# What happens to $SO(3)$ Lie group if I have angle defect?

$$SO(3)$$ group is defined as linear transformations that preserve euclidean norm. Now if I identify $$\phi$$ (angle in $$xy$$ plane whose axis of rotation is $$z$$-axis) $$0$$ to $$2\pi\beta$$ where $$\beta$$ can vary from $$0$$ to $$1$$. What happens to our spherical symmetry? Will $$SO(3)$$ group still preserve the euclidean norm?

According to me as soon as I picked the $$xy$$ plane ($$z$$-axis of rotation) the spherical symmetry is broken symmetry due to both polar angle as well as azimuthal angle is lost. Is my reasoning right?

Edit: My terminology might be non-standard so what I meant by angle defect(or deficit) is $$2\pi(1-\beta)$$

In my case, it will look like

with the open section stuck together. Also since all the line element for above case is $$ds^2=d\theta ^2+\beta^2\sin^2(\theta)d\phi^2$$. It's killing vector, which forms the generator of $$SO(3)$$ will be same as that of $$S^2$$ with $$\phi$$ replaced by $$\beta\phi$$ i.e.

$$K_1=\cos(\beta\phi)\frac{\partial}{\partial \theta}-\cot\theta\sin(\beta \phi)\frac{\partial}{\partial(\beta\phi)}$$ $$K_2=-\sin(\beta\phi)\frac{\partial}{\partial(\theta)}-\cot(\theta)\cos(\beta\phi)\frac{\partial}{\partial(\beta\phi)}$$ $$K_3=\frac{1}{\beta}\frac{\partial}{\partial \phi}$$

And oddly enough they do satisy their usual commutation relation $$[K_i,K_j]=\epsilon_{ijk}K_k$$ So there seems to be no issue with spherical symmetry since commutation relation is smoking gun to fire away any issue of coordinate representation. But clearly looking at the would-be distorted image of $$\beta$$ sphere is a clear no to spherical symmetry? What is going on? Is our intuition of spherical symmetry failing in the above case?

• My terminology may be non-standard so to remove any ambiguity I have added as much info as possible in the question. Jul 15, 2020 at 16:35
• The symmetry of a prolate spheroid (Aussie football) is not SO(3). Jul 15, 2020 at 17:56