We are interested in the investigating of paths of conformal deformations of a metric defined in a compact surface, aiming the study of the connectedness of the set of metrics without conjugate points.
It is known that the set of Anosov metrics, in the topology $C^{2}$, is in the interior of the metrics without conjugate points. But it is not known if this set is connected or contractile. Hamilton showed, using the Ricci flow, that given any metric on a compact surface of genus greater than one, there exists a differentiable curve that starts at the metric and ends at a surface 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 one 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 that can be continuously deformed into Anosov metrics without focal points until reaching a metric of negative curvature.