- #1
lichen1983312
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I am looks at problems that use the line integrals ##\frac{i}{{2\pi }}\oint_C A ## over a closed loop to evaluate the Chern number ##\frac{i}{{2\pi }}\int_T F ## of a U(1) bundle on a torus . I am looking at two literatures, in the first one the torus is divided like this
then the Chern number is computed as the difference ##\frac{i}{{2\pi }}\left( {\oint_C {{A^I}} - \oint_C {{A^{II}}} } \right)##, this can be done because a smooth ##A## on both patches can be defined, since both patch I and II are contractible.
However, I am also looking at another litherature, the patches are divided as
The author says although the two patches A and B are not contractible, a smooth vector bundle over A and B can still be found:
and the Chern number is evaluated in the same manner as the first case. I cannot find out why a smooth vector bundle on A or B must exist, can anybody explain ?
thanks very much .
then the Chern number is computed as the difference ##\frac{i}{{2\pi }}\left( {\oint_C {{A^I}} - \oint_C {{A^{II}}} } \right)##, this can be done because a smooth ##A## on both patches can be defined, since both patch I and II are contractible.
However, I am also looking at another litherature, the patches are divided as
The author says although the two patches A and B are not contractible, a smooth vector bundle over A and B can still be found:
and the Chern number is evaluated in the same manner as the first case. I cannot find out why a smooth vector bundle on A or B must exist, can anybody explain ?
thanks very much .