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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

I will gladly accept hish/her answer.

Community
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Albert
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1 Answers1

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The answer is that the data of a manifold just really does not carry higher order information. Choosing different charts maintains the class of differentiable functions, and allows you to define first derivatives consistently as derivations of the algebra of smooth functions, but there really is not such a thing as a second derivative all on its lonesome. Jets are the best thing we have as a stand in, given the data of a smooth manifold.

Steven Gubkin
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