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In differential geometry, the cotangent space is a vector ... vector space being considered, for example, ... a direct definition of the cotangent space without ...
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
For example, this is a way to describe the phase space of a pendulum. The state of the pendulum is determined by its position (an angle) and its momentum (or equivalently, its velocity, since its mass is constant). The entire state space looks like a cylinder, which is the cotangent bundle of the circle.
Examples of coordinate-free statements are that the tangent space corresponds to the velocities of the points , while the cotangent space corresponds to momenta. (Velocities and momenta can be connected; for the most general, abstract case, this is done with the rather abstract notion of the tautological one-form .)
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 ...
Just as a complex structure on a vector space V allows a decomposition of V C into V + and V − (the eigenspaces of J corresponding to +i and −i, respectively), so an almost complex structure on M allows a decomposition of the complexified tangent bundle TM C (which is the vector bundle of complexified tangent spaces at each point) into TM ...
At each point p of the manifold M, the forms α and β are elements of an exterior power of the cotangent space at p. When the exterior algebra is viewed as a quotient of the tensor algebra, the exterior product corresponds to the tensor product (modulo the equivalence relation defining the exterior algebra).
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 .