Is the induced Fisher information metric equal to the Fisher metric of a submanifold?

Question

If $M = \{p_\theta : \theta \in \Theta \subset \mathbb R^d \}$ is a statistical manifold parametrized by $\theta$, with the Fisher information metric \begin{equation*} g_{ij}(\theta) = \int_{\mathcal X} p_\theta(x) \partial_{i}\log(p_\theta(x)) \partial_{j}\log(p_\theta(x)) dx \end{equation*} and we take a submanifold $N \subset M$, then $N$ can be considered as a Riemannian submanifold with the induced metric. But if $N$ is a submanifold parametrized by some parameters $\eta \in H\subset \mathbb R^k$ with $k\le d$, then there is another construction, which is the Fisher metric of $N$ with parameters $\eta$. My question: are these two metrics the same?