# Related constructions and facts

The tensor product of two vector bundles with the same base space $${(E,M,\mathbb{K}^{m})}$$ and $${(E^{\prime},M,\mathbb{K}^{n})}$$ is another vector bundle

$$\displaystyle (E\otimes E^{\prime},M,\mathbb{K}^{mn}).$$

The direct product of two vector bundles $${(E,M,\mathbb{K}^{m})}$$ and $${(E^{\prime},M^{\prime},\mathbb{K}^{n})}$$ is another vector bundle

$$\displaystyle (E\times E^{\prime},M\times M^{\prime},\mathbb{K}^{m+n}).$$

If we form the direct product of two vector bundles with the same base space, we can then restrict the base space to the diagonal via the pullback by $${f\colon M\times M\rightarrow M}$$ defined by $${(x,x)\mapsto x}$$. The resulting vector bundle is called the Whitney sum (AKA direct sum bundle), and is denoted

$$\displaystyle (E\oplus E^{\prime},M,\mathbb{K}^{m+n}).$$

The total Whitney class of a real vector bundle $${(E,M,\mathbb{R}^{n})}$$ is defined as

$$\displaystyle w(E)\equiv1+w_{1}(E)+w_{2}(E)+\cdots+w_{n}(E).$$

The series is finite since $${w_{i}(E)}$$ vanishes for $${i>n}$$, and is thus an element of $${H^{*}(M,\mathbb{Z}_{2})}$$. The total Whitney class is multiplicative over the Whitney sum, i.e.

$$\displaystyle w(E\oplus E^{\prime})=w(E)w(E^{\prime}).$$

The total Chern class is defined similarly, and has the same multiplicative property.

The flag manifold $${F_{n}(\mathbb{K}^{\infty})}$$ is a limit of the finite-dimensional flag manifold $${F_{n}(\mathbb{K}^{k})}$$, which is all ordered $${n}$$-tuples of orthogonal lines in $${\mathbb{K}^{k}}$$ through the origin. The name is due to the fact that an ordered $${n}$$-tuple of orthogonal lines in $${\mathbb{K}^{k}}$$ is equivalent to an n-flag, a sequence of subspaces $${V_{1}\subset\cdots\subset V_{n}}$$ in $${\mathbb{K}^{k}}$$ where each $${V_{i}}$$ has dimension $${i}$$.