Given a smooth manifold $M$, there is a vector bundle over $M$, denoted $\tau M$, known as the second-order tangent bundle. The fiber $\tau\_mM$ at $m\in M$ is the collection of linear operators $A\_m:C^\infty(M)\rightarrow\mathbb{R}$ that satisfy


$$ A\_m(f^3)=3f(m)A\_m(f^2)-3f^2(m)A\_m(f) $$


for each $f\in C^\infty(M)$. It isn't too hard to show that each section of $\tau M$ is locally of the form


$$A\_m=a^i(m)\partial\_i+b^{ij}(m)\partial^2\_{ij},$$


where $a^i$ and $b^{ij}$ are smooth functions, $b^{ij}=b^{ji}$, $\partial\_i=\frac{\partial}{\partial x^i}$, $\partial^2\_{ij}=\frac{\partial^2}{\partial x^i\partial x^j}$, and I'm using the Einstein summation convention.


For each $m\in M$ There is a short exact sequence


$$T\_mM\rightarrow \tau\_m M\rightarrow T\_mM\odot T\_mM$$


where $\odot$ denotes the symmetric tensor product. The first arrow is the inclusion map that sends a vector at $m$ to its corresponding directional derivative operator $C^\infty(M)\rightarrow\mathbb{R}$. The second arrow is given by $A\_m\mapsto \hat{A}\_m$, where $\hat{A}\_m$ is defined by the formula


$$ \hat{A}\_m(\mathbf{d}\_mf,\mathbf{d}\_mg)=A\_m(fg)-f(m) A\_m(g)-g(m) A\_m(f), $$


and $f,g\in C^\infty(M)$. 


My question: One way to split the sequence, and thereby identify $\tau M$ with $TM\oplus(TM\odot TM)$, is to assign to each $m\in M$ a subspace $S\_m\subset \tau\_mM$ that is complementary to $T\_mM\subset \tau\_m M$. Is there a name for such an assignment?
I'd just like to know if there is an established name so that I can more easily search for what people already know about such things. 



Making the identification $\tau M\approx TM\oplus (TM\odot TM)$
Given a connection on the tangent bundle, you can define the second covariant derivative $\nabla^2f$ by
$$ \nabla^2f[X, Y] := \partial\_X\partial\_Y f - \partial\_{\nabla\_X Y} f.$$
Then $\nabla^2f$ is a symmetric tensor *provided that $\nabla$ was torsion-free*, which Robert pointed out but I missed in the first moment.


(This is because
$$\nabla^2f[Y, X] = \partial\_X\partial\_Y f + \partial\_{[Y, X]} f - \partial\_{\nabla\_Y, X} = \partial\_X\partial\_Y f - \partial\_{\nabla\_X, Y}+ T(X, Y),$$
where $T$ is the torsion tensor of nabla.)


Now given such a torsion-free connection, you can associate to an element polynomial $p \in TM \odot TM$ the operator 
$$Pf = \langle p, \nabla^2 f\rangle.$$
This gives a splitting of your sequence, since the principal symbol of $P$ will be $p$ again.


Conversely, if $S: TM \odot TM \longrightarrow \tau M$ is such a splitting, set $\Gamma$ to be the projection onto $TM$ along the image of $S$ (i.e. $\Gamma = \iota^{-1}(\mathrm{id} - S \circ \pi)$ with $\pi:\tau M \longrightarrow TM \odot TM$ the projection). Then 
$$\nabla\_X Y := \Gamma(\partial\_X \partial\_Y)$$
is a torsion-free connection.


