Hi everyone, sorry if my question may be wrong (i'm an engineer not a mathematician!). I was wondering, given a total space E of a trivial bundle, such that E=MxF, where $gm_{ij}$, and $gf_{ij}$ are the metric tensors on the base space M and the fiber F resp., what is the metric tensor of E?

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extradata. One can always put a metric (if the manifold admits partitions of unity, which it will in all practical applications) but there are uncountably many possible metrics. Gunnar's comment is true in the sense that thisdoesgive a metric on $M \times F$, and this is certainly the most natural, given that you have metrics on $M$ and $F$ separately, but it's certainly not the only possibility. It really depends on what you are studying. – Spiro Karigiannis Dec 13 '11 at 19:23nota product. One way to get a metric on the total space of a fibre bundle, given metrics on the base and the fibre, is by using a connection (called an Ehresmann connection in this context.) You can probably find this in Kobayashi-Nomizu. – Spiro Karigiannis Dec 13 '11 at 19:25