# Is this reasoning correct? Connection with torsion on SO(3)

Let say I'm in $SO(3)$. In the tangent space at the identity I define a inner product, i.e. $$g_{0}\left(X,\,Y\right)=\frac{1}{2}\mbox{Tr}\left(X^{T}Y\right),$$ Then I define a leftinvariant metric defined for every point $R$ in $SO(3)$ through the left action, i.e. $$g\left(X,\,Y\right)=g_{0}\left(L_{R^{-1}*}\left(X\right),\,L_{R^{-1}*}\left(Y\right)\right).$$ It's clearly left invariant by construction. Now let's say I take the following base $\left\{\left(E_{1}\right)_{R},...,\left(E_{n}\right)_{R}\right\}$ for the point of the tangent space in $R$ where $$\left(E_{1}\right)_{R}=L_{R*}\left(\left(E_{1}\right)_{\mathbb{1}}\right)$$ where of course $\left\{ \left(E_{1}\right)_{\mathbb{1}},...,\left(E_{n}\right)_{\mathbb{1}}\right\}$ is the canonical base at the identity. Now it should be that $$g\left(\left(E_{i}\right)_{R},\,\left(E_{j}\right)_{R}\right) = \frac{1}{2}\mbox{Tr}\left(\left(R\left(E_{i}\right)_{\mathbb{1}}\right)^{T}\left(R^{-1}\right)^{T}\left(R^{-1}\right)\left(R\left(E_{j}\right)_{\mathbb{1}}\right)\right)= g\left(\left(E_{i}\right)_{\mathbb{1}},\,\left(E_{j}\right)_{\mathbb{1}}\right),$$ So that the coefficient of the metric in the base given by $\left\{ \left(E_{1}\right)_{R},...,\left(E_{n}\right)_{R}\right\}$ are constant. Being that the Christoffel Symbols are identically zero and I have that $\nabla_{X}Y$ is nothing else that the partial derivative in the point.

Defining the torsion $$T\left(X,Y\right)=\nabla_{X}Y-\nabla_{Y}X-\left[X,Y\right].$$

I then have $$T\left(X,Y\right)=-\left[X,Y\right],$$ and therefore in the base $\left\{ \left(E_{1}\right)_{R},...,\left(E_{n}\right)_{R}\right\}$ the coefficient of the tensor of torsion are $$T_{bc}^{a}=-\epsilon_{bc}^{a},$$ where in general since it works for every Lie Group the coefficient should be che structure constante of the lie algebra for the specific base choosen. Is this reasoning right? What did I get wrong? I found in a book the same construction to find the Curvature of the Levi-Civita connection and so -since the Levi-Civita Connection has zero torsion- I'm quite sure I've done something wrong somewhere....

• @Dac0: All coordinate computations involving metrics, Christoffel symbols, and curvature tensor components rely on a coordinate frame, i.e., a set of vector fields $(\partial/\partial x_{i})$ for some local coordinate system $(x_{i})$. Your orthonormal frame is not a coordinate frame; that is, there exists no coordinate system on $SO(3)$ whose coordinate frame is left-invariant. (If there were, the fields would commute, implying $SO(3)$ is Abelian.) :) – Andrew D. Hwang Aug 16 '16 at 21:54