2D Measuring Surface: Hedgehogs, Skyrmions

This chapter presents a new method to characterize topological features, using a 2D measuring surface rather that a 1D measuring path. The 2D measuring surface leads to a 2D topological charge, which is different from the 1D winding number. Based on this 2D topological charge, the chapter discusses hedgehogs (or Bloch points) as topological defects, and skyrmions as topological solitons. These concepts are developed first for 3D polar order, and then extended to 3D nematic order, with certain necessary modifications.