Let M,N be real smooth manifolds and p:M-->N a smooth map. Then smooth functions on M form a module over the ring of smooth functions on N (via pullback). Is it know whether this module is flat when p is a submersion or fiber bundle?

Recall that flatness is equivalent to the following: whenever h<sub>1</sub> ... h<sub>k</sub> are smooth functions on N and g<sub>1</sub> ... g<sub>k</sub> are smooth functions on M such that:

h<sub>1</sub>g<sub>1</sub> + ... + h<sub>k</sub>g<sub>k</sub> = 0 (as function on M)

then there are functions G<sub>1</sub> ... G<sub>r</sub> on M and a<sub>i,j</sub> on N such that:

g<sub>i</sub>= &Sigma;<sub>j</sub> a<sub>i,j</sub>G<sub>j</sub>  for all i

and &Sigma;<sub>i</sub> h<sub>i</sub> a<sub>i,j</sub>= 0  for all j.