Flatness of Gaussian curvature and area of ideal triangles

Let M be a C k, k ≥ 4, compact surface of genus greater than two whose curvature is negative in all points but along a simple closed geodesic γ(t) where the curvature is zero at every point. We show that the area of ideal triangles having a lifting of 7 as an edge is infinite. This provides a family of surfaces having ideal triangles of infinite area whose geodesic flows are equivalent to Anosov flows, in contrast with the well-known examples of surfaces with flat strips which also have ideal triangles of infinite area. By the CAT-comparison theory we can deduce, using these surfaces as models, that a C∞ compact surface of non-positive curvature having one geodesic along which the curvature is zero has ideal triangles of infinite area. © 1997, Sociedade Brasileira de Matemática.