Special orthogonal group

The special orthogonal group isEvery element of $SO(2)$ is a rotation in the plane. Every element of $SO(3)$ fixes an axis and restricts to a planar rotation on its perpendicular plane.

The SO(4) group consists of real orthogonal four-by-four matrices of determinant one. It preserves orientation and the Euclidean metric. Its Spin(4) double cover identifies it with the quotient of two copies of SU(2) by their simultaneous central sign.

Tensor products $V_{j_L}\otimes V_{j_R}$ give the irreducible representations of the Spin(4) double cover. The simultaneous central sign acts by $(-1{)}^{2j_L+2j_R}$, so exactly the integer-sum pairs descend to SO(4). Their dimensions are $(2j_L+1)(2j_R+1)$. Under the diagonal spatial rotation subgroup, the Clebsch-Gordan decomposition for SU2 gives spins from $|j_L-j_R|$ through $j_L+j_R$ in steps of one.

The subgroup of three-dimensional proper rotations preserving a chosen unoriented axis consists of all rotations about it and all half-turns about perpendicular axes. It is isomorphic to $O(2)$; the continuous molecular point-group notation is $D_{\infty }$. This uncountable group is different from the abstract countable infinite dihedral group $\mathbb{Z}⋊C_2$.

The SO(3) group consists of real three-by-three orthogonal matrices with determinant one. It acts by rotation in three dimensions. Its group manifold is ${\mathbb{RP}}^3$, since the Adjoint double cover from SU(2) to SO(3) identifies antipodal points of the three-sphere.

The Adjoint double cover from SU(2) to SO(3) has kernel $\{I,-I\}$. Under SU(2) as the three-sphere, multiplication by $-I$ is the antipodal map, so the quotient is Real projective space ${\mathbb{RP}}^3$. Equivalently, an axis-angle ball of radius $\pi$ has opposite boundary points identified. The fundamental group is ${\mathbb{Z}}_2$.

Every real orthogonal transformation of odd dimension with determinant one has eigenvalue $+1$. Nonreal eigenvalues come in conjugate pairs with product one, while real eigenvalues are $+1$ or $-1$. There are an odd number of real eigenvalues and an even number of negative ones, leaving at least one positive eigenvalue. The corresponding eigenvector is fixed.

The map $A\mapsto A^{T}A$ from real matrices to symmetric matrices has derivative $H\mapsto R^{T}H+H^{T}R$ at $R\in O(n)$, which is surjective. Thus $SO(n)$ is a submanifold of dimension $n(n-1)/2$, and

For $v\in S^n$, the stabilizer $S_v=\{R\in SO(n+1):Rv=v\}$ is conjugate to the block subgroup $SO(n)$ acting on $v^{\perp }$. Hence it is a submanifold of dimension $n(n-1)/2$, with

For distinct $v,w\in S^n$, the submanifolds $S_v,S_w\subseteq SO(n+1)$ are not transverse. They both contain the identity. If $v,w$ are independent, the bivector $v\wedge w$ is a nonzero normal direction common to both tangent spaces there; if $w=-v$, the two stabilizers coincide.

A rotation in three dimensions is an element of $SO(3)$. It fixes an axis and acts as a two-dimensional rotation on the plane perpendicular to that axis.