In mathematics, a parallelization [1] of a manifold of dimension n is a set of n global smooth linearly independent vector fields.
In mathematics, a parallelization [1] of a manifold of dimension n is a set of n global smooth linearly independent vector fields.
Given a manifold of dimension n, a parallelization of is a set of n smooth vector fields defined on all of such that for every the set is a basis of , where denotes the fiber over of the tangent vector bundle .
A manifold is called parallelizable whenever it admits a parallelization.
Proposition. A manifold is parallelizable iff there is a diffeomorphism such that the first projection of is and for each the second factor—restricted to —is a linear map .
In other words, is parallelizable if and only if is a trivial bundle. For example, suppose that is an open subset of , i.e., an open submanifold of . Then is equal to , and is clearly parallelizable. [2]
In differential geometry, the tangent bundle of a differentiable manifold is a manifold which assembles all the tangent vectors in . As a set, it is given by the disjoint union of the tangent spaces of . That is,
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 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.
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In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear.
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In mathematics, a differentiable manifold of dimension n is called parallelizable if there exist smooth vector fields
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In mathematics, the Melnikov method is a tool to identify the existence of chaos in a class of dynamical systems under periodic perturbation.
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