- #1
lichen1983312
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I am trying to follow Nakahara's book about Holonomy.
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?
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?