On a Riemannian manifold $M$, there is a canonical horizontal lift $X^{\mathrm{hor}}$ of vector fields $X$ to $TM$, which is characterized by the two properties that

- $X^{\mathrm{hor}}$ is a horizontal vector
- $d \pi (X^{\mathrm{hor}}) = X$.

I wonder how one would define the horizontal lift of higher order differential operators. For example they should satisfy the property $$ P^{\mathrm{hor}} (f u) = f P^{\mathrm{hor}} u$$ for any function $u \in C^\infty(TM$) and any $f$ of the form $f(X) = \tilde{f}(|X|)$ for some function $\tilde{f} \in C^\infty(\mathbb{R})$. One would like to say that $P^\mathrm{hor}$ maps functions that are "constant in horizontal directions" to zero but that doesn't make sense as the horizontal distribution is never integrable unless $M$ is flat (in which case, obviously, we have a nice definition of the horizontal lift of any differential operator).

The problem is clearly that the horizontal lift does not induce an algebra homomorphism on the algebra of differential operators because if two vector fields commute on $M$, then their horizontal lift usually does not commute. So there must be some kind of "quantization" going on.

More specifically: It is clear that there are probably many choices to lift differential operators, but it would be nice to list some properties in the line of those stated above that characterize a horizontal lift of any differential operator uniquely.