I'm learning differential geometry. I'm curious that when we learned analysis, we learned higher order derivative, while in differential geometry, first order derivative is generalized to element of tangent space. Then my question is what's the higher order derivative in differential geometry? Are there some literatures including this area? Or maybe this is just a trivial question because there is nothing interesting in higher order derivative.

1$\begingroup$ Jet bundle. Or, more generally, there are the Weil functors. $\endgroup$– user40276Oct 17, 2014 at 10:14

$\begingroup$ The derivative is on the tangent bundle $df : TM \to TN$, so the 2nd order derivative is a map of the tangent bundle of the tangent bundle, $d^2f : T^2M \to T^2N$. And so on. $\endgroup$– Ryan BudneyOct 17, 2014 at 15:05
1 Answer
There are several possibilities:
1.The most naive one is the following: Consider a smooth map $f\colon M\to N.$ Then its derivative $df\colon TM\to TN$ is again smooth, and you can again differentiate. Clearly, you take a lot of useless information around, so this is usually not the method used by differential geometers to solve any problems.
Another possibility is by defining socalled jet bundles in a similar way for higher order derivatives as the tangent bundle is defined for first order derivatives. There are many books on this, check for example wikipedia. The "problem" with jet bundles is, that they are no vector bundles, and one somehow loses the (mulit)linear algebra tools..
Differential geometer often use connections on vector bundles to define higher order derivatives. As the most basic example, one should mention the LeviCivita connection $\nabla$ (defined by a Riemannian metric on the manifold). With its help you can derivate vector fields along vector fields. A nice and simple application of its use is the following: (Locally) shortest curves are exactly those curves which satisfy the second order differential equation: $$\nabla_\gamma' \gamma'=0.$$

1$\begingroup$ Concerning your remark 2: multilinear algebra is not lost with jet bundles since jets of consecutive order are affine bundles over symmetric multilinear tensors. $\endgroup$ Oct 17, 2014 at 17:04

$\begingroup$ @Michael: You are of course right, that one can recover the multilinear algebra. Nevertheless, this is not totally transparent, and quite often people prefer to use connections in order to work with higher order derivatives. $\endgroup$ Oct 20, 2014 at 7:49