A Riemannian metric on a manifold $X$ defines a function on the symplectic space $T^\*X$ whose Hamiltonian flow gives geodesics. Is there a similar interpretation of the Levi-Civita connection?



Intuition for Levi-Civita connection via Hamiltonian flows
The intuition is that the Levi-Civita connection corresponds to the linearization of the geodesic flow plus a simple projective-geometric construction:


Let $c(t)$ be an orbit of the geodesic flow (projecting down to a geodesic), consider the vertical subspaces $V(t)$ along $c(t)$ and bring them back to the tangent space of the cotangent bundle over the point c(0) by using the differential of the flow. You get a family of (Lagrangian) subspaces $l(t) := D\phi\_{-t}(V(t))$ that is "fanning" or "regular". 


Now forget you ever had a geodesic flow: all that you need is the curve of subspaces. A bit of differential projective geometry shows that you also get a second curve $h(t)$ of (Lagrangian subspaces) in $T\_{c(0)}(T^\*M)$that is transversal to $l(t)$. The subspace h(0) is the horizontal subspace of the connection and $T\_{c(0)}(T^\*M) = l(0) \oplus h(0)$ is the decomposition into vertical and horizontal subspaces.


I think this is nicely written in [this paper](http://www.sciencedirect.com/science/article/pii/S0196885808001012) ;-) 


However, this is classical: it's just the geometry behind the Schwartzian derivative. There are plenty of references in the paper if you're interested in this. 


**Addendum on the geometry of the Schwartzian derivative.**


Since the Schwartzian derivative has many "standard" geometric interpretations, for completeness sake I'll sketch the (new?) one that I'm refering to. I will do it in a way just a bit more conceptual than was done by Duran and myself in the paper. Actually, it's just a "high brow" version of what was done in the paper, but I will do it here just for curves on the projective line. The reader may amuse himself/herself at extending this to curves on the Grassmannian of $n$-planes in $\mathbb{R}^{2n}$ and checking against the paper.


Consider the action of the linear group $GL(2;\mathbb{R})$ on the projective line and lift it to an action on its cotangent bundle. The moment map of this action takes values on the set of nilpotent matrices. In fact, if you take away the zero section, the moment map takes value in the coadjoint orbit formed by $2 \times 2$ nilpotent matrices of rank one. In higher dimensions, one needs to take away a bit more than the zero section and "rank one" is replanced by "the kernel and the image of the matrix (seen as linear transformation) are the same". 


We will also need a neat thing about the geometry of the tangent space of the projective line: there is a canonical way to identify non-zero tangent vectors to non-zero covectors. In fact, the tangent space of the projective line at a line $\ell$ is the space of linear maps between $\ell$ and the quotient space $\mathbb{R}^2/\ell$, which is the tensor product $(\mathbb{R}^2/\ell) \otimes \ell^\* $. If the map is invertible, its inverse is a map from $\mathbb{R}^2/\ell$ to $\ell$ and, therefore, an element of $ \ell \otimes (\mathbb{R}^2/\ell)^\*$, which is the cotangent space at $\ell$. 


Now consider a curve $\gamma(t)$ on the projective line whose derivative never vanishes. We lift the curve to the curve $\dot{\gamma}(t)$ on the tangent bundle and use the isomorphism described above to obtain a curve $\Gamma(t)$ on the cotangent of the projective line. Use the moment map to obtain a curve $F(t)$ of nilpotent matrices. Note that everything we have done is projective-equivariant. 


Finally we come to the little miracle: the time derivative of $F(t)$ is a curve of reflections $\dot{F}(t)$ (i.e., $\dot{F}(t)^2 = I$) whose -1 eigenspace is the curve of lines $\gamma(t)$ and whose $1$-eigenspace defines a "horizontal curve" $h(t)$ equivariantly attached to $\gamma(t)$. This is the construction that yields the Levi-Civita connection (and what is behind the formalisms of Grifone and Foulon for connections of second order ODE's on manifolds).


Differentiate $F(t)$ a second time to find the Schwartzian derivative. Geometrically, it just describes how the curve $h(t)$ moves with respect to $\gamma(t)$. For comparison, recall that the curvature of a connection is obtained by differentiating (i.e., bracketing) horizonal vector fields and projecting onto the vertical bundle.


