# 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 $M$.

## Contents

### 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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