Suppose $(M,g)$ and $(N,g')$ are smooth Riemannian manifolds and $J^r(M,N)$ is the smooth manifold of $r$-jets $j^r_xf$ of smooth maps $f:M\to N$.

Is there an 'induced' Riemannian metric $g''$ on $J^r(M,N)$?

Of course the term 'induced' is conceptual vague here...

That's because, I don't want to restrict the question in a particular direction. However $J^r(M,N)$ gets other things induced from $M$ and $N$. For example atlases and chart transitions of $M$ and $N$ induces an atlas and chart transitions of $J^r(M,N)$.

In general and for any $m \in \mathbb{N}$, the jet $J^r(\cdot,\cdot)$ is a functor $J^r(\cdot,\cdot): \mathbf{M}_m \times \mathbf{M}\to \mathbf{M}$ from the product of the category of smooth $m$-dimensional manifolds and local diffeomorphisms with the category of smooth manifolds and smooth maps into the latter.

So 'maybe' with induced matric I mean, if $J^r(\cdot,\cdot)$ is still a functor from the product of the category of smooth $m$-dimensional riemannian manifolds with local (isometric) diffeomorphisms with the category of smooth riemannian manifolds and (ismometric) smooth maps into the latter.

I used parantheses around the term 'isometric' here because, the more general situation is of course the preferred one.