Toponogov's triangle comparison theorem and its generalizations are important tools for studying the topology of Riemannian manifolds. In these theorems, one assumes that the curvature of a given manifold is bounded from below by the curvature of a model surface. The models are either of constant curvature, or, in the generalizations, rotationally symmetric about some point. One concludes that geodesic triangles in the manifold correspond to geodesic triangles in the model surface which have the same corresponding side lengths, but smaller corresponding angles. In addition, a certain rigidity holds: Whenever there is equality in one of the corresponding angles, the geodesic triangle in the surface embeds totally geodesically and isometrically in the manifold.
In this paper, we discuss a condition relating the geometry of a Riemannian manifold to that of a model surface which is weaker than the usual curvature hypothesis in the generalized Toponogov theorems, but yet is strong enough to ensure that a geodesic triangle in the manifold has a corresponding triangle in the model with the same corresponding side lengths, but smaller corresponding angles. In contrast, it is interesting that rigidity fails in this setting.
"Replacing the lower curvature bound in Toponogov's comparison theorem by a weaker hypothesis." Tohoku Math. J. (2) 69 (2) 305 - 325, 2017. https://doi.org/10.2748/tmj/1498269628