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 complimentary 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.

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.