In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion).

Let ${\displaystyle (M,g)}$ be a Riemannian manifold, and ${\displaystyle S\subset M}$ a Riemannian submanifold. Define, for a given ${\displaystyle p\in S}$, a vector ${\displaystyle n\in \mathrm {T} _{p}M}$ to be normal to ${\displaystyle S}$ whenever ${\displaystyle g(n,v)=0}$ for all ${\displaystyle v\in \mathrm {T} _{p}S}$ (so that ${\displaystyle n}$ is orthogonal to ${\displaystyle \mathrm {T} _{p}S}$). The set ${\displaystyle \mathrm {N} _{p}S}$ of all such ${\displaystyle n}$ is then called the normal space to ${\displaystyle S}$ at ${\displaystyle p}$.

Just as the total space of the tangent bundle to a manifold is constructed from all tangent spaces to the manifold, the total space of the normal bundle ${\displaystyle \mathrm {N} S}$ to ${\displaystyle S}$ is defined as

The conormal bundle is defined as the dual bundle to the normal bundle. It can be realised naturally as a sub-bundle of the cotangent bundle.

More abstractly, given an immersion ${\displaystyle i:N\to M}$ (for instance an embedding), one can define a normal bundle of ${\displaystyle N}$ in ${\displaystyle M}$, by at each point of ${\displaystyle N}$, taking the quotient space of the tangent space on ${\displaystyle M}$ by the tangent space on ${\displaystyle N}$. For a Riemannian manifold one can identify this quotient with the orthogonal complement, but in general one cannot (such a choice is equivalent to a section of the projection ${\displaystyle p:V\to V/W}$).

Thus the normal bundle is in general a quotient of the tangent bundle of the ambient space ${\displaystyle M}$ restricted to the subspace ${\displaystyle N}$.

Formally, the normal bundle to ${\displaystyle N}$ in ${\displaystyle M}$ is a quotient bundle of the tangent bundle on ${\displaystyle M}$: one has the short exact sequence of vector bundles on ${\displaystyle N}$:

where ${\displaystyle \mathrm {T} M\vert _{i(N)}}$ is the restriction of the tangent bundle on ${\displaystyle M}$ to ${\displaystyle N}$ (properly, the pullback ${\displaystyle i^{*}\mathrm {T} M}$ of the tangent bundle on ${\displaystyle M}$ to a vector bundle on ${\displaystyle N}$ via the map ${\displaystyle i}$). The fiber of the normal bundle ${\displaystyle \mathrm {T} _{M/N}{\overset {\pi }{\twoheadrightarrow }}N}$ in ${\displaystyle p\in N}$ is referred to as the normal space at ${\displaystyle p}$ (of ${\displaystyle N}$ in ${\displaystyle M}$).