Normal bundle

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 ${\displaystyle (M,g)}${\displaystyle (M,g)} be a Riemannian manifold, and ${\displaystyle S\subset M}${\displaystyle S\subset M} a Riemannian submanifold. Define, for a given ${\displaystyle p\in S}${\displaystyle p\in S}, a vector ${\displaystyle n\in \mathrm {T} _{p}M}${\displaystyle n\in \mathrm {T} _{p}M} to be normal to ${\displaystyle S}${\displaystyle S} whenever ${\displaystyle g(n,v)=0}${\displaystyle g(n,v)=0} for all ${\displaystyle v\in \mathrm {T} _{p}S}${\displaystyle v\in \mathrm {T} _{p}S} (so that ${\displaystyle n}${\displaystyle n} is orthogonal to ${\displaystyle \mathrm {T} _{p}S}${\displaystyle \mathrm {T} _{p}S}). The set ${\displaystyle \mathrm {N} _{p}S}${\displaystyle \mathrm {N} _{p}S} of all such ${\displaystyle n}${\displaystyle n} is then called the normal space to ${\displaystyle S}${\displaystyle S} at ${\displaystyle p}${\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^[1] ${\displaystyle \mathrm {N} S}${\displaystyle \mathrm {N} S} to ${\displaystyle S}${\displaystyle S} is defined as

${\displaystyle \mathrm {N} S:=\coprod _{p\in S}\mathrm {N} _{p}S}${\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 ${\displaystyle i:N\to M}${\displaystyle i:N\to M} (for instance an embedding), one can define a normal bundle of ${\displaystyle N}${\displaystyle N} in ${\displaystyle M}${\displaystyle M}, by at each point of ${\displaystyle N}${\displaystyle N}, taking the quotient space of the tangent space on ${\displaystyle M}${\displaystyle M} by the tangent space on ${\displaystyle N}${\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 ${\displaystyle p:V\to V/W}${\displaystyle p:V\to V/W}).

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

Formally, the normal bundle^[2] to ${\displaystyle N}${\displaystyle N} in ${\displaystyle M}${\displaystyle M} is a quotient bundle of the tangent bundle on ${\displaystyle M}${\displaystyle M}: one has the short exact sequence of vector bundles on ${\displaystyle N}${\displaystyle N}:

${\displaystyle 0\to \mathrm {T} N\to \mathrm {T} M\vert _{i(N)}\to \mathrm {T} _{M/N}:=\mathrm {T} M\vert _{i(N)}/\mathrm {T} N\to 0}${\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 ${\displaystyle \mathrm {T} M\vert _{i(N)}}${\displaystyle \mathrm {T} M\vert _{i(N)}} is the restriction of the tangent bundle on ${\displaystyle M}${\displaystyle M} to ${\displaystyle N}${\displaystyle N} (properly, the pullback ${\displaystyle i^{*}\mathrm {T} M}${\displaystyle i^{*}\mathrm {T} M} of the tangent bundle on ${\displaystyle M}${\displaystyle M} to a vector bundle on ${\displaystyle N}${\displaystyle N} via the map ${\displaystyle i}${\displaystyle i}). The fiber of the normal bundle ${\displaystyle \mathrm {T} _{M/N}{\overset {\pi }{\twoheadrightarrow }}N}${\displaystyle \mathrm {T} _{M/N}{\overset {\pi }{\twoheadrightarrow }}N} in ${\displaystyle p\in N}${\displaystyle p\in N} is referred to as the normal space at ${\displaystyle p}${\displaystyle p} (of ${\displaystyle N}${\displaystyle N} in ${\displaystyle M}${\displaystyle M}).

