Let M be a compact manifold on wich acts a torus T. Denote $M^T$ the invariant subset of M by the action of T.
Let F be a submanifold of $M^T$. Denote N the normal bundle of $M^T$ in M restricted to F.
Let $X \in \mathfrak{t}=Lie(T) $. Denote $ M(X_{|M})$, the zero set in M of the fundamental vector field of X, and denote by $ N(X_{|N})$ the zero set in N of the fundamental vector field of X.
Why this implication is true:
$N(X_{|N})\neq F \Rightarrow $ the inclusion $M^T \subset M(X_{|M})$ is strict ?
Thanks!