Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Small deformations of polygons   We describe the first-order variations of the angles of Euclidean, spherical
or hyperbolic polygons under infinitesimal deformations such that the lengths
of the edges do not change. Using this description, we introduce a
vector-valued quadratic invariant $b$ on the space of those isometric
deformations which, for convex polygons, has a remarkable positivity property.
  We give two geometric applications. The first is an isoperimetric statement
for hyperbolic polygons: among the convex hyperbolic polygons with given edge
lengths, there is a unique polygon with vertices on a circle, a horocycle, or
on one connected component of the space of points at constant distance from a
geodesic, and it has maximal area. The second application is a new proof of the
infinitesimal rigidity of convex polyhedra in the Euclidean space, and a new
rigidity result for polyhedral surfaces in the Minkowski space.
  Finally we indicate how the invariant $b$ can be used to define natural
metrics on the space of convex spherical (or hyperbolic) polygons with fixed
edge lengths. Those metrics are related to known (and interesting) metrics on
the space of convex polygons with given angles in the plane.

1
The categories of this paper are: cs
11
The categories of this paper are: hep-lat
19
The categories of this paper are: eess
3
The categories of this paper are: physics
31
The categories of this paper are: ao-sci
33
The categories of this paper are: plasm-ph
21
The categories of this paper are: adap-org
12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
14
The categories of this paper are: math-ph
30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
29
The categories of this paper are: funct-an
6
The categories of this paper are: astro-ph
25
The categories of this paper are: patt-sol
26
The categories of this paper are: bayes-an