nLab
special orthogonal group

Definition

The special orthogonal group $\mathrm{SO}(n)$ is one of the classical Lie groups. It is the connected component of the neutral element in the orthogonal group $O(n)$.

For instance for $n=2$ we have $\mathrm{SO}(2)$ the circle group.

It is the first step in the Whitehead tower of $O(n)$

$$\cdots \to \mathrm{Fivebrane}(n)\to \mathrm{String}(n)\to \mathrm{Spin}(n)\to \mathrm{SO}(n)\to \mathrm{O}(n).$$\cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n) \,.

Orientation of manifolds

For $X$ an $n$-dimensional manifold a lift of the classifying map $X\to ℬO(n)$ of the $O(n)$-principal bundle to which the tangent bundle $TX$ is associated is the same as a choice of orientation of $X$.

Related concepts

$\cdots \to$ Fivebrane group $\to$ string group $\to$ spin group $\to$ special orthogonal group $\to$ orthogonal group.

group	symbol	universal cover	symbol	higher cover	symbol
orthogonal group	$\mathrm{O}(n)$	Pin group	$\mathrm{Pin}(n)$	Tring group	$\mathrm{Tring}(n)$
special orthogonal group	$\mathrm{SO}(n)$	Spin group	$\mathrm{Spin}(n)$	String group	$\mathrm{String}(n)$
Lorentz group	$\mathrm{O}(n,1)$	$$	$\mathrm{Spin}(n,1)$	$$	$$
anti de Sitter group	$\mathrm{O}(n,2)$	$$	$\mathrm{Spin}(n,2)$	$$	$$
Narain group	$O(n,n)$
Poincaré group	$\mathrm{ISO}(n,1)$	$$	$$	$$	$$
super Poincaré group	$\mathrm{sISO}(n,1)$	$$	$$	$$	$$