We consider the quadratic vector field $V$ $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end {cases}\;\;\;\;(V)$$

where $P,Q \in \mathbb{R}[x,y]$ are polynomials of degree $2$ with $P(0,0)=Q(0,0)=0$.

We denote by $(W)$ the linear radial vector field $$W=x\partial_x+y\partial_y\;\;\;\;(W)$$

Consider the $1\_$form $$\psi=\frac{1}{x^2+y^2}(ydx-xdy)$$

Let $C$ be the algebraic curve $C=\{(x,y)\mid yP(x,y)-xQ(x,y)=0\}$

We consider the Riemannian metric on $\mathbb{R}^2 \setminus C$ whose orthonormal frame is the following:$$\{V/\psi(V), W/(x^2+y^2)\}$$

For such orthonormal frame, the determinant of the corresponding tensor metric $$\begin{pmatrix}E&F\\F&G \end{pmatrix}$$ is identically $1$, that is $EG-F^2=1$. Furthermore the curvature is zero for the linear center $V=y\partial_x-x\partial_y$.

Question: Is it true to say that $V$ has a center at origin if and only if the Gaussian curvature of the above metric is zero?

Note that a center is a singularity which is surrounded by a band of closed orbits. for quadratic vector fields they are classified at this paper.

The motivation for this post is mentioned in this answer

**Remark:** The initial motivation is mentioned in page 3, item 5 of this arxiv note.