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Dec 19, 2014 at 12:23 comment added Jjm But if one connection is defined in a bundle and the other one in the dual bundle, then the description above makes no sense, since $X$, $Y$ and $Z$ are fields. I still do not understand the setting completely.
Dec 19, 2014 at 11:49 comment added user64142 Thanks for your answer, however, this is not what I want. Maybe because I didn't explain my question in a right way. Consider the dual connections over vector bundler and its dual space, I think they don not need the metrics. Since we talk about connections, I think the metric in the definition mentioned is also unnecessary.
Dec 19, 2014 at 9:29 history answered Jjm CC BY-SA 3.0