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The Grassmannian bundle of a vector bundle $E$ is a smooth manifold where each fiber over the base space is replaced by the Grassmannian (of specified rank) of the fiber. I am interested in defining a natural metric and local co-ordinates on this space to compute Riemannian quantities like the gradient and Hessian on it. Is there a good reference for such computations? Ideally, I am looking for something like this but any step towards it would be appreciated.

If it helps, my base space is another Grassmannian.

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    $\begingroup$ For the metric on each fiber, it's natural to take the unique (up to scaling) metric invariant under the unitary / orthogonal group. But an overall metric requires more data than the metric on the fiber and the metric on the base, and I believe there is no canonical choice of this except in special cases. $\endgroup$
    – Will Sawin
    Commented Jun 27, 2022 at 2:22
  • $\begingroup$ Thanks for your answer! So in my application it turns out that a certain flag manifold can be fibered with a Grassmannian both as a base space and the fiber on it (though of different dimensions). This makes a canonical choice of metric and other computations quite simple. $\endgroup$
    – mathuser128
    Commented Jun 28, 2022 at 3:06

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