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Concept
Cotangent bundle
Summary
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 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.
There are several equivalent ways to define the cotangent bundle. One way is through a diagonal mapping Δ and germs.
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 be the sheaf of germs of smooth functions on M×M which vanish on the diagonal. Then the quotient sheaf consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms. The cotangent sheaf is defined as the 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.
Smooth sections of the cotangent bundle are called (differential) one-forms.
A smooth morphism of manifolds, induces a pullback sheaf on M. There is an induced map of vector bundles .
The tangent bundle of the vector space is , and the cotangent bundle is , where denotes the dual space of covectors, linear functions .
Given a smooth manifold embedded as a hypersurface represented by the vanishing locus of a function with the condition that the tangent bundle is
where is the directional derivative . By definition, the cotangent bundle in this case is
where Since every covector corresponds to a unique vector for which for an arbitrary
Since the cotangent bundle X = T*M is a vector bundle, it can be regarded as a manifold in its own right.
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Ontological neighbourhood
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Differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart.
Sheaf (mathematics)
In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts).
One-form (differential geometry)
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold is a smooth mapping of the total space of the tangent bundle of to whose restriction to each fibre is a linear functional on the tangent space. Symbolically, where is linear. Often one-forms are described locally, particularly in local coordinates. In a local coordinate system, a one-form is a linear combination of the differentials of the coordinates: where the are smooth functions.
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