# Is the exponential map to the indefinite special orthogonal groups $SO^+(p,q)$ surjective?

Is the exponential map to the identity component of the special indefinite orthogonal groups $$\mathrm{exp} \colon \mathfrak{so}(p,q) \to SO^+(p,q)$$ surjective?

• No; I think this should already be false for $SO(2, 1)$. Dec 13, 2015 at 18:30
• @QiaochuYuan: I think it's actually true for $SO^+(2,1)=PSL(2)$, even though it is not true for $SL(2)$ (which is its 2-fold cover) Dec 13, 2015 at 18:40
• Hmm. Maybe? I hadn't thought about it beyond the double cover. In any case, the argument that works for the definite orthogonal groups (the spectral theorem) fails here. Dec 13, 2015 at 18:41

1. For the special orthogonal group $$SO(d)$$ and the restricted Lorentz group $$SO^+(d,1)\cong SO^+(1,d)$$, the exponential maps \begin{align}\exp: so(d)~~\longrightarrow~~& SO(d), \cr \exp: so(d,1)~~\longrightarrow~~& SO^+(d,1),\end{align}\tag{1} are surjective. See also this related Phys.SE post and links therein.

2. Simplest counterexample. The exponential map $$\exp: sl(2,\mathbb{R})\oplus sl(2,\mathbb{R})~~\longrightarrow~~ SL(2,\mathbb{R})\times SL(2,\mathbb{R})\tag{2}$$ for the split group $$SO^+(2,2)~\cong~[SL(2,\mathbb{R})\times SL(2,\mathbb{R})]/\mathbb{Z}_2 \tag{3}$$ is not surjective. Here the $$\mathbb{Z}_2$$-action identifies \begin{align} (g_L,g_R)~~\sim~~&(-g_L,-g_R), \cr g_L,g_R~\in~& SL(2,\mathbb{R})~:=~\{g\in {\rm Mat}_{2\times 2}(\mathbb{R}) \mid \det g~=~1\}.\end{align}\tag{4} One may show that a pair $$(g_L,g_R)~\in~SL(2,\mathbb{R})\times SL(2,\mathbb{R})\tag{5}$$ with $${\rm tr}(g_L)<-2\quad\text{and}\quad{\rm tr}(g_R)>2\tag{6}$$ (or vice-versa $$L\leftrightarrow R$$) is not in the image of the exponential map, even after $$\Bbb{Z}_2$$-modding.

3. More generally, one may prove for the indefinite orthogonal groups $$SO^+(p,q)$$, where $$p,q\geq 2$$, that the exponential map $$\exp: so(p,q)~~\longrightarrow~~ SO^+(p,q)\tag{7}$$ is not surjective, cf. e.g. Ref. 1.

References:

1. D.Z. Dokovic & K.H. Hofmann, Journal of Lie Theory 7 (1997) 171. The pdf file is available here.
• Is the exponential map also injective on $SO(p)$?
– ABIM
Mar 2, 2018 at 16:23
• No, already the case $d=2$ is not injective. Mar 2, 2018 at 16:30
• A solid reference for surjectivity of the exponential map $exp: o(n,1)\to SO^+(n,1)$ is Theorem 2.1 in M.Moskowitz, "The Surjectivity of the Exponential Map for Certain Lie Groups" Annali di Matematica pura ed applicata (IV), Vol. CLXVI (1994), pp. 129-143 May 17 at 15:58
• @Moishe Kohan: Thanks. 2 days ago