know whether the following question is true. Let $\xi$ and $\eta$ be sums of $k$ complex line bundles over a generalized Bott manifold $B$. [*Are two bundles $\xi$ and $\eta$ isomorphic if their total Chern classes are equal?*]{} The answer is true when $B$ is a generalized Bott tower and $\eta$ is the trivial bundle, see [@CMS]. In this section we provide two more affirmative answers to the question. They are Proposition \[proposition:trivial bundle\] and Proposition \[proposition:isomorphism of sum of two line bundle\]. We first need the following lemma. We sometimes confuse Bott tower with its last stage Bott manifold when they are clear from the context.

\[lemma:2-1\] Let $B_n$ and $B_n'$ be two $n$ stage Bott towers. If the associated matrices to them are $$\left(
      \begin{array}{ccccc}
        0    & \ast   & \ast & b_1 & a_1 \\
             & \ddots & \ast & \vdots & \vdots \\
             &   & 0 & b_{n-2} & a_{n-2} \\
            &   &   & 0 & 0 \\
         &  &  & & 0 \\
      \end{array}
    \right) {\textrm and }     \left(
      \begin{array}{ccccc}
        0    & \ast   & \ast & a_1 & b_1 \\
             & \ddots & \ast & \vdots & \vdots \\
             &   & 0 & a_{n-2} & b_{n-2} 