In metric geometry, comparison triangles are constructions used to define higher bounds on curvature in the framework of locally geodesic metric spaces, thereby playing a similar role to that of higher bounds on sectional curvature in Riemannian geometry.
be the euclidean plane,
be the hyperbolic plane.
denote the spaces obtained, respectively, from
is the unique complete, simply-connected, 2-dimensional Riemannian manifold of constant sectional curvature
be a metric space.
Such a triangle, when it exists, is unique up to isometry.
, it can be ensured by the additional condition
(i.e. the length of the triangle does not exceed that of a great circle of the sphere
The interior angle of
is called the comparison angle between
Using inverse trigonometry, one has the formulas:
Comparison angles provide notions of angles between geodesics that make sense in arbitrary metric spaces.
The Alexandrov angle, or outer angle, between two nontrivial geodesics
The following similar construction, which appears in certain possible definitions of Gromov-hyperbolicity, may be regarded as a limit case when
in a metric space
, the Gromov product of
is half of the triangle inequality defect:
is the metric graph obtained by gluing three segments
is the union of the three unique geodesic segments
Furthermore, there is a well-defined comparison map
One way to formulate Gromov-hyperbolicity is to require
not to change the distances by more than a constant
Another way is to require the insizes of triangles
to be bounded above by a uniform constant
Equivalently, a tripod is a comparison triangle in a universal real tree of valence
Such trees appear as ultralimits of the
[1] In various situations, the Alexandrov lemma (also called the triangle gluing lemma) allows one to decompose a geodesic triangle into smaller triangles for which proving the CAT(k) condition is easier, and then deduce the CAT(k) condition for the bigger triangle.
This is done by gluing together comparison triangles for the smaller triangles and then "unfolding" the figure into a comparison triangle for the bigger triangle.
This metric geometry-related article is a stub.