TITLE: Geometric information on transferred structure
QUESTION [7 upvotes]: 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 ?

REPLY [2 votes]: 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.)