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.
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 so-called 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 Levi-Civita 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: $$\nabla_\gamma' \gamma'=0.$$