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On $\mathbb{R}^n$ one can write a general nonlinear $k$th order differential equation as $F(x,\partial^{\alpha}u)=0$, where $u\colon \mathbb{R}^n\to \mathbb{R}$ and $\alpha$ ranges over the multi-indices of length $\leq k$. I am interested in verifying a way to think about these operators in a coordinate invariant way. In Lee's Smooth Manifolds, Ch. 22, this is done for first order equations on manifolds by considering $F$ to be a function on the 1-Jet bundle, defined as the Whitney sum $(M\times \mathbb{R})\oplus(T^* M)$, and defining $u$ to be a solution if $(u,du)\in \Gamma(J^1N)$ lies in the set $F^{-1}(0)$.

In trying to extend this to higher order operators but have struggled to find introductory-level material on Jet bundles.

As I understand it, an obstacle to extending the $k=1$ definition to higher orders is the lack of an invariant definition of a Hessian. However, if we also endow $M$ with the structure of a Riemannian Manifold, I think one can extend the $k=1$ construction by considering the "$k$-Jet bundle'' to be $$ J^kM:=\bigoplus_{l=0}^k T^{l}T^*M $$ and a $k$th order non-linear differential operator to be a function $F\colon J^kM\to \mathbb{R}$. We would then define $u\in C^{\infty}(M)$ to be a solution to the equation associated to $F$ if the section $(u,\nabla u,\nabla^2 u,\cdots,\nabla^{k}u)\in \Gamma(J^{k}M)$ lies in $F^{-1}(U)$, where $\nabla$ is the total covariant derivative.

I am wondering if anyone knows of references which define nonlinear operators in this way and/or if this agrees with other ways of defining nonlinear operators. Thanks!

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    $\begingroup$ Higher order PDEs are indeed hard to work with on a manifold. Jet bundles are awkward to use. Writing the PDE with respect to the Levi-Civita connection is better but you still have to be careful since "partial derivatives" no longer commute. $\endgroup$
    – Deane
    Aug 5 at 20:55

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