Differentiable Manifolds/Integral curves and Lie derivatives

Integral curves

Definition 8.1:

Let $M$ be a manifold, let $\mathbf {V} \in {\mathfrak {X}}(M)$, and let $I\subseteq \mathbb {R}$ be an interval. An integral curve for $\mathbf {V}$ is a function $\gamma :I\to M$ such that

$\forall x\in I:\gamma _{x}'=\mathbf {V} (\gamma (x))$

Theorem 8.2:

Let $M$ be a manifold of class ${\mathcal {C}}^{n}$, $n\in \mathbb {N} \cup \{\infty \}$ (it is important that $n\geq 1$ here), let $\mathbf {V} \in {\mathfrak {X}}(M)$ be differentiable of class ${\mathcal {C}}^{j}$, $j\in \mathbb {N} \cup \{\infty \}$ and let $p\in M$. Then there exists an interval $I\subseteq \mathbb {R}$ and an integral curve $\gamma :I\to M$ of $\mathbf {V}$ such that $0\in I$ and $\gamma (0)=p$.

Proof:

Let $p\in M$ be arbitrary, and let $(O,\phi )$ be contained in the atlas of $M$ such that $p\in O$.

Lemma 2.3 stated that if we denote $\phi =(\phi _{1},\ldots ,\phi _{d})$, then the $\phi _{k}$, $k\in \{1,\ldots ,d\}$ are contained in ${\mathcal {C}}^{n}(M)$. From $\mathbf {V}$ being differentiable of class ${\mathcal {C}}^{j}$ with a $j\in \mathbb {N} \cup \{\infty \}$, it follows that the functions $\mathbf {V} \phi _{k}$, $k\in \{1,\ldots ,d\}$ are contained in ${\mathcal {C}}^{n}(M)$.

Thus the Picard–Lindelöf theorem is applicable, and it tells us, that each of the initial value problems

$y_{k}'(x)=(\mathbf {V} \phi _{k}\circ y_{k})(x)$, $k\in \{1,\ldots ,d\}$

$y_{k}(0)=\phi _{k}(p)$

has a solution $y_{k}:I_{k}\to \mathbb {R}$, where each $I_{k}\subseteq \mathbb {R}$ is an interval containing zero. We now choose

$I:=\bigcap _{k=1}^{d}I_{k}$

and

$\gamma :I\to M,\gamma (x):=\phi ^{-1}(y_{1}(x),\ldots ,y_{d}(x))$

We note that

$(\phi _{k}\circ \gamma )(x)=\phi _{k}(\phi ^{-1}(y_{1}(x),\ldots ,y_{d}(x)))=\phi (\phi ^{-1}(y_{1}(x),\ldots ,y_{d}(x)))_{k}=y_{k}(x)$

Therefore we have for each $x\in I$ and $k\in \{1,\ldots ,d\}$:

$\gamma _{x}'(\phi _{k})=(\phi _{k}\circ \gamma )'(x)=y_{k}'(x)=(\mathbf {V} \phi _{k}\circ \gamma )(x)=\mathbf {V} (\gamma (x))(\phi _{k})$

Because of theorem 2.7 then follows:

$\gamma '_{x}=\sum _{k=1}^{d}\gamma _{x}'(\phi _{k})\left({\frac {\partial }{\partial \phi _{k}}}\right)_{\phi ^{-1}(x)}=\sum _{k=1}^{d}\mathbf {V} (\phi ^{-1}(x))(\phi _{k})\left({\frac {\partial }{\partial \phi _{k}}}\right)_{\phi ^{-1}(x)}=\mathbf {V} (\gamma (x))$$\Box$

Lie derivatives

In the following, we will define so-called Lie derivatives, for

• ${\mathcal {C}}^{n}(M)$ functions and
• for vector fields.

Definition 8.3:

Let $M$ be a manifold of class ${\mathcal {C}}^{n}$, $\mathbf {V} \in {\mathfrak {X}}(M)$ and $\varphi \in {\mathcal {C}}^{n}(M)$. The Lie derivative of $\varphi$ in the direction of $\mathbf {V}$, denoted by ${\mathfrak {L}}_{\mathbf {V} }\varphi$, is defined as follows:

${\mathfrak {L}}_{\mathbf {V} }\varphi :M\to \mathbb {R} ,{\mathfrak {L}}_{\mathbf {V} }\varphi (p):=\mathbf {V} \varphi (p)$

Definition 8.4:

Let $M$ be a manifold and $\mathbf {V} ,\mathbf {W} \in {\mathfrak {X}}(M)$. The Lie derivative of $\mathbf {W}$ in the direction of $\mathbf {V}$, denoted by ${\mathfrak {L}}_{\mathbf {V} }\mathbf {W}$, is defined as follows:

${\mathfrak {L}}_{\mathbf {V} }\mathbf {W} :M\to TM,L_{\mathbf {V} }\mathbf {W} (p):=[V,W](p)$

So we simply defined the Lie derivative of a function in the direction of a vector field as the function defined like in definition 5.1, and the Lie derivative of a vector field in the direction of the other vector field as the lie bracket of first the first vector field and then the other (the order is important here because the Lie braket is anti-symmetric (see theorem ? and definition ?)). Since we already had symbols for these, why have we defined new symbols? The reason is that in certain circumstances, the Lie derivatives really are derivatives in the sense of limits of differential quotients, as is explained in the next chapter.