# Concrete definition of nonlinear differential equation on a Riemannian Manifold

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!

• 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. Aug 5 at 20:55