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