A Morse function $f: M \rightarrow \mathbb R$ on a connected closed manifold $M$ is called $\mathit{perfect}$ with respect to the field $\mathbb F$ if all of the Morse inequalities are equalities, i.e. the number of critical points of $f$ with index $k$ coincides with the $k$-th Betti number of $M$ with respect to $\mathbb F$-coefficients for all $k$.


Now assume that $f: M \rightarrow \mathbb R$ is a perfect Morse function on a closed connected Riemannian manifold and that $N \subset M$ is a closed submanifold, such that $N$ contains all the critical points of $f$ and the restriction $f|\_N$ is a Morse function on $N$. If in addition the gradient $\nabla f$ is tangent to $N$ along $N$, i.e. $\nabla f|\_N \in \Gamma(TN)$, then $N$ is a union of flow lines and the critical points of $f|\_N$ are exactly the critical points of $f$. 


Now my question is: Is it possible to deduce perfectness of $f|\_N$ from perfectness of $f$ in this setting, or are there additional conditions under which this is possible? 



Restrictions of perfect Morse functions to submanifolds
Isn't there a counterexample with $(M,N)=(\mathbb CP^2,\mathbb RP^2)$?


