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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

• General definition

• Conormal bundle

• Stable normal bundle

• Dual to tangent bundle

• For symplectic manifolds

• References

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 bundlecite-ref-1[1] 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.

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 bundlecite-ref-2[2] 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}

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

If Y ⊆ ⊆ X {\displaystyle Y\subseteq X} is a smooth submanifold of a manifold X {\displaystyle X} , we can pick local coordinates ( x 1 , … … , x n ) {\displaystyle (x_{1},\dots ,x_{n})} around p ∈ ∈ Y {\displaystyle p\in Y} such that Y {\displaystyle Y} is locally defined by x k + 1 = ⋯ ⋯ = x n = 0 {\displaystyle x_{k+1}=\dots =x_{n}=0} ; then with this choice of coordinates

T p X = R { ∂ ∂ ∂ ∂ x 1 | p , … … , ∂ ∂ ∂ ∂ x k | p , … … , ∂ ∂ ∂ ∂ x n | p } T p Y = R { ∂ ∂ ∂ ∂ x 1 | p , … … , ∂ ∂ ∂ ∂ x k | p } T X / Y p = R { ∂ ∂ ∂ ∂ x k + 1 | p , … … , ∂ ∂ ∂ ∂ x n | p } {\displaystyle {\begin{aligned}\mathrm {T} _{p}X&=\mathbb {R} {\Big \lbrace }{\frac {\partial }{\partial x_{1}}}{\Big |}_{p},\dots ,{\frac {\partial }{\partial x_{k}}}{\Big |}_{p},\dots ,{\frac {\partial }{\partial x_{n}}}{\Big |}_{p}{\Big \rbrace }\\\mathrm {T} _{p}Y&=\mathbb {R} {\Big \lbrace }{\frac {\partial }{\partial x_{1}}}{\Big |}_{p},\dots ,{\frac {\partial }{\partial x_{k}}}{\Big |}_{p}{\Big \rbrace }\\{\mathrm {T} _{X/Y}}_{p}&=\mathbb {R} {\Big \lbrace }{\frac {\partial }{\partial x_{k+1}}}{\Big |}_{p},\dots ,{\frac {\partial }{\partial x_{n}}}{\Big |}_{p}{\Big \rbrace }\\\end{aligned}}}

and the ideal sheaf is locally generated by x k + 1 , … … , x n {\displaystyle x_{k+1},\dots ,x_{n}} . Therefore we can define a non-degenerate pairing

( I Y / I Y 2 ) p × × T X / Y p ⟶ ⟶ R {\displaystyle (I_{Y}/I_{Y}^{\ 2})_{p}\times {\mathrm {T} _{X/Y}}_{p}\longrightarrow \mathbb {R} }

that induces an isomorphism of sheaves T X / Y ≃ ≃ ( I Y / I Y 2 ) ∨ ∨ {\displaystyle \mathrm {T} _{X/Y}\simeq (I_{Y}/I_{Y}^{\ 2})^{\vee }} . We can rephrase this fact by introducing the conormal bundle T X / Y ∗ ∗ {\displaystyle \mathrm {T} _{X/Y}^{*}} defined via the conormal exact sequence

0 → → T X / Y ∗ ∗ ↣ ↣ Ω Ω X 1 | Y ↠ ↠ Ω Ω Y 1 → → 0 {\displaystyle 0\to \mathrm {T} _{X/Y}^{*}\rightarrowtail \Omega _{X}^{1}|_{Y}\twoheadrightarrow \Omega _{Y}^{1}\to 0} ,

then T X / Y ∗ ∗ ≃ ≃ ( I Y / I Y 2 ) {\displaystyle \mathrm {T} _{X/Y}^{*}\simeq (I_{Y}/I_{Y}^{\ 2})} , viz. the sections of the conormal bundle are the cotangent vectors to X {\displaystyle X} vanishing on T Y {\displaystyle \mathrm {T} Y} .

When Y = { p } {\displaystyle Y=\lbrace p\rbrace } is a point, then the ideal sheaf is the sheaf of smooth germs vanishing at p {\displaystyle p} and the isomorphism reduces to the definition of the tangent space in terms of germs of smooth functions on X {\displaystyle X}

T X / { p } ∗ ∗ ≃ ≃ ( T p X ) ∨ ∨ ≃ ≃ m p m p 2 {\displaystyle \mathrm {T} _{X/\lbrace p\rbrace }^{*}\simeq (\mathrm {T} _{p}X)^{\vee }\simeq {\frac {{\mathfrak {m}}_{p}}{{\mathfrak {m}}_{p}^{\ 2}}}} .

