Internal Constraints in Finite Elasticity: Manifolds or not

An internal constraint for an elastic material is described either by a submanifold [InlineEquation not available: see fulltext.] of the space of deformation gradients or by a submanifold [InlineEquation not available: see fulltext.] of the space of symmetric strain tensors. It was proved in Podio-Guidugli and Vianello (Arch. Ration. Mech. Anal. 105(2):105-121, 2001) that the dimension of an isotropic constraint [InlineEquation not available: see fulltext.] is always 8, or, equivalently, that the dimension of an isotropic constraint [InlineEquation not available: see fulltext.] is always 5, rigidity and conformality being the only exceptions. Recently, this statement has been questioned in Carroll (Int. J. Eng. Sci. 47:1142-1148, 2009), where it is suggested that isotropic constraints might exist which do not conform to the above prescriptions. It is shown here that this is not the case, because the proposed counterxamples lack a proper manifold structure.