 Solution

Let $p:\mathbb{P}(E)\to X$ be the projectivization of a real vector bundle, let $L_E$ be its tautological bundle, and putThe mod-two projective bundle formula says that $H^{\ast }(\mathbb{P}(E);{\mathbb{F}}_2)$ is a free $H^{\ast }(X;{\mathbb{F}}_2)$-module on $1,u,\dots ,u^{d-1}$. The Projective bundle definition of Stiefel–Whitney classes is the unique relation

Apply the splitting principle for real vector bundles. After an injective pullback, writeIf $x_i=w_1(L_i)$, the projective-bundle relation factors assoThe line summands of $E\oplus E^{\prime }$ are the union of the two lists, henceComparing the degree-$k$ components proves the Whitney product formula for Stiefel–Whitney classesInjectivity of the splitting pullback returns the identity to $X$.

For real line bundles, the transition functions take values in $O(1)=\{\pm 1\}$. Tensor product multiplies these signs, while the identification $\{\pm 1\}\cong \mathbb{Z}/2$ turns multiplication into addition. The corresponding degree-one characteristic classes therefore satisfy the First Stiefel–Whitney class of a tensor product of real line bundles formulaEquivalently, this follows from the classification of real line bundles by $H^1(X;{\mathbb{F}}_2)$.

Now take $M={\mathbb{RP}}^n$ and $E=k{\gamma }_{\mathbb{R}}^{1,n+1}$. Sincea line in $E_x$ is the fixed line ${\gamma }_x$ tensored with a line in ${\mathbb{R}}^k$. Thus the projectivization of copies of the real tautological line bundle isLet $x$ and $v$ be the degree-one generators pulled back from the first and second factors. The mod-two cohomology ring of real projective space and the Künneth theorem giveThe tautological line $L_E$ is the tensor product of the two tautological lines, so $w_1(L_E)=x+v$. In the alternative generator $u=w_1(L_E)$, the same ring is

The stable tangent-bundle identitygivesFor the vertical part of the tangent bundle of a projectivized real vector bundle,Each of the $k$ line summands on the right has first Stiefel–Whitney classThe Whitney product formula for Stiefel–Whitney classes therefore yieldsthe Total Stiefel–Whitney class of the projectivization of copies of the tautological line.