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Let $(M,g)$ be a Riemannian manifold and let $\lbrace e_1,...,e_n\rbrace$ be a locally frame field on $M$ and $\omega _1 ,...,\omega _n$ be the dual $1$-forms of it. If $\omega _{ij}$ be the connection $1$-forms with respect to the mentioned frame and metric $g$. Then I would like to know what can be the structure equations? I would be grateful if you could give me a link or introduce a book containing these equations.

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This book must be useful: An Introduction to Differentiable Manifolds and Riemannian Geometry written by William Munger Boothby Page 319. Read online in Google Link.

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I cannot resist to recommend the book of Élie Cartan "La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile", because it makes you understand clearly how to go from heavy system of coordinates on moving frames to the $1$-form framework. Furthermore the structure equations can be easily checked in this case (although in $\mathbb{R}^3$). His papers on the subject should also be very clear. Otherwise, from an abstract point of view, one can simply use twice the exterior derivative $d$ to get the structure equations.

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There is a detailed discussion of Cartan's structural equations, in Postnikov's book ''Geometry IV, Riemannian Geometry'' (Chapter 4, pages 44-50) (http://www.springer.com/us/book/9783540411086)

Another reference is the book Hamilton's Ricci Flow by B.Chow, P.Lu and L.Ni (see pages 21-23) (http://bookstore.ams.org/gsm-77)

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