Abstract
Using the idea of decomposability of gauge potential and internal structure put forward by Duan Yi-Shi, we decompose gauge potential of SO( n ) group in the unit vector field by using geometric algebra method, we get the general decomposition form and discuss the properties of this decomposition. In this paper, we give the form of decomposition of SU(2) group and U(1) group with unit vector field, which is exactly the result given by famous physicist Fadeev in 1999. The local topological structure of Gauss-Bonnet-Chern density is discussed by using the general form of decomposition of SO( n ) gauge potential. The global topological structure of the density is Gauss-Bonnet-Chern theorem, and the Morse theoretical form of Euler-Poincare characteristic is easily obtained from the topological structure. A new circulation condition is obtained by using the normal potential decomposition of SU(2) gauge potential to study –1/2 Bose-Einstein condensate, which is also a generalization of the Mernin-Ho relation. Finally, using the relation between Torsion tensor and U(1) gauge theory of three-dimensional Riemannian geometry discovered by Duan Yi-Shi, the relation between dislocation line and link number is studied by using U(1) gauge potential decomposition.
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