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Usually what is meant under a fibre metric is that one is given a (smooth) vector bundle $\pi:Y\rightarrow X$, and on each fibre $Y_x$ an algebraic inner product $g_x$ that varies smoothly from point to point, in other words, a fibre metric is a positive definite smooth map $g:Y\times_X Y\rightarrow\mathbb R$ which is linear in each variable separately.

I have been thinking about something, and to properly formulate it, I need the notion of a fibre metric on non-linear fibre bundle. Specifically, let $\pi:Y\rightarrow X$ be a locally trivial fibre bundle, such that for each $x\in X$, $g_x$ is a Riemannian metric on the fibre $Y_x$, such that this family of metrics varies smoothly with $x$.

This does not seem too hard to formulate myself, but if a formalism already exists for this (and I assume it does), I'd like to conform to existing terminology/notations and not to spend time reinventing the kettle.

I am especially interested in that each fibre $Y_x$ also has a linear connexion $\Gamma_x$ on it that is the Levi-Civita connexion of the metric $g_x$, and I'd like to know how this "fibre connexion" appears in the context of the total space and the fibred structure.

I am first and foremost looking for references that develop such matters.

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    $\begingroup$ This situation arises in family index theory. If you have a fibrewise metric and a splitting $TY=T^vY\oplus T^hY$ with $T^vY=\ker d\pi$ and $T^hY\cong\pi^*TX$, then there is a natural connection (due to Bismut) on the bundle $T^vY\to Y$ that restricts to the Levi-Civita connection on each fibre. This is discussed in the book by Berline, Getzler and Vergne in Section 10.1. Is that what you are looking for? $\endgroup$
    – Sebastian Goette
    Commented Oct 16, 2020 at 10:13

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