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).
Definition
Riemannian manifold Let ( M , g ) {\displaystyle (M,g)} be a Riemannian manifold, and S ⊂ M {\displaystyle S\subset M} a Riemannian submanifold. Define, for a given p ∈ S {\displaystyle p\in S} , a vector n ∈ T p M {\displaystyle n\in \mathrm {T} _{p}M} to be normal to S {\displaystyle S} whenever g ( n , v ) = 0 {\displaystyle g(n,v)=0} for all v ∈ T p S {\displaystyle v\in \mathrm {T} _{p}S} (so that n {\displaystyle n} is orthogonal to T p S {\displaystyle \mathrm {T} _{p}S} ). The set N p S {\displaystyle \mathrm {N} _{p}S} of all such n {\displaystyle n} is then called the normal space to S {\displaystyle S} at p {\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 N S {\displaystyle \mathrm {N} S} to S {\displaystyle S} is defined as
N S := ∐ p ∈ S N p S {\displaystyle \mathrm {N} S:=\coprod _{p\in S}\mathrm {N} _{p}S} . 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 (of M {\displaystyle M} ).
General definition More abstractly, given an immersion i : N → M {\displaystyle i:N\to M} (for instance an embedding), one can define a normal bundle of N {\displaystyle N} in M {\displaystyle M} , by at each point of N {\displaystyle N} , taking the quotient space of the tangent space on M {\displaystyle M} by the tangent space on N {\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 p : V → V / W {\displaystyle p:V\to V/W} ). Thus the normal bundle is in general a quotient of the tangent bundle of the ambient space M {\displaystyle M} restricted to the subspace N {\displaystyle N} . Formally, the normal bundle to N {\displaystyle N} in M {\displaystyle M} is a quotient bundle of the tangent bundle on M {\displaystyle M} : one has the short exact sequence of vector bundles on N {\displaystyle N} :
0 → T N → T M | i ( N ) → T M / N := T M | i ( N ) / T N → 0 {\displaystyle 0\to \mathrm {T} N\to \mathrm {T} M\vert _{i(N)}\to \mathrm {T} _{M/N}:=\mathrm {T} M\vert _{i(N)}/\mathrm {T} N\to 0}
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