Curvature of the elastic deformations in a nematic sample

In this work we study the geometry of the elastic deformations of the uniaxial nematic liquid crystals at the bulk. We will show that, at this region of the sample, the elastic terms of the free energy can be separated as the sum of two kinds of elastic deformations, the first is proportional to the Gaussian curvature obtained from the director field of a three-dimensional nematic sample and the second is composed by those terms that cannot be expressed as resulting from this curvature. To achieve these results we will construct the metric of an unixial nematic sample using the fact that the director gives the direction of the anisotropy of the system. With this approach we will give analytical and geometrical arguments to show that the elastic terms determined by , and are contained in a curvature term, while the terms fixed by the splay elastic term, , and the bend elastic term, , are not. The novelty here is that while and do not contribute the bulk elastic energy of a nematic sample, they have an important contribution to the curvature of the system.