Morse's index formula in VMO for compact manifolds with boundary

In this paper, we study Vanishing Mean Oscillation vector fields on a compact manifold with boundary. Inspired by the work of Brezis and Nirenberg, we construct a topological invariant — the index — for such fields, and establish the analogue of Morse's formula. As a consequence, we characterize the set of boundary data which can be extended to nowhere vanishing VMO vector fields. Finally, we show briefly how these ideas can be applied to (unoriented) line fields with VMO regularity, thus providing a reasonable framework for modeling a surface coated with a thin film of nematic liquid crystals.