  We construct test function spaces for geometric finite elements. Geometric
finite elements (GFE) are generalizations of Lagrangian finite elements to
situations where the unknown function maps into a nonlinear space. Test
functions for such spaces arise as variations of GFE functions wherever the GFE
function space has a local manifold structure. For any given GFE function
$u_h$, the test functions form a linear space that depends on $u_h$. They
generalize Jacobi fields in the same way that the GFE interpolation functions
generalize geodesic curves. Having test function spaces allows to extend the
GFE method to boundary value problems that do not have a minimization
formulation.
