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Let $(M,g)$ be a Riemannian manifold such that the parallel transport along every simple closed curve is the identity operator. A smooth function $f:M\to \mathbb{R}$ is called a quadric function if for every two points $x,y \in M$ we have $H(f)(x)=P_{xy}^{*}H(f)(y)$ where $P_{xy}$ is the parallel transport from $x$ to $y$.(With respect to an arbitrary curve which joins $x$ to $y$).

Is the product of two affine functions, a quadric function?

Can we write every quadric function as a linear combination of terms in the form $fg$ where $f$ and $g$ are affine functions?

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    $\begingroup$ Your assumption means that the metric g is locally flat. Now you can solve your own problem. In any case, this is more suitable for the MSE, since it is not at research level. $\endgroup$
    – Misha
    Commented Jul 8, 2014 at 5:34
  • $\begingroup$ @Misha How can we globalize? $\endgroup$
    – Ali Taghavi
    Commented Jul 8, 2014 at 5:41
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    $\begingroup$ Suppose you are dealing with complete manifolds. The you are looking at quadratic/linear functions on $R^n$ which are invariant under the action of the Bieberbach group of covering transformations. Such functions hardly ever exist (you can classify when). $\endgroup$
    – Misha
    Commented Jul 8, 2014 at 10:49

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