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I have a problem with Kuranishi's theorem in deformation theory. I'll try to formulate it in general terms, and then describe the particular situation.

Let $\pi : M \to S$ be a smooth fiber bundle - i.e. $M$ and $S$ are smooth manifolds, and $\pi$ is a surjective submersion. There is an associated surjective morphism of vector bundles $ \pi_* : T_M \to \pi^* T_S$. I want to find a lifting of $\pi^* T_S$ into $T_M$.

Suppose I can find a smooth map $f : S \to M$ which satisfies $\pi \circ f = id_S$. This induces an injective map $T_S \to f^*T_M$. Can I lift this to a map $\pi^* T_S \to T_M$? How about if some extra data is given, like a metric on $S$, or a family of metrics $g_s$ on the fibers $T_M |_{M_s}$ (where $s$ is a parameter in $S$)?

Basically I'm trying to use Kuranishi's theorem to get something like Siu's canonical lifts. In this situation $M$ is the product of a fixed smooth manifold and the space of its complex structures, and $S$ is a complex manifold (open ball, even). The map $\pi$ is the passing to the quotient by the action of the group of diffeomorphisms. If we fix a hermitian metric $h$ on $M_0$, then Kuranishi gives a map $f : S \to M$ which satisfies the above hypothesis. I'm told that Kuranishi should induce a lifting of $\pi^*T_S$ into $T_M$, but I can't seem to figure out how.

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    I'm a bit confused. A map $\pi^* T_S\to T_M$ splitting the map $T_M\to\pi^*T_S$ is called an Ehresmann connection on $M\to S$. Ehresmann connections are sections of an affine $A$ bundle over $S$ (modeled by the vector bundle $\pi^* T_S^* \otimes K$, where $K$ is the kernel of $\pi^*$). Hence a $C^\infty$ Ehresmann connection always exists, and you can also extend a section of $A$ from a submanifold. Is $C^\infty$ not good enough?2011-04-11
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    Sure, $C^\infty$ is good enough, but there are lots of smooth splittings $T_M \to \pi^*T_S$ (any metric on $M$ gives one) and I want one induced by the map $f : S \to M$ somehow. Can we use $f$ to construct a specific Ehresmann connection?2011-04-11

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