Higher-Order Differential Operators as Vector Fields On a $C^\infty$ manifold $M$, one can produce the tangent space $T_p M$ at a point by equivalence classes of tangents to smooth curves through the point $p$.
When realised this way, the tangent vectors are local derivations of functions defined at $p$. Globally,then, vector fields on $M$ act on global functions as differential operators. Adapted to some chart $(U, x^1, \ldots, x^n)$, a vector field $V$ looks like
$$
\sum_{i=1}^{n} v^i \frac{\partial}{\partial x^i}
$$
where the $v^i$ are functions on $U$. A lot of stuff comes from this, Lie theory, etc. All this is ok for me.
But what about, say, acceleration vectors of curves through a point? These have the same geometrical interpretation that velocity vectors do. I can construct a curve in some local coordinates with any arbitrary acceleration that I like. The resulting space has the same dimension as the tangent space, too. Presumably one could also extend such things to global objects. With some fooling around, these can be made to correspond (locally) to expressions of the form
$$
\sum_{i=1}^{n} v^i \frac{\partial^2}{\partial^2 x^i}?
$$
Going in the other direction, note that things of the above type close under the same Lie bracket as regular vector fields. Physically, acceleration fields look like forces, so one might expect such a thing to be physically meaningful -- yet I've never heard of them.
Obviously this all carries on to derivative vectors of all orders. Do these sit in some enveloping algebra for vector fields? Some multivector construction?
I would greatly appreciate any references, please, if you have them.
 A: The velocity of a particle on a curved space is a covariant notion but acceleration isn't. They appear to be very similar concepts in flat space because we represent them in the same way, but on curved spaces they are very different - not because they are represented in different ways: they are both tangent fields along the curve of travel - but they are obtained in different ways.
Consider a moving particle on a manifold, it traces out a curve $c$. This is simply a map from the real line to the manifold and so we can take it's derivative directly via the tangent functor. Now if it were to return another map from the real line to the manifold we could take the derivative again and so obtain the acceleration. However this is not the case. What is returned is a tangent field along the curve $c$. We cannot derive this directly by the tangent functor. What we need to use is the covariant derivative. But this requires a connection.
This is rather like gauging a theory with a global symmetry to a local symmetry in particle physics (ie QED) which forces the introduction of a connection.
Once we've introduced the connection, we can talk about curvature. This is basically a manifestation of Einstein's equivalence principle: to talk about acceleration is to talk about a connection which is to talk about curvature which is to talk about gravity.
