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Let $(M,g)$ be a Riemannian manifold and $\pi:P \to M$ be $S^1$- principal fiber bundle endowed with a connection $\Gamma$. For every $p\in P$ we have, $$T_pP \simeq T_pV\oplus\Gamma_p$$ Where $V$ is the vertical bundle. Using the isomorphism $T_pV\simeq iR$ and $\Gamma_p\simeq T_{\pi( p ) }M$ we can define a Riemannian metric $g_{\Gamma}$ on $P$. We suppose that for an open subset $U$ of $M$ there exists the 1-forms $\omega_i\in A^1(U)$ such that on $U$ we have $g=\sum_{i=1}^n\omega_i^2$, So the induced metric on $\pi^{-1}(U)$ is defined by $$g_{\pi^{-1}(U)}=\sum_{i=1}^n(\pi^*{\omega_i})^2+\omega^2$$

where $\omega\in A^1 ( P )$.

If $(\xi_1,…,\xi_{n+1})$ is the dual basis of $(\omega_1,…,\omega_n,\omega)$ then how we can wright the components of the Riemann curvature of $g_\Gamma$ in this basis.

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    $\begingroup$ This really seems like homework. I think it would be more suitable for MathStackExchange than MathOverflow. Have you tried asking this question there? $\endgroup$ Sep 24, 2014 at 15:39
  • $\begingroup$ @Robert:thanks for your comment, I will try MathStackExchange. $\endgroup$
    – Amin
    Sep 24, 2014 at 21:10

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