The inner topological structure and defect control of magnetic skyrmions

We prove that the integrand of magnetic skyrmions can be expressed as curvature tensor of Wu-Yang potential. Taking the projection of the normalized magnetization vector on the 2-dim material surface, and according to Duan's decomposition theory of gauge potential, we reveal that every single skyrmion is just characterized by Hopf index and Brouwer degree at the zero point of this vector field. Our theory meet the results that experimental physicists have achieved by many technologies. The inner topological structure expression of skyrmion with Hopf index and Brouwer degree will be indispensable mathematical basis of skyrmion logic gates.