We are interested in the investigation paths of conformal deformations of a metric defined on a compact surface, aiming to study the connectedness of the set of metrics without conjugate points. It is known that the set of Anosov metrics, in the 𝐶2 topology, is in the interior of the set of metrics without conjugate points. But it is not known if this set is connected or contractible. Hamilton showed, using the Ricci flow, that given any metric on a compact surface of genus greater than 1, there exists a differentiable curve of metrics that starts at the given metric and ends at a metric with negative curvature. However, it is not known whether, when the initial metric has no conjugate points, this property is preserved along the curve. Our study has two main objectives.The first is to present a family of compact surfaces of genus greater than 1 that, despite having a finite number of simply connected regions that admit positive curvature, do not present focal points, and whose metrics are Anosov. The second goal is to demonstrate that this family contains a subfamily whose metrics can be continuously deformed through Anosov metrics without focal points until reaching a metric of negative curvature.