Cotangent bundle

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may be generalized to categories with more structure than smooth manifolds, such as complex manifolds, or (in the form of cotangent sheaf) algebraic varieties or schemes. In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not in general isomorphic in other categories. In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may be generalized to categories with more structure than smooth manifolds, such as complex manifolds, or (in the form of cotangent sheaf) algebraic varieties or schemes. In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not in general isomorphic in other categories. Smooth sections of the cotangent bundle are differential one-forms. Let M be a smooth manifold and let M×M be the Cartesian product of M with itself. The diagonal mapping Δ sends a point p in M to the point (p,p) of M×M. The image of Δ is called the diagonal. Let I {displaystyle {mathcal {I}}} be the sheaf of germs of smooth functions on M×M which vanish on the diagonal. Then the quotient sheaf I / I 2 {displaystyle {mathcal {I}}/{mathcal {I}}^{2}} consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms. The cotangent sheaf is the pullback of this sheaf to M: By Taylor's theorem, this is a locally free sheaf of modules with respect to the sheaf of germs of smooth functions of M. Thus it defines a vector bundle on M: the cotangent bundle. See also: bundle of principal parts (which generalizes the above constructions to higher orders.) A smooth morphism ϕ : M → N {displaystyle phi colon M o N} of manifolds, induces a pullback sheaf ϕ ∗ T ∗ N {displaystyle phi ^{*}T^{*}N} on M. There is an induced map of vector bundles ϕ ∗ ( T ∗ N ) → T ∗ M {displaystyle phi ^{*}(T^{*}N) o T^{*}M} . The tangent bundle of the vector space R n {displaystyle mathbb {R} ^{n}} is T R n = R n × R n {displaystyle T,mathbb {R} ^{n}=mathbb {R} ^{n} imes mathbb {R} ^{n}} , and the cotangent bundle is T ∗ R n = R n × ( R n ) ∗ {displaystyle T^{*}mathbb {R} ^{n}=mathbb {R} ^{n} imes (mathbb {R} ^{n})^{*}} , where ( R n ) ∗ {displaystyle (mathbb {R} ^{n})^{*}} denotes the dual space of covectors, linear functions v ∗ : R n → R {displaystyle v^{*}:mathbb {R} ^{n} o mathbb {R} } . Given a smooth manifold M ⊂ R n {displaystyle Msubset mathbb {R} ^{n}} embedded as the vanishing locus of a smooth function f {displaystyle f} , its tangent bundle is: where d f x {displaystyle df_{x}} is the covector defined by the directional derivative d f x ( v ) = ∂ f ∂ v ( x ) = ∇ f ( x ) ⋅ v {displaystyle df_{x}(v)={ frac {partial f}{partial v}}(x)= abla !f(x)cdot v} . Its cotangent bundle consists of pairs ( x , v ∗ mod   R d f x ) {displaystyle (x,v^{*},{ ext{mod}} mathbb {R} ,df_{x})} , where f ( x ) = 0 {displaystyle f(x)=0} and we take the covector v ∗ {displaystyle v^{*}} in the quotient space of ( R n ) ∗ {displaystyle (mathbb {R} ^{n})^{*}} modulo the line generated by d f x {displaystyle df_{x}} . Of course, the dot product identifies the quotient space ( R n ) ∗ / R d f x {displaystyle (mathbb {R} ^{n})^{*}/mathbb {R} ,df_{x}} with the orthogonal space to the gradient ∇ f ( x ) {displaystyle abla f(x)} , so the two bundles are isomorphic.