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An immersion of a smooth $d$-manifold in ${\mathbb{R}}^{d+1}$ supplies a rank-one normal bundle $\nu$ with $TM\oplus \nu \cong {\epsilon }^{d+1}$. The Whitney product formula for Stiefel–Whitney classes therefore requires $w(\nu )=w(TM{)}^{-1}$. A real line bundle has no positive-degree classes beyond $w_1$, so every higher-degree part of the inverse total class must vanish. For ${\mathbb{RP}}^2\times {\mathbb{RP}}^2$, the stabilized tangent bundle of real projective space gives $w(TM{)}^{-1}=(1+a)(1+b)$ in ${\mathbb{F}}_2[a,b]/(a^3,b^3)$. Its nonzero degree-two part $ab$ excludes an immersion in ${\mathbb{R}}^5$ even though the normal line need not be orientable.

Local defining functions $f_{\alpha }$ for a properly embedded hypersurface $Y\subset X$ have nowhere-zero smooth ratios $f_{\alpha }/f_{\beta }$ on overlaps. These ratios define a real line bundle with a global transverse section whose local representatives are the $f_{\alpha }$ and whose zero set is $Y$. The bundle is naturally isomorphic to the normal bundle of $Y$.

An immersion $M^4\to {\mathbb{R}}^5$ would inject $TM$ into the trivial rank-five bundle. Taking its orthogonal complement gives a normal real line bundle $\nu$ with $TM\oplus \nu \cong {\epsilon }^5$. The Whitney product formula for Stiefel–Whitney classes requiresThe real line bundle need not be orientable, but it has no Stiefel–Whitney classes above degree one.

Let $a,b$ be the pulled-back degree-one generators from the two real projective planes. Their mod-two cohomology ring, by the Künneth theorem, is ${\mathbb{F}}_2[a,b]/(a^3,b^3)$. To compute the tangent classes, at a line $L$ the projective tangent space is $\mathrm{Hom}(L,L^{\perp })$. Adding $\mathrm{Hom}(L,L)\cong \mathbb{R}$ gives $\mathrm{Hom}(L,{\mathbb{R}}^3)\cong 3L$, with the real line identified with its dual using a metric. Thus the stabilized tangent bundle of real projective space gives $T{\mathbb{RP}}^2\oplus {\epsilon }^1\cong 3\gamma$ andThe tangent bundle of the product is the direct sum of the pulled-back tangent bundles. Since $(1+a+a^2)(1+a)=1$ in this truncated ring, and similarly for $b$,The degree-two class $ab$ is nonzero by the product-ring basis. It cannot be a class of the rank-one bundle $\nu$. Equivalently $w_1(TM{)}^2+w_2(TM)=ab\ne 0$, violating the codimension-one Stiefel–Whitney immersion obstruction. Hence ${\mathbb{RP}}^2\times {\mathbb{RP}}^2$ admits no immersion in ${\mathbb{R}}^5$.

Let $E$ have rank $r\ge 1$, let $p:\mathbb{P}(E)\to X$ be its projectivization of a real vector bundle, and let $\lambda \subset p^{\ast }E$ be its tautological bundle. Put $u=w_1(\lambda )\in H^1(\mathbb{P}(E);{\mathbb{F}}_2)$. The Projective bundle definition of Stiefel–Whitney classes starts from the isomorphismIn particular, $p^{\ast }$ is injective. Expand $u^r$ uniquely in this basis. The coefficients define the Stiefel–Whitney classes through the relationSet $w_0(E)=1$ and $w_i(E)=0$ for $i>r$; for a rank-zero vector bundle set $w(E)=1$. Uniqueness also proves naturality of the Stiefel–Whitney classes under a pullback vector bundle.

