therefore we can put this forumula to the one above and get:

det $\displaystyle d( (g_j\circ f \circ (f_i)^{-1})\circ f_i) \circ((g_j\circ f \circ (f_i)^{-1}) \circ f_j)^{-1})>0$

Do you see, why this is greater than 0?

Regards