Difference between revisions of "Vector field on a manifold"
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Godbillon, "Géométrie différentielle et mécanique analytique" , Hermann (1969) {{MR|0242081}} {{ZBL|0653.53001}} {{ZBL|0284.53018}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968) {{MR|0229177}} {{ZBL|0155.30701}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S. Lang, "Introduction to differentiable manifolds" , Interscience (1967) pp. App. III {{MR|1931083}} {{MR|1532744}} {{MR|0155257}} {{ZBL|1008.57001}} {{ZBL|0103.15101}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> K. Nomizu, "Lie groups and differential geometry" , Math. Soc. Japan (1956) {{MR|0084166}} {{ZBL|0071.15402}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M.M. Postnikov, "Introduction to Morse theory" , Moscow (1971) (In Russian) {{MR|0315739}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962) {{MR|0145455}} {{ZBL|0111.18101}} </TD></TR></table> |
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− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German) {{MR|0666697}} {{ZBL|0495.53036}} </TD></TR></table> |
Revision as of 17:02, 15 April 2012
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 |
Comments
References
[a1] | W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German) MR0666697 Zbl 0495.53036 |
Vector field on a manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_field_on_a_manifold&oldid=11561