Shrinking-gluing under lower curvature bound

Details Shrinking-gluing under lower curvature bound Shrinking-gluing under lower curvature bound

2011-10-25

arxiv.org/abs/1110.5498v3

Mathematics

Differential Geometry

27 pages

Scientific paper

We show that the gluing of (singular) length metric spaces which preserves (Hausdorff) volume and lower curvature bound has to be along the boundary isometrically. A consequence of this is the converse of Petrunin's Gluing Theorem: if the gluing of two Alexandrov spaces is an Alexandrov space, then the gluing is along the boundary and by isometry. Another form for the main theorem is: a distance non-increasing onto map between Alexandrov spaces preserves volume if and only if it is the projection map of some gluing along the boundary isometrically.

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