Homogenization of integral functionals is studied
under the constraint that admissible maps have to take their values
into a given smooth manifold. The notion of tangential
homogenization is defined by analogy with the tangential
quasiconvexity introduced by Dacorogna et al. [Calc. Var. Part. Diff. Eq. 9 (1999) 185–206]. For energies with superlinear or linear growth, a
Γ-convergence result is established in Sobolev spaces, the
homogenization problem in the space of functions of bounded
variation being the object of [Babadjian and Millot, Calc. Var. Part. Diff. Eq.
36 (2009) 7–47].