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). 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 ( M , g ) {displaystyle (M,g)} be a Riemannian manifold, and S ⊂ M {displaystyle Ssubset M} a Riemannian submanifold. Define, for a given p ∈ S {displaystyle pin S} , a vector n ∈ T p M {displaystyle nin 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 vin 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 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 icolon N o M} (for instance an embedding), one can define a normal bundle of N in M, by at each point of N, taking the quotient space of the tangent space on M by the tangent space on 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 V → V / W {displaystyle V o V/W} ). Thus the normal bundle is in general a quotient of the tangent bundle of the ambient space restricted to the subspace. Formally, the normal bundle to N in M is a quotient bundle of the tangent bundle on M: one has the short exact sequence of vector bundles on N: where T M | i ( N ) {displaystyle TMvert _{i(N)}} is the restriction of the tangent bundle on M to N (properly, the pullback i ∗ T M {displaystyle i^{*}TM} of the tangent bundle on M to a vector bundle on N via the map i {displaystyle i} ). The fiber of the normal bundle T M / N ↠ π N {displaystyle T_{M/N}{overset {pi }{ woheadrightarrow }}N} in p ∈ N {displaystyle pin N} is referred to as the normal space at p {displaystyle p} (of N {displaystyle N} in M {displaystyle M} ). If Y ⊆ X {displaystyle Ysubseteq 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 pin 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