A quadratic  system is  a  polynomial vector  field on the plane in the form $$X_{\alpha}\;\;\;\;\;\;\begin{cases}\dot x=P(x,y)\\  \dot y=Q(x,y) \end{cases}$$

where  $P,Q$  are polynomials of degree $2$. The  coefficients of $P,Q$ are  denoted by $\alpha$. So the $\alpha$ in $X_{\alpha}$ is  indicating to coefficients.


A  center is  a  singularity of this  vector  field which is  surrounded by a band of closed orbit. All quadratic vector fields  with center are classified: they correspond to a  finite  number of  algebraic conditions in $\alpha$, These  algebraic  conditions are denoted by $Cent(\alpha)=0$.
see "integrability of plane  quadratic vector fields" Expos. Math(1990)3-25. 

>Are there  a  family of (polynomial) Riemannian  metrics $g_{\alpha}$ with the following properties:

>The  gaussian  curvature of $g_{\alpha}$ is  not  zero except at a  finite  number of  algebraic curves which  consist either a singular points or points  at which $X_{\alpha}$ is transverse to the equation $\kappa_{\alpha}=0$. Moreover $\kappa_{\alpha}$  is  identically  zero if  $Cent(\alpha)=0$.

If the  answer would  be  yes, then $H(2)$, the maximum  number of  limit  cycles  of  a  quadratic system,  would  be  finite.


This  question is  already  discussed at the comment-conversation of  the  following post:

http://mathoverflow.net/questions/160945/limit-cycles-as-closed-geodesicsin-negatively-curved-space