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Oct 15, 2013 at 17:57 comment added Matthias Ludewig "Real $G$-vector bundles are generally understood to be $O(n)$-bundles"... I highly doubt that, as this reduction from $GL(n)$ to $O(n)$ is equivalent to choosing a metric.
Oct 12, 2013 at 17:17 history edited Peter May CC BY-SA 3.0
correction
Oct 12, 2013 at 17:15 comment added Peter May Yes, that is what I meant to say. Thanks. Answer edited
Oct 12, 2013 at 14:13 comment added Tom Goodwillie Peter, just to be clear, your $E$ is the principal bundle associated to a vector bundle, yes?
Oct 12, 2013 at 13:36 history answered Peter May CC BY-SA 3.0