Now, we take $f$ to be some special function to obtain some classical classes.<\/p>\n
Sometimes, we would prefer to \\emph{normalize} the function by conceder
\n\\[
\n\\tr\\left[f\\left(\\frac{\\sqrt{-1}}{2\\pi}R^E\\right)\\right],
\n\\]
\nsince $R^E$ is a anti-symmetric matrix, we want to make the eigenvalue be real, and $2\\pi$ just a unitization, such that it becomes rational even integer.
\n\u00a0
\n\\begin{excs}
\n the product of closed forms is still a closed form.
\n\\end{excs}<\/p>\n
\\begin{examp}[Chern form and Chern classes]
\n Let $E$ be a complex vector bundle, and set
\n \\[
\n c(E,\\nabla^E)=\\det\\left(\\I+\\frac{\\sqrt{-1}}{2\\pi}R^E\\right)
\n =\\exp\\left(\\tr\\left[\\log\\left(\\I+\\frac{\\sqrt{-1}}{2\\pi}R^E\\right)\\right]\\right),
\n \\]
\n note that
\n \\[
\n \\I+\\frac{\\sqrt{-1}}{2\\pi}R^E
\n \\]
\n is invertible, and
\n \\[
\n \\exp(\\tr A)=\\det(\\exp(A)).
\n \\]
\n Note also that the power series
\n \\begin{align*}
\n \\log(1+x)&=x+\\frac{x^2}{2}+\\cdots\\\\
\n \\exp(x)&=1+x+\\frac{x^2}{2!}+\\cdots,
\n \\end{align*}
\n substitute $x$ with $\\frac{\\sqrt{-1}}{2\\pi}R^E$, then it only has finite terms and this shows that for any integer $k\\geq0$, $\\tr[(R^E)^k]$ is a linear combination of various products of $c_i(E,\\nabla^E)$’s, this established the fundamental importance of Chern classes in the theory of characteristic classes of complex vector bundles.<\/p>\n
We write
\n \\begin{align*}
\n c(E,\\nabla^E)&=\\exp\\left(\\tr\\left[\\log\\left(\\I+\\frac{\\sqrt{-1}}{2\\pi}R^E\\right)\\right]\\right)\\\\
\n &=1+c_1(E,\\nabla^E)t+c_2(E,\\nabla^E)t^2+\\cdots,
\n \\end{align*}
\n the $c_i(E,\\nabla^E)$ are closed $2$-form, called \\emph{Chern form}, and $c(E,\\nabla^E)\\bigoplus c_i(E,\\nabla^E)$ called the \\emph{total Chern form}, and the cohomology class associated to $k$-th Chern form $c_k(E,\\nabla^E)$ are called the $k$-th Chern classes of second type, denoted as $c_k(E)$, and $\\bigoplus c_k(E)$ are called \\emph{total Chern classes}.
\n\\end{examp}
\n\\begin{examp}[Pontrjagin classes of real vector bundle]
\nLet $E$ be a \\emph{real} vector bundle of $M$, define
\n\\[
\np(E,\\nabla^E)=\\det\\left(I-\\left(\\frac{\\sqrt{-1}}{2\\pi}\\right)^2\\right)^{\\frac{1}{2}}.
\n\\]
\nExpand $\\sqrt{1-x}$, we can write
\n\\[
\np(E,\\nabla^E)=1+P_1(E,\\nabla^E)t+\\cdots,
\n\\]
\nhere $p_k(E,\\nabla^E)$ is closed $4k$-form.
\n\\end{examp}
\nSince any real vector bundle can be complexificated to be $E\\otimes\\C$, and $\\nabla^E$ can be extend to a complex-linear operator $\\nabla^E_\\C$, then
\n\\[
\nc_{2k}(E\\otimes \\C)=(-1)^kp_k(E).
\n\\]
\n\\begin{excs}
\n Prove that claim that
\n \\[
\n c_{2k}(E\\otimes \\C)=(-1)^kp_k(E).
\n \\]
\n Hint: try the consider it from their froms.
\n\\end{excs}
\nSimilarly, if we write
\n\\[
\n\\log\\left(\\det\\left(I-\\left(\\frac{R^E}{2\\pi}\\right)^2\\right)^{\\frac{1}{2}}\\right)
\n=\\tr\\left(\\frac{1}{2}\\log\\left(I-\\left(\\frac{R^E}{2\\pi}\\right)^2\\right)\\right),
\n\\]
\nand from the power series expansion formulas for $\\log(\\sqrt{1-x})$, one deduces that for any integer $k\\geq0$, $\\tr[(R^E)^{2k}]$ can be written as a linear combination of various products of $p_i(E,\\nabla^E)$’s.<\/p>\n
This establishes the fundamental importance of Pontrjagin classes in the theory of characteristic classes of real vector bundles. <\/p>\n","protected":false},"excerpt":{"rendered":"
Now, we take $f$ to be some special func …<\/p>\n