Vector field on a manifold
A section of the tangent bundle . The set of differentiable vector fields forms a module over the ring of differentiable functions on .
For a chart of the manifold one defines the -th basic vector field according to the formula
where is the -th basic tangent vector to at the point . Any vector field can be uniquely represented in the form
where are the components of in . Since a vector field can be regarded as a derivation of the ring (see example 2), the set of vector fields forms a Lie algebra with respect to the commutation operation (the Lie bracket).
For the chart and , the function is defined by the formula
where is the partial derivative with respect to . Note that ; is called the derivative of in the direction .
For the chart and , the commutator (Lie bracket) of the vector fields
is defined by the formula
It satisfies the relations
Each vector field induces a local flow on — a family of diffeomorphisms of a neighbourhood ,
such that for and
is the integral curve of the vector field through , i.e.
where is the tangent vector to at . Conversely, a vector field is associated with a local flow , which is a variation of the mapping ; here
Each vector field defines a Lie derivation of a tensor field of type with values in a vector space (infinitesimal transformation of ), corresponding to the local flow ; its special cases include the action of the vector field on ,
and the Lie bracket
A vector field without singularities generates an integrable one-dimensional differential system as well as a Pfaffian system associated with it on .
A generalization of the concept of a vector field on a manifold is that of a vector field along a mapping , which is a section of the bundle induced by , as well as a tensor field of type , which is a section of the bundle associated with with the aid of the functor .
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Vector field on a manifold. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Vector_field_on_a_manifold&oldid=24590