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

${\displaystyle {\begin{aligned}\mathrm {T} _{p}X&=\mathbb {R} {\Big \lbrace }{\frac {\partial }{\partial x_{1}}}{\Big |}_{p},\dots ,{\frac {\partial }{\partial x_{k}}}{\Big |}_{p},\dots ,{\frac {\partial }{\partial x_{n}}}{\Big |}_{p}{\Big \rbrace }\\\mathrm {T} _{p}Y&=\mathbb {R} {\Big \lbrace }{\frac {\partial }{\partial x_{1}}}{\Big |}_{p},\dots ,{\frac {\partial }{\partial x_{k}}}{\Big |}_{p}{\Big \rbrace }\\{\mathrm {T} _{X/Y}}_{p}&=\mathbb {R} {\Big \lbrace }{\frac {\partial }{\partial x_{k+1}}}{\Big |}_{p},\dots ,{\frac {\partial }{\partial x_{n}}}{\Big |}_{p}{\Big \rbrace }\\\end{aligned}}}${\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 ${\displaystyle x_{k+1},\dots ,x_{n}}${\displaystyle x_{k+1},\dots ,x_{n}}. Therefore we can define a non-degenerate pairing

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

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

${\displaystyle 0\to \mathrm {T} _{X/Y}^{*}\rightarrowtail \Omega _{X}^{1}|_{Y}\twoheadrightarrow \Omega _{Y}^{1}\to 0}${\displaystyle 0\to \mathrm {T} _{X/Y}^{*}\rightarrowtail \Omega _{X}^{1}|_{Y}\twoheadrightarrow \Omega _{Y}^{1}\to 0},

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

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

${\displaystyle \mathrm {T} _{X/\lbrace p\rbrace }^{*}\simeq (\mathrm {T} _{p}X)^{\vee }\simeq {\frac {{\mathfrak {m}}_{p}}{{\mathfrak {m}}_{p}^{\ 2}}}}${\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 ${\displaystyle \mathbf {R} ^{N}}${\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 ${\displaystyle X}${\displaystyle X}, any two embeddings in ${\displaystyle \mathbf {R} ^{N}}${\displaystyle \mathbf {R} ^{N}} for sufficiently large ${\displaystyle N}${\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 ${\displaystyle {N}}${\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,

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

in the Grothendieck group. In case of an immersion in ${\displaystyle \mathbf {R} ^{N}}${\displaystyle \mathbf {R} ^{N}}, the tangent bundle of the ambient space is trivial (since ${\displaystyle \mathbf {R} ^{N}}${\displaystyle \mathbf {R} ^{N}} is contractible, hence parallelizable), so ${\displaystyle [\mathrm {T} N]+[\mathrm {T} _{M/N}]=0}${\displaystyle [\mathrm {T} N]+[\mathrm {T} _{M/N}]=0}, and thus ${\displaystyle [\mathrm {T} _{M/N}]=-[\mathrm {T} N]}${\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 ${\displaystyle X}${\displaystyle X} is embedded in to a symplectic manifold ${\displaystyle (M,\omega )}${\displaystyle (M,\omega )}, such that the pullback of the symplectic form has constant rank on ${\displaystyle X}${\displaystyle X}. Then one can define the symplectic normal bundle to ${\displaystyle X}${\displaystyle X} as the vector bundle over ${\displaystyle X}${\displaystyle X} with fibres

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

where ${\displaystyle i:X\rightarrow M}${\displaystyle i:X\rightarrow M} denotes the embedding and ${\displaystyle (\mathrm {T} X)^{\omega }}${\displaystyle (\mathrm {T} X)^{\omega }} is the symplectic orthogonal of ${\displaystyle \mathrm {T} X}${\displaystyle \mathrm {T} X} in ${\displaystyle \mathrm {T} M}${\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.^[3]

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

${\displaystyle i^{*}(\mathrm {T} M)\cong \mathrm {T} X/\nu \oplus (\mathrm {T} X)^{\omega }/\nu \oplus (\nu \oplus \nu ^{*})}${\displaystyle i^{*}(\mathrm {T} M)\cong \mathrm {T} X/\nu \oplus (\mathrm {T} X)^{\omega }/\nu \oplus (\nu \oplus \nu ^{*})}

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

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