In projective geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space V = (V,Q) on the associated projective space P(V). Explicitly, the projective orthogonal group is the quotient group
PO(V) = O(V)/ZO(V) = O(V)/{±I}
where O(V) is the orthogonal group of (V) and ZO(V)={±I} is the subgroup of all orthogonal scalar transformations of V – these consist of the identity and reflection through the origin. These scalars are quotiented out because they act trivially on the projective space and they form the kernel of the action, and the notation "Z" is because the scalar transformations are the center of the orthogonal group.
The projective special orthogonal group, PSO, is defined analogously, as the induced action of the special orthogonal group on the associated projective space. Explicitly:
PSO(V) = SO(V)/ZSO(V)
where SO(V) is the special orthogonal group over V and ZSO(V) is the subgroup of orthogonal scalar transformations with unit determinant. Here ZSO is the center of SO, and is trivial in odd dimension, while it equals {±1} in even dimension – this odd/even distinction occurs throughout the structure of the orthogonal groups. By analogy with GL/SL and GO/SO, the projective orthogonal group is also sometimes called the projective general orthogonal group and denoted PGO.
Like the orthogonal group, the projective orthogonal group can be defined over any field and with varied quadratic forms, though, as with the ordinary orthogonal group, the main emphasis is on the real positive definite projective orthogonal group; other fields are elaborated in generalizations, below. Except when mentioned otherwise, in the sequel PO and PSO will refer to the real positive definite groups.
Like the spin groups and pin groups, which are covers rather than quotients of the (special) orthogonal groups, the projective (special) orthogonal groups are of interest for (projective) geometric analogs of Euclidean geometry, as related Lie groups, and in representation theory.
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In mathematics, a covering group of a topological group H is a covering space G of H such that G is a topological group and the covering map p : G → H is a continuous group homomorphism. The map p is called the covering homomorphism. A frequently occurring case is a double covering group, a topological double cover in which H has index 2 in G; examples include the spin groups, pin groups, and metaplectic groups.
In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids. Projective polyhedra are also referred to as elliptic tessellations or elliptic tilings, referring to the projective plane as (projective) elliptic geometry, by analogy with spherical tiling, a synonym for "spherical polyhedron".
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