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Consider $\psi:SU(n)\times U(1)\to U(n)$, $(w,z)\mapsto \bar{z}\cdot w$. One can show that $\psi$ serves as a projection and $SU(n)\times U(1)$ is a principal $\mathbb Z_n$-bundle over $U(n)$.

Suppose we are given a bi-invariant metric $g$ on $SU(n)$ and the canonical metric $h = d\theta^2$ on $U(1)$. I want to show that the product metric $g\oplus h$ on $SU(n)\times U(1)$ induces a bi-invariant metric $G$ on $U(n)$ in such a way that $\psi$ is a local isometry. How do I do that?

Finally, given $f\in C^\infty(U(n))$ with $\Delta_{U(n)}f = \mu f$ for some $\mu>0$, can I show that also $\Delta_{SU(n)\times U(1)}\psi^*f = \mu \psi^*f$?

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    $\begingroup$ $\phi$ is a surjective Lie group homomorphism. Its kernel is discrete, so it induces an isomorphism on the Lie algebras and a bijection on the respective sets of invariant metrics. $\endgroup$
    – Antonius
    Commented Nov 10 at 5:49

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