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
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where is the
-th basic tangent vector to
at the point
. Any vector field
can be uniquely represented in the form
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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
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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
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is defined by the formula
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It satisfies the relations
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in particular,
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Each vector field induces a local flow on
— a family of diffeomorphisms of a neighbourhood
,
![]() |
such that for
and
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is the integral curve of the vector field through
, i.e.
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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
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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
,
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and the Lie bracket
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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) |
[2] | D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968) |
[3] | S. Lang, "Introduction to differentiable manifolds" , Interscience (1967) pp. App. III |
[4] | K. Nomizu, "Lie groups and differential geometry" , Math. Soc. Japan (1956) |
[5] | M.M. Postnikov, "Introduction to Morse theory" , Moscow (1971) (In Russian) |
[6] | S. Helgason, "Differential geometry and symmetric spaces" , Acad. Press (1962) |
Comments
References
[a1] | W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German) |
Vector field on a manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_field_on_a_manifold&oldid=24590