Quasiconvexity in the Riemannian setting

We introduce a notion of quasiconvexity for continuous functions $f$ defined on the vector bundle of linear maps between the tangent spaces of a smooth Riemannian manifold $(M,g)$ and ${\mathbb{R}}^m$, naturally generalizing the classical Euclidean definition. We prove that this condition characterizes the sequential lower semicontinuity of the associated integral functional $$F(u,\mathrm{\Omega })={\int }_{\mathrm{\Omega }}f(du)d\mu$$ with respect to the weak${}^{\ast }$ topology of $W^{1,\mathrm{\infty }}(\mathrm{\Omega },{\mathbb{R}}^m)$, for every bounded open subset $\mathrm{\Omega }\subseteq M$.