The Oseen–Frank Energy Functional on Manifolds

We observe that for a unit tangent vector field u ∈ TM on a 3-dimensional Riemannian manifold M, there is a unique unit cotangent vector field A ∈ TM associated to u such that we can define the curl of u by dA. Through a unit cotangent vector field A ∈ TM, we define the Oseen–Frank energy functional on 3-dimensional Riemannian manifolds. Moreover, we prove partial regularity of minimizers of the Oseen–Frank energy on 3-dimensional Riemannian manifolds.