# Tangent space describes the manifold's first order characteristic. Is there something like tangent space describes higher order characteristic?

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.

• Jet bundle. Or, more generally, there are the Weil functors. Oct 17, 2014 at 10:14
• 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. Oct 17, 2014 at 15:05

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.
2. 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: $$\nabla_\gamma' \gamma'=0.$$