*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 some 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 talk, I will discuss a condition relating the geometry
of a Riemannian manifold to that of a model surface which is
weaker than 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.

(This
is joint work with Yutaka Ikeda.)