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.

Definitions

Comparison triangles

be the euclidean plane be the unit 2-sphere be the hyperbolic plane is the unique complete, simply-connected, 2-dimensional Riemannian manifold.

be a metric space. A comparison triangle is a geodesic triangle.

Such a triangle, when it exists, is unique up to isometry (i.e. the length of the triangle does not exceed that of a great circle).

Comparison angles

The interior angle is called the comparison angle

Alexandrov angles

Comparison angles provide notions of angles between geodesics that make sense in arbitrary metric spaces. The Alexandrov angle, or outer angle

Comparison tripods

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, the Gromov product is half of the triangle inequality, the comparison tripod.

is isometric is called the center is called the center, its points the internal points, and its diameter the insize.

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Equivalently, a tripod is a comparison triangle in a universal real tree. Such trees appear as ultralimits.

The CAT(k) condition

The Alexandrov lemma

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.

References

• M Bridson & A Haefliger - Metric Spaces Of Non-Positive Curvature,