I'm interested in finding a global (coordinate-free) description of the Levi-Civita connection on a (possibly infinite-dimensional) Riemannian manifold X.
I'm not looking for a description of this object as a differential operator.
Instead, I'm looking for a splitting of the natural map
$\alpha = (\pi_{T(TX)}, D\pi_{TX}): T(TX) \to TX \oplus TX$, where $\pi_{T(TX)}$ is the structure map of the double tangent bundle and $D\pi_{TX}$ is the map on tangent bundles induced by the structure map $\pi_{TX} : TX \to X$.
Noting that $TX \oplus TX= (\pi_{TX})^* (TX)$, a splitting of $\alpha$ is the analogue, for the vector bundle $TX$, of the standard notion of a connection on a principle bundle: it's a way of lifting tangent vectors on the manifold $X$ up to tangent vectors on the bundle $TX$.
Lang (in GTM 160, Differential and Riemannian Manifolds) explains how to obtain this splitting using the metric spray, which is a map $F: TX \to T(TX)$ that splits both of the above maps $T(TX) \to TX$ (and satisfies another "quadratic" condition). Lang gives a global description of $F$ as the vector field on $TX$ corresponding, under the metric, to the 1-form -dK, where $K(v) = (1/2)\langle v,v\rangle$ is the kinetic energy functional on $TX$. However, he doesn't really give a coordinate-free extension of F to the desired splitting. From studying the discussion in Lang, it seems to me that there is a unique splitting $H: T(TX) \to TX \oplus TX$ satisfying $F(v+w) = (F(v) + H(w,v)) + (F(w) + H(v,w))$ and such that in any local chart U on X, H has the form $H(x, v, w) = (x, v, w, B(x, v, w))$ (as a map $U\times E \times E \to (U \times E) \times (E\times E)$) with $B(x, -, -)$ a symmetric bilinear mapping. Here E is the Hilbert space on which X is modeled.
The parentheses in the expression $(F(v) + H(w,v)) + (F(w) + H(v,w))$ are important: inside the parentheses, + means addition in the fibers of the map $D\pi_{TX}$, whereas outside the parentheses, + means addition in a fiber of $\pi_{T(TX)}$. Note that H itself is definitely not symmetric, so I don't think it's clear from the global formula that H exists.
Establishing existence of the map H seems to depend on the rather ugly change-of-coordinate formulas for the "quadratic part" of the spray F, given by Lang.
Lang mentions that the book Symmetric Spaces (Loos, 1969) gives some discussion of this material in terms of second-order jet bundles, and I suspect that may be what I'm looking for. However, this book is hard to come by. I can't find any previews on-line, and it's not in our library. Lang also mentions Pohl's paper "Differential geometry of higher order" (Topology 1 1962 169–211) but I couldn't see anything about the Levi-Civita connection in there.
Does anyone know if Loos has what I'm looking for? Are there other discussions of these ideas in the literature? Does anyone have other suggestions for how to think about the splitting H?
I'll point out, as motivation, that the splitting $H$ gives a decomposition of $T(TX)$ as a direct sum $\pi^* (TX) \oplus \pi^* (TX)$ of bundles over $TX$ (because the kernel of $\alpha$ is isomorphic to $\pi^* (TX)$, and so this is one way to think about the standard fact that $TX$ is an orientable manifold, with a Riemannian metric inherited from the one on $X$.
Best Answer
See my response (number 4) to the MO question:
Exponential map and covariant derivative
There is a Math Review article by Kuranishi there of the paper "Sprays" by Ambrose, Singer, and Palais (in which sprays were first defined). I think the approach taken there (and described in Kuranishi's review) is pretty much what Dan Ramras is asking for.