Geometric information on transferred structure Let $(M,g)$ be a Riemannian manifold, let $\Omega^*(M)$ denote the cochain complex of differential forms on $M$ and $H^*(M)$ its cohomology considered as a chain complex with trivial differential. We have a map $f:\Omega^*(M) \to H^*(M)$ given by projecting a form to its harmonic part and taking its cohomology class and we have a map $h: H^*(M) \to \Omega^*(M)$ which sends a cohomology class to its harmonic representative. Then $fh$ is the identity and $hf$ is chain homotopic to the identity. Hence, we have the right setting to transfer the $C_{\infty}$-algebra structure of $\Omega^*(M)$ with structure maps $m_1=d$, $m_2=\wedge$, and $0=m_3=m_4=...$ to obtain an $C_{\infty}$-algebra structure on $H^*(M)$. We can get formulas for the structure maps of the transferred structure on $H^*(M)$ by taking sums over trees and putting the chain homotopies in the right internal edges, etc...
The topological information information obtained from this transferred structure is understood: up to homotopy, the transferred structure contains rational homotopy information. 
However, my question is the following: What kind of geometric information is contained in the transferred structure (which depends on the metric $g$ and it involves choices) up to isomorphism ?
 A: As far as I understand it, Theorem 7 of http://arxiv.org/abs/0811.1655 states that the $C_\infty$-algebra structures on $H^\ast(M)$ up to isomorphism correspond to rational homotopy types with the given cohomology algebra. This implies that the answer to your question is "any geometric information which can be read off from the rational homotopy type". This is a significant amount, see the book "Algebraic Models in Geometry" by Felix, Oprea and Tanré, or this Oberwolfach Report. 
(Of course, it may just be that your interpretation of "isomorphism" of $C_\infty$-algebras is stricter than Kadeishvili's.)
A: I am sorry but I will not answer your question rather it reminds me some beautiful results which go back to H. Whitney and is related to the transfer of the wedge product of forms.
If you pick a triangulation $T$ of your manifold $M$ and consider the rational cochain complex $C_T(M)$ associated to $T$. You have an integration map:
$I:\Omega(M)\rightarrow C_T(M)$
and you H. Whitney has defined an inverse map $W:C_T(M)\rightarrow \Omega(M)$ (we have to use $L_2$-forms in fact). 
We have $IW=Id$ and $WI$ is homotopic to the identity. These maps are not multiplicative with respect to the cup product and the wedge product. But H. Whitney has given a very nice combinatorial formula for a product on $C_T(M)$, let call it the Whitney product and let denote it by $\bullet$, we have:
$$c\bullet c'=I(Wc\wedge Wc').$$
What H. Whitney has done is to give a very nice and combinatorial formula for the product of the $C_{\infty}$-transfer of the algebra structure of differential forms on cochains.
But the story does not end here, if you want to add some geometrical data to the story you can build a combinatorial Hodge operator on cochains. And now look at the limit of all these structures when you refine the triangulation. In fact the combinatorial Hodge operator converges to the smooth Hodge operator and for the Whitney product all higher homotopy products will vanish in the limit. This was studied in details by S. O'Wilson in "Cochain algebra on manifolds and convergence under refinement", Topology and its Appl, 154 no. 9 (2007) 1898-1920.  
