I am trying to follow Nakahara's book about Holonomy.(adsbygoogle = window.adsbygoogle || []).push({});

if parallel transporting a vector around a loop induces a linear map (an element of holonomy group)

[tex]{P_c}:{T_p}M \to {T_p}M[/tex]

the holonomy group should be a subgroup of

[tex]GL(m,R)[/tex]

then the book said for a metric connection, the property

[tex]{g_p}({P_c}(X),{P_c}(Y)) = {g_p}(X,Y)[/tex]

makes the holonomy group a subgroup of [tex]O(m)[/tex] if the manifold is Riemannian; and a subgroup of [tex]O(m-1)[/tex] if the manifold is Lorentzian.

The author must think this is very straightforward and didn't explain why. Can anybody help?

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# A Question about Holonomy of metric connecton

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