I think that a good way to understand the Levi-Civita connection is to say that is is the Ehresmann connection in $TTM$ obtained from the linearization of the geodesic flow by a natural geometric construction.

I described this construction in my answer to this MO question, but I'll do so again with some improvements.

**Dynamic construction.**

Let $c(t)$ be an orbit of the geodesic flow in $TM$, consider the vertical subspaces $V(t)$ in $TTM$ 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 $l(t) := D\phi_{-t}(V(t))$ in the symplectic vector space $T_{c(0)}TM$.

Now forget you ever had a geodesic flow: all that you need is the curve of subspaces. A bit of differential projective geometry---described below---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.

**Projective construction.**

Now I'll describe as succintly as possible the projective-geometric construction
that underlies both the Levi-Civita connection and the Schwartzian derivative.
For the detais of what follows see this paper What's new in the description here is that I *explicitly* use the Springer resolution (Duran and I used implicitly in the paper).

First we need two remarks on the geometry of the Grassmannian $G_n(\mathbb{R}^{2n})$ of $n$-dimensional subspaces in $\mathbb{R}^{2n}$

**1.** The tangent space of $G_n(\mathbb{R}^{2n})$ at a subspace $\ell$
is canonically identified with the space of linear maps from $\ell$ to $\mathbb{R}^{2n}/\ell$ or, equivalently, with the space $(\mathbb{R}^{2n}/\ell) \otimes \ell^*$. Since $\mathbb{R}^{2n}/\ell$ and $\ell$ have the same dimension, we may distinguish a class of differentiable curves $\gamma$ on the Grassmannian by requiring that at each instant $t$ their velocities are invertible linear maps from $\gamma(t)$ to $\mathbb{R}^{2n}/\gamma(t)$. These curves are called *fanning* or *regular*.

Using that the cotangent space of $G_n(\mathbb{R}^{2n})$ at a subspace $\ell$
is canonically isomorphic to $\ell \otimes (\mathbb{R}^{2n}/\ell)^*$, we can lift
every fanning curve $\gamma(t)$ to a curve on the cotangent bundle of the Grassmannian by $t \mapsto (\dot{\gamma}(t))^{-1}$.

**2.** Consider the action of the linear group $GL(2n;\mathbb{R})$ on the Grassmannian $G_n(\mathbb{R}^{2n})$ and lift it to an action on its cotangent bundle. The moment map of this action takes values on the set of nilpotent matrices.

Now consider a fanning curve $\gamma(t)$ on the Grassmannian $G_n(\mathbb{R}^{2n})$ and lift it to the curve $(\dot{\gamma}(t))^{-1}$ on its cotangent bundle. Use the moment map to obtain a curve $F(t)$ of nilpotent matrices. Note that everything we have done is $GL(2n,\mathbb{R})$-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 subspaces $\gamma(t)$ 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.

Semi-Riemannian Geometryon page 61. $\endgroup$7more comments