I have a compact matrix manifold of $SU(2) \times U(1)$ I need to construct an invariant metric on it in relation to my Lie group $SU(2) \times U(1)$
I'm Trying to construct an invariant metric on my Lie group:
$$a_{kl}(x) = -Tr[\frac{\partial (g_{su(2)}(x), g_{u(1)}(x))}{\partial x^k} \cdot g(x)^{-1} (\frac{\partial (g_{su(2)}(x), g_{u(1)}(x))}{\partial x^l} \cdot g(x)^{-1})^+]$$
where $x$ - coordinates of $g$ on a manifold (parameters of my Lie group); $g\in SU(2) \times U(1); g_{su(2)} \in SU(2); g_{u(1)} \in U(1)$
Or
$$ds^2 = Tr[(dg_{su(2)}, dg_{u(1)}) \cdot (dg_{su(2)}, dg_{u(1)})^+] = Tr[dg_{su(2)} \cdot dg_{su(2)}^+] \cdot Tr[dg_{u(1)} \cdot dg_{u(1)}^+]$$
If I represent $g_{su(2)}$ as $\begin{bmatrix}\alpha& \beta\\-\beta^*& \alpha^*\end{bmatrix}$ where $\alpha = x_0 + i x_3; \beta = x_2 + i x_1$ And then parametrize: $$x_0 = cos\chi ; x_1 = sin\chi\cdot cos\theta ; x_2 = sin\chi\cdot sin\theta\cdot cos\phi ; x_3 = sin\chi\cdot sin\theta\cdot sin\phi$$
I will get for $ds^2_{su(2)} =Tr[dg_{su(2)} \cdot dg_{su(2)}^+] = d\chi^2 + sin^2\chi\cdot(d\theta^2 + sin^2\theta\cdot d\phi^2)$ - metric of $S^3$
Similarly $ds^2_{u(1)} = d\psi^2$ Therefore: $$ds^2_{su(2) \times u(1)} = Tr_{su(2)}\cdot Tr_{u(1)} = [d\chi^2 + sin^2\chi\cdot(d\theta^2 + sin^2\theta\cdot d\phi^2)]\cdot d\psi^2$$
Well, are these arguments correct? And how i can correctly represent this metric of $SU(2)\times U(1)$ as metric tensor?
And if this arguments are incorrect, please, tell me how I can build this invariant metric yet? Maybe this is written in some literature?
Thank you