To prove the Whitney product formula for Stiefel–Whitney classes, apply the splitting principle for real vector bundles simultaneously to $E$ and $E^{\prime }$. Its injective pullback on mod-two cohomology follows by iterating the Projective bundle definition of Stiefel–Whitney classes. It suffices to compute for a split vector bundle $E={⨁}_{i=1}^r{L}_i$. Put $x_i=w_1(L_i)$.

On $\mathbb{P}(E)$, the coordinate projections of the tautological inclusion $\lambda \hookrightarrow p^{\ast }E$ give sections of the real line bundles ${\lambda }^{\ast }\otimes p^{\ast }{L}_i$ with no common zero. The mod-two Euler class of a real line bundle is its first Stiefel–Whitney class, and the First Stiefel–Whitney class of a tensor product of real line bundles gives $u+p^{\ast }{x}_i$. Multiplication of Euler classes under a direct sum, and vanishing of the Euler class of a vector bundle with a nowhere-zero section, therefore giveComparing with the unique monic relation defining the Stiefel–Whitney classes shows that $w_j(E)$ is the $j$th elementary symmetric polynomial in the $x_i$. Thus $w(E)={\prod }_i(1+x_i)$. A direct sum joins the two lists, so injectivity of the splitting pullback gives

For the immersion obstruction, use the notation $g=(f,f)$: the printed $f\times f$ here means the map from one copy of Real projective space into the product. Write $x=w_1({\gamma }_{\mathbb{R}})$ and let $z$ be the mod-two reduction of $c_1({\gamma }_{\mathbb{C}}^{\ast })$. The mod-two cohomology ring of real projective space and the cohomology ring of complex projective space giveNontriviality in degree two forces $f^{\ast }z=x^2$. Necessarily $k\ge 1$, since the degree-two cohomology of ${\mathbb{CP}}^0$ vanishes.

The standard tangent bundle identities for Real projective space and Complex projective space areFor the underlying real vector bundle of a complex line bundle, its total Stiefel–Whitney class is $1+c_1\mathrm{mod}2$. HenceIf $g$ is homotopic to an immersion $j$, its normal bundle $\nu$ has rank $4k-(2k+1)=2k-1$, andThe Whitney product formula for Stiefel–Whitney classes and homotopy invariance of cohomology yieldThe division is legitimate in the truncated cohomology ring, since $1+x$ is a unit. The degree-$2k$ component isIt must vanish because $2k>\mathrm{rank}\nu$, while $x^{2k}\ne 0$. Therefore $k$ must be odd.

Choose a locally finite cover $(U_{\alpha })$ such that on every chart meeting $Y$ there is a smooth defining function $f_{\alpha }$ withand take $f_{\alpha }=1$ on charts disjoint from $Y$. On an overlap, the supplied division lemma extendssmoothly across $Y$. After shrinking the charts, this extension is nowhere zero. The identities $g_{\alpha \beta }{g}_{\beta \gamma }=g_{\alpha \gamma }$ make these functions transition functions for a real line bundle $L\to X$.

Choose local frames $e_{\alpha }$ with $e_{\beta }=g_{\alpha \beta }{e}_{\alpha }$. Then the local sectionsagree on overlaps and define a global section. Its zero set is exactly $Y$. Along $Y$, its vertical derivative is represented by the nonzero covector $d{f}_{\alpha }$, so $s$ is transverse to the zero section, as in the transverse intersection theorem. This is the defining line bundle of a properly embedded hypersurface.

Over Real projective space ${\mathbb{RP}}^n$, the real tautological line bundle isOn the standard chart $U_i=\{[x]:x_i\ne 0\}$, let $s_i([x])=x/x_i$. A local trivialization sends $([x],t)$ to $([x],t{s}_i([x]))$, with inverse $([x],v)\mapsto ([x],v_i)$. These maps prove that this is a real line bundle. Its unit sphere bundle is the antipodal map quotient $S^n\to {\mathbb{RP}}^n$. Its mod-two Euler class of a real line bundle is the degree-one generator of the mod-two cohomology ring of real projective space.