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All cotangent spaces at points on a connected manifold have the same dimension, equal to the dimension of the manifold.All the cotangent spaces of a manifold can be "glued together" (i.e. unioned and endowed with a topology) to form a new differentiable manifold of twice the dimension, the cotangent bundle of the manifold.
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 .
Let M be the projectivization of the cotangent bundle of N: thus M is fiber bundle over N whose fiber at a point x is the space of lines in T*N, or, equivalently, the space of hyperplanes in TN. The 1-form λ does not descend to a genuine 1-form on M.
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 . But there the one form defined is the sum of y i d x i {\displaystyle y_{i}\,dx_{i}} , and the differential is the canonical symplectic form, the ...
Canonical coordinates are defined as a special set of coordinates on the cotangent bundle of a manifold.They are usually written as a set of (,) or (,) with the x ' s or q ' s denoting the coordinates on the underlying manifold and the p ' s denoting the conjugate momentum, which are 1-forms in the cotangent bundle at point q in the manifold.
In complex geometry, ... It can be defined in terms of the complex structure ... is the inner product on the exterior products of the cotangent space of ...
A covector or cotangent vector has components that co-vary with a change of basis in the corresponding (initial) vector space. That is, the components must be transformed by the same matrix as the change of basis matrix in the corresponding (initial) vector space. The components of covectors (as opposed to those of vectors) are said to be ...