Vector field on a manifold

$ M $

A section of the tangent bundle $ \tau ( M) $. The set of differentiable vector fields forms a module over the ring $ F $ of differentiable functions on $ M $.

Example 1.

For a chart $ x _ {U} $ of the manifold $ M $ one defines the $ i $- th basic vector field $ \partial / \partial x ^ {i} $ according to the formula

$$ \frac \partial {\partial x ^ {i} } ( p) = \ \left . \frac \partial {\partial x ^ {i} } \right | _ {p} ,\ \ p \in U , $$

where $ \partial / \partial x _ {i} \mid _ {p} $ is the $ i $- th basic tangent vector to $ M $ at the point $ p $. Any vector field $ X $ can be uniquely represented in the form

$$ X = \sum _ { i } \xi ^ {i} ( p) \frac \partial {\partial x ^ {i} } ( p), $$

where $ \xi ^ {i} ( p) $ are the components of $ X $ in $ x _ {U} $. Since a vector field can be regarded as a derivation of the ring $ F $( 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 $ x _ {U} $ and $ f \in F $, the function $ Xf $ is defined by the formula

$$ ( Xf) ( p) = \sum _ { i } \xi ^ {i} ( p) \left . D _ {i} ( f( x _ {U} ^ {-1})) \right | _ {x _ {U} ( p) } = $$

$$ = \ \sum _ { i } \left . \xi ^ {i} ( p) \frac \partial {\partial x ^ {i} } \right | _ {p} ( f ), $$

where $ D _ {i} $ is the partial derivative with respect to $ x ^ {i} $. Note that $ \xi ^ {i} ( p)= ( X x ^ {i} ) ( p) $; $ Xf $ is called the derivative of $ f $ in the direction $ X $.

Example 3.

For the chart $ x _ {U} $ and $ f \in F $, the commutator (Lie bracket) $ [ X, Y] $ of the vector fields

$$ X = \sum _ { i } \xi ^ {i} \frac \partial {\partial x ^ {i} } \ \textrm{ and } \ \ Y = \sum _ { i } \eta ^ {i} \frac \partial {\partial x ^ {i} } $$

is defined by the formula

$$ ([ X, Y ] f )( p) = ( X( Yf )) ( p) - ( Y( Xf ))( p) = $$

$$ = \ \sum _ { i,k } \left . \left ( \xi ^ {k} \frac{\partial \eta ^ {i} }{\partial x ^ {k} } - \eta ^ {k} \frac{\partial \xi ^ {i} }{\partial x ^ {k} } \right ) \partial \frac{f}{\partial x ^ {i} } \right | _ {p} . $$

It satisfies the relations

$$ [ X, Y] = - [ Y, X], $$

$$ [[ X, Y] , Z] + [[ Y, Z], X] + [[ Z, X], Y] = 0; $$

in particular,

$$ \left [ \frac \partial {\partial x ^ {i} } , \frac \partial {\partial x ^ {j} } \right ] = 0. $$

Each vector field $ X $ induces a local flow on $ M $— a family of diffeomorphisms of a neighbourhood $ U $,

$$ \Phi : (- \epsilon , + \epsilon ) \times U \rightarrow M, $$

such that $ \Phi ( 0, p)= p $ for $ p \in U $ and

$$ \Phi ( t, p) = \Phi _ {p} ( t): (- \epsilon , \epsilon ) \rightarrow M $$

is the integral curve of the vector field $ X $ through $ p $, i.e.

$$ \Phi ^ \star \left ( \frac \partial {\partial t } \right ) ( t) = \ X ( \Phi ( t, p) ), $$

where $ \Phi ^ {*} ( \partial / \partial t ) ( t) $ is the tangent vector $ d \Phi _ {p} ( t) $ to $ M $ at $ \Phi _ {p} ( t) $. Conversely, a vector field $ X $ is associated with a local flow $ \Phi ( t, p)= \Phi _ {t} ( p) $, which is a variation of the mapping $ \Phi _ {0} ( p) $; here

$$ ( Xf)( p) = \lim\limits _ {t \rightarrow 0 } \frac{f( \Phi _ {t} ( p) )- f( p) }{t} . $$

Each vector field defines a Lie derivation $ L _ {X} $ of a tensor field of type $ \lambda $ with values in a vector space (infinitesimal transformation of $ \lambda $), corresponding to the local flow $ \Phi ( t, p) $; its special cases include the action of the vector field on $ f \in F $,

$$ L _ {X} f = X f, $$

and the Lie bracket

$$ L _ {X} Y = [ X, Y] = \lim\limits _ {t \rightarrow 0 } \frac{Y- \Phi _ {t} ^ \star Y \Phi _ {-t} }{t} . $$

A vector field without singularities generates an integrable one-dimensional differential system as well as a Pfaffian system associated with it on $ M $.

A generalization of the concept of a vector field on a manifold is that of a vector field along a mapping $ \phi : N \rightarrow M $, which is a section of the bundle $ \tau _ \phi ( N) $ induced by $ \phi $, as well as a tensor field of type $ \lambda $, which is a section of the bundle $ \lambda [ \tau ] $ associated with $ \tau ( M) $ with the aid of the functor $ \lambda $.

References

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