Why is a connection on the bundle $SO(M)$ metric compatible?

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If we have an orientable manifold $M$ with a metric $g$ and signature $(r, s)$, we can define the principal-$SO(r, s)$ bundle $SO(M)$, the bundle of orthonormal frames of $TM$. This is a subset of the principal-$GL(n, mathbbR)$ bundle $Fr(M)$, the frame bundle of $M$.



As a Koszul connection $nabla$, any such connection is metric compatible. That is, for all $X, Y, Z in mathrmVect(M)$, we have



$$X g(Y, Z) = g(nabla_X Y, Z) + g(Y, nabla_Z X).$$



My question is this: how can we prove that all connections on the bundle $SO(M)$ are metric compatible?




differential-geometry riemannian-geometry connections principal-bundles tangent-bundle




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    If we have an orientable manifold $M$ with a metric $g$ and signature $(r, s)$, we can define the principal-$SO(r, s)$ bundle $SO(M)$, the bundle of orthonormal frames of $TM$. This is a subset of the principal-$GL(n, mathbbR)$ bundle $Fr(M)$, the frame bundle of $M$.



    As a Koszul connection $nabla$, any such connection is metric compatible. That is, for all $X, Y, Z in mathrmVect(M)$, we have



    $$X g(Y, Z) = g(nabla_X Y, Z) + g(Y, nabla_Z X).$$



    My question is this: how can we prove that all connections on the bundle $SO(M)$ are metric compatible?




    differential-geometry riemannian-geometry connections principal-bundles tangent-bundle




    share|cite|improve this question





















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      If we have an orientable manifold $M$ with a metric $g$ and signature $(r, s)$, we can define the principal-$SO(r, s)$ bundle $SO(M)$, the bundle of orthonormal frames of $TM$. This is a subset of the principal-$GL(n, mathbbR)$ bundle $Fr(M)$, the frame bundle of $M$.



      As a Koszul connection $nabla$, any such connection is metric compatible. That is, for all $X, Y, Z in mathrmVect(M)$, we have



      $$X g(Y, Z) = g(nabla_X Y, Z) + g(Y, nabla_Z X).$$



      My question is this: how can we prove that all connections on the bundle $SO(M)$ are metric compatible?




      differential-geometry riemannian-geometry connections principal-bundles tangent-bundle




      share|cite|improve this question











      If we have an orientable manifold $M$ with a metric $g$ and signature $(r, s)$, we can define the principal-$SO(r, s)$ bundle $SO(M)$, the bundle of orthonormal frames of $TM$. This is a subset of the principal-$GL(n, mathbbR)$ bundle $Fr(M)$, the frame bundle of $M$.



      As a Koszul connection $nabla$, any such connection is metric compatible. That is, for all $X, Y, Z in mathrmVect(M)$, we have



      $$X g(Y, Z) = g(nabla_X Y, Z) + g(Y, nabla_Z X).$$



      My question is this: how can we prove that all connections on the bundle $SO(M)$ are metric compatible?





      differential-geometry riemannian-geometry connections principal-bundles tangent-bundle





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









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