Abstract manifolds have a canonical tangent bundle, but do not have a normal bundle: only an embedding (or immersion) of a manifold in another yields a normal bundle. However, since every manifold can be embedded in R N {\displaystyle \mathbf {R} ^{N}} , by the Whitney embedding theorem, every manifold admits a normal bundle, given such an embedding.

There is in general no natural choice of embedding, but for a given manifold X {\displaystyle X} , any two embeddings in R N {\displaystyle \mathbf {R} ^{N}} for sufficiently large N {\displaystyle N} are regular homotopic, and hence induce the same normal bundle. The resulting class of normal bundles (it is a class of bundles and not a specific bundle because the integer N {\displaystyle {N}} could vary) is called the stable normal bundle.

The normal bundle is dual to the tangent bundle in the sense of K-theory: by the above short exact sequence,

[ T N ] + [ T M / N ] = [ T M ] {\displaystyle [\mathrm {T} N]+[\mathrm {T} _{M/N}]=[\mathrm {T} M]}

in the Grothendieck group. In case of an immersion in R N {\displaystyle \mathbf {R} ^{N}} , the tangent bundle of the ambient space is trivial (since R N {\displaystyle \mathbf {R} ^{N}} is contractible, hence parallelizable), so [ T N ] + [ T M / N ] = 0 {\displaystyle [\mathrm {T} N]+[\mathrm {T} _{M/N}]=0} , and thus [ T M / N ] = − − [ T N ] {\displaystyle [\mathrm {T} _{M/N}]=-[\mathrm {T} N]} .

This is useful in the computation of characteristic classes, and allows one to prove lower bounds on immersibility and embeddability of manifolds in Euclidean space.

Suppose a manifold X {\displaystyle X} is embedded in to a symplectic manifold ( M , ω ω ) {\displaystyle (M,\omega )} , such that the pullback of the symplectic form has constant rank on X {\displaystyle X} . Then one can define the symplectic normal bundle to X {\displaystyle X} as the vector bundle over X {\displaystyle X} with fibres

( T i ( x ) X ) ω ω / ( T i ( x ) X ∩ ∩ ( T i ( x ) X ) ω ω ) , x ∈ ∈ X , {\displaystyle (\mathrm {T} _{i(x)}X)^{\omega }/(\mathrm {T} _{i(x)}X\cap (\mathrm {T} _{i(x)}X)^{\omega }),\quad x\in X,}

where i : X → → M {\displaystyle i:X\rightarrow M} denotes the embedding and ( T X ) ω ω {\displaystyle (\mathrm {T} X)^{\omega }} is the symplectic orthogonal of T X {\displaystyle \mathrm {T} X} in T M {\displaystyle \mathrm {T} M} . Notice that the constant rank condition ensures that these normal spaces fit together to form a bundle. Furthermore, any fibre inherits the structure of a symplectic vector space.cite-ref-3[3]

By Darboux's theorem, the constant rank embedding is locally determined by i ∗ ∗ ( T M ) {\displaystyle i^{*}(\mathrm {T} M)} . The isomorphism

i ∗ ∗ ( T M ) ≅ ≅ T X / ν ν ⊕ ⊕ ( T X ) ω ω / ν ν ⊕ ⊕ ( ν ν ⊕ ⊕ ν ν ∗ ∗ ) {\displaystyle i^{*}(\mathrm {T} M)\cong \mathrm {T} X/\nu \oplus (\mathrm {T} X)^{\omega }/\nu \oplus (\nu \oplus \nu ^{*})}

(where ν ν = T X ∩ ∩ ( T X ) ω ω {\displaystyle \nu =\mathrm {T} X\cap (\mathrm {T} X)^{\omega }} and ν ν ∗ ∗ {\displaystyle \nu ^{*}} is the dual under ω ω {\displaystyle \omega } ,) of symplectic vector bundles over X {\displaystyle X} implies that the symplectic normal bundle already determines the constant rank embedding locally. This feature is similar to the Riemannian case.

cite-note-11. ↑ John M. Lee, Riemannian Manifolds, An Introduction to Curvature, (1997) Springer-Verlag New York, Graduate Texts in Mathematics 176 ISBN 978-0-387-98271-7

cite-note-22. ↑ Tammo tom Dieck, Algebraic Topology, (2010) EMS Textbooks in Mathematics ISBN 978-3-03719-048-7

cite-note-33. ↑ Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X