Induced Riemannian metric on Jet-Manifold

Question

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.

Answer 1

There are, of course, several different functorially induced metrics on $J^r(M,N)$ when $M$ and $N$ are endowed with given Riemannian metrics.

For example, $J^0(M,N)=M\times N$ and one can just take the product metric.

Meanwhile $J^1(M,N)$ can be regarded as a vector bundle over $M\times N$ with fiber $T^\ast_xM\otimes T_yN$ over $(x,y)\in M\times N$. The metrics on $M$ and $N$ induce an inner product on the bundle $T^\ast M\otimes TN$ (since you have innter products on each factor bundle separately), and the Levi-Civita connections on the two factors can be used to define a connection on this bundle. This information is enough to specify a metric on $J^1(M,N)$.

Added at the request of the OP: Given two bundles $E$ and $F$ over a base that are endowed with inner products and connections, the bundle $E\otimes F$ inherits both an inner product and a connection in the usual way: The inner product is the one for which the square length of $e\otimes f$ is just $\langle e,e\rangle_E\ \langle f,f\rangle_F$, and the connection is the tensor product connection. (You should look this up if you don't know it; I won't spell it out here.) Once you have a bundle $E$ with inner product $\langle,\rangle$ and connection $\nabla$ over a Riemannian manifold $M$, there is a unique metric on the total space $E$ that makes the horizontal space provided by the connection perpendicular to the fibers of $E\to M$, makes the projection to $M$ a Riemannian submersion, and restricts to each fiber to be the given inner product.

One can continue on this this way or just use an induction based on a certain natural inclusion of $J^{r+1}(M,N)$ into $J^1\bigl(M,J^r(M,N)\bigr)$ to finish the construction.

This construction is one choice among several possible ones, which I won't spell out here.