# Higher order differential on a manifold, connections

I am trying to understand how to define the second order differential of a map $f : M \rightarrow N$ between smooth manifolds.

I came across this exact same question : Higher-order derivatives in manifolds

The answers in the comments pointed towards jets, which I didn't know and shall learn about.I also thought that there should be a link with covariant derivatives : if you have a given covariant derivative on $M$, say the Levi-Civita connection of a certain Riemannian metric, isn't there any standard way to define higher order differentials, in a way that would yield a Taylor formula ? Or are jets really the only good way to look at these things ?

Any comment or reference appreciated. I know this is almost a duplicate (almost because I am asking also about the link with covariant derivatives, if there is one) but the original question seems inactive, so...

EDIT :

As I suspected, a torsion-free covariant derivative $\nabla$ allows one to define the second order differential. Indeed, a covariant derivative $\nabla$ allows one to differentiate not only sections of $TM$ (ie vector fields) but also sections of $TM^{\otimes p} \otimes T^*M^{\otimes q}$ along vector fields. So the second order differential of a function $f : M \rightarrow \mathbb{R}$ is defined by $$\mathrm{Hess}(f) = \nabla df$$ and it is a section of $S^2 T^*M$. Reference : Gallot Hulin Lafontaine

Related question : Definitions of Hessian in Riemannian Geometry

The treatment in the reference I found is rather quick though. I am betting that the exponential map will yield a Taylor formula to order 2 with this definition, and that a similar definition holds for a map between manifolds (you probably need a Riemannian metric in both manifolds). I will work out the details for myself, and may or may not answer my own question to close it. If someone finds a reference where the following points are treated (and not just mentionned !) :

• definition of all high order differentials in the context of Riemannian geometry

• Taylor formula with the exponential map

• case of a map between smooth manifolds