What does it mean by a Riemannian metric on a vector bundle?by petergreat Tags: riemannian metric, vector bundle 

#1
Mar1811, 05:32 PM

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It's really a question about convention. Does such a metric have to be linear on each fiber?




#2
Mar1811, 06:18 PM

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#3
Mar1811, 06:33 PM

P: 270





#4
Mar1811, 06:38 PM

P: 270

What does it mean by a Riemannian metric on a vector bundle?
In other words, are we allowed to "curve" the base space only or the entire space?




#5
Mar1811, 06:41 PM

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There is generally no natural Euclidean metric on a fiber. If you have a submanifold of another manifold then its tangent and normal bundles inherit a metric from the metric on the tangent space of the ambient manifold. 



#6
Mar1811, 06:44 PM

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#7
Mar1811, 06:51 PM

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#8
Mar1811, 06:57 PM

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#9
Mar1811, 07:00 PM

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P: 1,716

Different fibers are different vector spaces and generally have different metrics  which means there is no natural way to compare these vector spaces or their metrics. 



#10
Mar1811, 07:02 PM

P: 270

Thanks! That's clear now.



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