Gluing

Gluing of Intrinsic Metric Spaces

What kinds of gluing preserve the lower curvature bound in the comparison sense?

• "Gluing" - identification of points and the quotient space is equipped with the induced intrinsic metric.

• "Comparison sense" - Toponogov's Theorem holds. These spaces are so-called Alexandrov spaces with lower curvature bound. Consequently, they can not have geodesic bifurcations and must satisfy Bishop-Gromov Volume Comparison.

The following are some 2-dimensional examples of the gluing which produce Alexandrov spaces.

(A-1)- Two hemispheres of the same size are glued along their boundaries. This produces a sphere.

(A-2) - A flat metric cone is glued with a flat disk along their isometric circle boundaries.

(A-3) - The boundary of a hemisphere is glued with itself via an antipodal map. This produces the real projective space RP^2.

(A-4) - Three isosceles triangles are glued with one equilateral triangle. The glued edges have the same lengths.

Here are some examples which do not produce Alexandrov spaces.

(B-1) - Contrast to (A-1), the boundaries of two hemispheres of different sizes are identified via a stretching map between. The glued space does not satisfy Bishop-Gromov Volume Comparison for any lower curvature bound. However, there is no geodesic bifurcation in this space.

(B-2) - Three rectangles of the same size are glued along one common edge. There are geodesic bifurcations in the glued space.

(B-3) - An edge is identified with an interior segment. The glued space has geodesic bifurcations.

(B-4) - Edges are partially identified. The glued space looks like the Chinese character "凹", which means "concave".

How to distinguish these examples?

If the gluing is along a lower dimensional subset, then the projection map is naturally a 1-Lipschitz, volume preserving onto. The study of such map would give a characterization for the corresponding gluing (see my paper "Volume and gluing rigidity in Alexandrov geometry").