In the end of the proof of theorem1.9, we have
\n\\[
\n tr \\left[ f(R^E)\\right]-tr \\left[ f(\\tilde{R}^E)\\right]=-d\\int_0^1tr \\left[ \\frac{d\\nabla_t^E}{dt}f'(R_t^E)\\right]dt.
\n\\]
\nThe transgressed term
\n\\[
\n -d\\int_0^1tr \\left[ \\frac{d\\nabla_t^E}{dt}f'(R_t^E)\\right]dt
\n\\]
\nis usually called a \\emph{Chern-Simons term}. In many interesting cases, it is a closed form. Of course mathematician interests this case.
\n\\begin{examp}
\nLet $M$ be a compact oriented 3-dimension manifold. Then $TM$ is trivial bundle[Stiefel]. One can choose a fixed global basis $e_1$, $e_2$, $e_3$ of $TM$. Then
\n\\begin{gather*}
\n\\big(\\forall X \\in \\Gamma(TM)\\big)\\quad \\exists f_1,f_2,f_3 \\in C^{\\infty}(M)\\\\s.t.\\qquad X=f_1e_1+f_2e_2+f_3e_3.
\n\\end{gather*}<\/p>\n
Let $d^{TM}$ denote the trivial connection on $TM$ defined by
\n\\[
\n d^{TM}(f_1e_1+f_2e_2+f_3e_3)=df_1\\cdot e_1+df_2\\cdot e_2+df_3\\cdot e_3.
\n\\]
\nThen any connection $\\nabla^{TM}$ on $TM$ can be written as
\n\\[
\n\\nabla^{TM}= d^{TM}+A, \\quad A\\in \\Omega^1(M; End(M)).
\n\\]
\nSet
\n\\[
\n \\nabla^{TM}_t=d^{TM}+tA,\\quad t\\in[0,1].
\n\\]
\nAnd, take $f(x)=-x^2$. By $\\dim(M)=3$ and $(R^E)^2 \\in \\Omega^4(M)=\\{0\\}$, one can obtain $(R^E)^2=0$.
\nThen
\n\\begin{align*}
\n0=&tr \\left[ f(d^{TM})\\right]-tr \\left[ f(R^{TM})\\right]=-d\\int_0^1tr \\left[ \\frac{d\\nabla_t^{TM}}{dt}f'(R_t^{TM})\\right]dt\\\\
\n=&-d\\int_0^1tr\\left[A\\left(-2(d^{TM}+tA)^2 \\right) \\right]dt\\\\
\n=&2d\\int_0^1tr\\left[A\\wedge\\left(t(d^{TM}\\circ A+A\\circ d^{TM})+t^2A\\wedge A \\right) \\right]dt\\\\
\n=&2d\\int_0^1tr\\left[tA\\wedge(d^{TM}A)+t^2A\\wedge A\\wedge A\\right]dt\\\\
\n=&d\\left\\{tr\\left[A\\wedge (d^{TM}A)+\\frac{2}{3}A\\wedge A\\wedge A\\right]\\right\\}.
\n\\end{align*}
\nThe term ‘$tr\\left[A\\wedge (d^{TM}A)+\\frac{2}{3}A\\wedge A\\wedge A\\right]$’ is (up to a constant) the \\emph{Chern-simons form}.
\n\\end{examp} <\/p>\n","protected":false},"excerpt":{"rendered":"
In the end of the proof of theorem1.9, w …<\/p>\n