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In differential geometry, the cotangent space is a vector space associated with a point on a smooth (or differentiable) manifold ; one can define a cotangent ...
The cotangent bundle has a canonical symplectic 2-form on it, as an exterior derivative of the tautological one-form, the symplectic potential. Proving that this form is, indeed, symplectic can be done by noting that being symplectic is a local property: since the cotangent bundle is locally trivial, this definition need only be checked on R n ...
In mathematics, a coordinate-induced basis is a basis for the tangent space or cotangent space of a manifold that is induced by a certain coordinate system. Given the coordinate system x a {\displaystyle x^{a}} , the coordinate-induced basis e a {\displaystyle e_{a}} of the tangent space is given by
The cotangent bundle of a manifold is locally modeled on a space similar to the first example. It can be shown that we can glue these affine symplectic forms hence this bundle forms a symplectic manifold. A less trivial example of a Lagrangian submanifold is the zero section of the cotangent bundle of a manifold. For example, let
denote the tangent bundle and cotangent bundle, respectively, of the smooth manifold . , denote the tangent spaces of , at the points , , respectively. denotes the cotangent space of at the point .
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 [ 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} :
The total complex of this simplicial object is the cotangent complex L B/A. The morphism r induces a morphism from the cotangent complex to Ω B / A called the augmentation map . In the homotopy category of simplicial A -algebras (or of simplicial ringed topoi), this construction amounts to taking the left derived functor of the Kähler ...
The dual space of a vector space is the set of real valued linear functions on the vector space. The cotangent space at a point is the dual of the tangent space at that point and the elements are referred to as cotangent vectors; the cotangent bundle is the collection of all cotangent vectors, along with the natural differentiable manifold ...