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 .

Example 1.

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).

Example 2.

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 .

Example 3.

For the chart and , the commutator (Lie bracket) of the vector fields

is defined by the formula

It satisfies the relations

in particular,

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 .

References

[1]	C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969) MR0242081 Zbl 0653.53001 Zbl 0284.53018
[2]	D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968) MR0229177 Zbl 0155.30701
[3]	S. Lang, "Introduction to differentiable manifolds" , Interscience (1967) pp. App. III MR1931083 MR1532744 MR0155257 Zbl 1008.57001 Zbl 0103.15101
[4]	K. Nomizu, "Lie groups and differential geometry" , Math. Soc. Japan (1956) MR0084166 Zbl 0071.15402
[5]	M.M. Postnikov, "Introduction to Morse theory" , Moscow (1971) (In Russian) MR0315739
[6]	S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962) MR0145455 Zbl 0111.18101

References

[a1]	W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German) MR0666697 Zbl 0495.53036