Consider the rank-${n \choose m}$ vector bundle $\pi\colon E:=\bigwedge^m(TN)\to N$ over a smooth Finsler manifold $N$ and equip each fibre $E_q := \pi^{-1}(q)$ with a norm that depends smoothly on $q\in N$ (induced by an arbitrary Riemannian metric on $N$).

Let $A\colon \bigwedge^m(TN)\to \mathbb{R}$ be a function with the property that the restriction to each fibre $E_q$ is a convex and absolutely homogeneous extension of a continuous function $a_q:GC_m(T_qN)\to \mathbb{R}$. Here $GC_m(V)$ is the set of simple or totally decomposable $m$-vectors in $V$. More precisely, I am looking at the $m$-dimensional Busemann--Hausdorff area density as defined in [Section 4.1, p. 18] and its extension (for $m=2$ in [Theorem 1, p. 3]).

How can I show that $A$ is continuous on $E$?

Indeed, I have shown that $A_q := A\big|_{E_q}$ is continuous on each fibre by using that bounded, convex functions are locally Lipschitz (e.g. [Clarke-Optimization and nonsmooth analysis, Proposition 2.2.6, p. 34]). I would like to compare elements from different fibres now to show continuity on the complete total space $E$. Intuitively it feels like it is clear because the fibres vary smoothly with $q$ (because $N$ is smooth) and on each fibre $A$ is continuous. I fail to make this intuition rigorous using local trivialisations though.

Any hint or comment is welcome.

4more comments