I know of the theorem that says that in any fiber bundle with structure group G having finitely many connected component, the structure group can be reduced to a maximal compact subgroup of G.(adsbygoogle = window.adsbygoogle || []).push({});

But here I am reading a thesis in which the author says:"[Since U(n,n)] ishomotopicto its maximal compact subgroup U(n) x U(n), the structure group can be reduced to U(n) x U(n)."

The context here is a vector bundle with structure group G=U(n,n).

So I am wondering, is there a result about reduction of str. groups of vector bundles that says something like "if H<G has the same homotopy type as G, then we can reduce to H" ??

Thx

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# When can one reduce the structure group?

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