This is when studying about Chern classes from Kobayashi and Nomizu.

Let $ \pi:E\rightarrow M$ be a complex vector bundle with fibre $ \mathbb{C}^r$ and Group $ G=GL(r,\mathbb{C})$ .

Let $ p:P\rightarrow M$ be associated principal $ G$ bundle. Let $ \mathfrak{g}=\mathfrak{gl}(r,\mathbb{C})$ denote the Lie algebra of $ G$ .

Given $ B\in \mathfrak{g}$ , the determinant $ \text{det}\left(\lambda I_r-\frac{1}{2\pi\sqrt{-1}} B\right)$ is $ \sum_{k=0}^r a_k\lambda^{r-k}$ for some $ a_k\in \mathbb{C}$ .

Given $ B\in \mathfrak{g}$ , we have $ r$ elements in $ \mathbb{C}$ . Varying $ B$ over $ \mathfrak{g}$ , gives $ r$ functions $ f_k:\mathfrak{g}\rightarrow \mathbb{C}$ .

We have $ $ \text{det}\left(\lambda I_r-\frac{1}{2\pi\sqrt{-1}} X\right)=\sum_{k=0}^r f_k(X) \lambda ^{r-k}$ $ These $ f_k:\mathfrak{gl}(r,\mathbb{C})\rightarrow \mathbb{C}$ are homogeneous, degree $ k$ polynomial functions on $ \mathfrak{gl}(r,\mathbb{C})$ . I can recall what are polynomial functions on a vector space if some one needs it.

These are $ GL(r,\mathbb{C})$ invariant i.e., $ f_k(X)=f_k(DXD^{-1})$ for all $ D\in Gl(r,\mathbb{C})$ . These $ f_k$ gives a symmetric, multilinear, $ Gl(r,\mathbb{C})$ invariant mappings $ f_k\in I^k(G)$ .

Let $ \Gamma$ be a connection on $ P(M,G)$ and $ \Omega$ be its curvature form. This $ f_k$ gives a $ 2k$ form $ f_k(\Omega)$ on $ P$ . Let $ \gamma_k$ be the unique closed $ 2k$ -form on $ M$ such that $ p^*(\gamma_k)=f_k(\Omega)$ .

We then have $ $ \sum_{k=0}^r f_k(\Omega)=\sum_{k=0}^rp^*(\gamma_k)=p^*(1+\gamma_1+\cdots+\cdots+\gamma_r)$ $ . Then, they (Kobayashi and Nomizu, page no $ 307$ ) write $ $ \text{det}\left(I_r-\frac{1}{2\pi \sqrt{-1}} \Omega\right)= p^*(1+\gamma_1+\cdots+\cdots+\gamma_r)$ $ . I see that they are just replacing $ X$ with $ \Omega$ . But what does it mean to say determinant of $ \Omega$ ?