# Differentiable Manifolds/Product manifolds and Lie groups
## The product manifold

Definition 10.1:
Let $X,Y,Z,W$ be sets and $f:X\to Z$ and $g:Y\to W$ be functions. Then the cartesian product of $f$ and $g$, denoted by $f\times g$, is defined to be the function
    $f\times g:X\times Y\to Z\times W,(f\times g)(x,y)=(f(x),g(y))$
Definition 10.2:
Let $M$ be a manifold with atlas $\{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}$. The product manifold of $M$ is defined to be the product space $M\times M$ with the atlas
    $\{(O_{\upsilon }\times O_{\mathrm {o} },(\phi _{\upsilon },\phi _{\mathrm {o} }))|\upsilon ,\mathrm {o} \in \Upsilon \}$
Lemma 10.3:
Let $O\subseteq \mathbb {R} ^{2d}$ be open. Then for each $(x,y)\in O$, there exists $\epsilon _{(x,y)}>0$ such that
    $B_{\epsilon _{(x,y)}}(x)\times B_{\epsilon _{(x,y)}}(y)\subset O$
Proof:
Let $(x,y)\in O$ be arbitrary. We choose $\delta >0$ such that $B_{\delta }((x,y))\subseteq O$. Then we define $\epsilon _{(x,y)}:={\frac {\delta }{2}}$. Then, by the triangle inequality we have for every $(z,w)\in B_{\epsilon _{(x,y)}}(x)\times B_{\epsilon _{(x,y)}}(y)$:
    $\|(z,w)-(x,y)\|=\|(z,0)+(0,w)-(x,0)-(0,y)\|\leq \|(z,0)-(x,0)\|+\|(0,w)-(0,y)\|=\|z-x\|+\|w-y\|<2\epsilon _{(x,y)}=\delta$
Therefore, $(z,w)\in B_{\delta }((x,y))\subseteq O$, and since $(z,w)\in B_{\epsilon _{(x,y)}}(x)\times B_{\epsilon _{(x,y)}}(y)$ was arbitrary:
    $B_{\epsilon _{(x,y)}}(x)\times B_{\epsilon _{(x,y)}}(y)\subseteq O$$\Box$
Lemma 10.4:
Let $O,U\subseteq \mathbb {R} ^{d}$ be open sets. Then $O\times U$ is open in $\mathbb {R} ^{2d}$.
Proof:
Due to the openness of $O$ and $U$ for each point $(x,y)$ in $O\times U$, we find $\epsilon _{1},\epsilon _{2}\in \mathbb {R} _{>0}$ such that $B_{\epsilon _{1}}(x)\subseteq O$ and $B_{\epsilon _{2}}(x)\subseteq U$. If we define $\epsilon :=\min\{\epsilon _{1},\epsilon _{2}\}$, we have for all $(z,w)\in B_{\epsilon }((x,y))$:
    $\|x-z\|^{2}=\|(x,y)-(z,w)\|^{2}-\|y-w\|^{2}\leq \|(x,y)-(z,w)\|^{2}$
and
    $\|y-w\|^{2}=\|(x,y)-(z,w)\|^{2}-\|x-z\|^{2}\leq \|(x,y)-(z,w)\|^{2}$
and since the function ${\sqrt {\cdot }}$ is monotonely increasing on $[0,\infty )$, it follows
    $\|x-z\|<\epsilon$
and
    $\|y-w\|<\epsilon$
and thus:
    $(z,w)\in O\times U$$\Box$
Lemma 10.5: Let $X,Y,Z,W$ be sets and let $f:X\to Z$ and $g:Y\to W$ be functions. If $A\subseteq Z$ and $B\subseteq W$, then
    $(f\times g)^{-1}(A\times B)=f^{-1}(A)\times g^{-1}(B)$
Proof: See exercise 2.
Theorem 10.6: The product manifold of a manifold $M$ of class ${\mathcal {C}}^{n}$, where $n\in \mathbb {N} _{0}\cup \{\infty \}$ with atlas $\{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}$ really is a manifold; i. e. $\{(O_{\upsilon }\times O_{\mathrm {o} },\phi _{\upsilon }\times \phi _{\mathrm {o} })|\upsilon ,\mathrm {o} \in \Upsilon \}$ really is an atlas of class ${\mathcal {C}}^{n}$.
Proof:
1. We show that for all $\upsilon ,\mathrm {o} \in \Upsilon$ the set $(\phi _{\upsilon }\times \phi _{\mathrm {o} })(O_{\upsilon }\times O_{\mathrm {o} })$ is open.
We have:
    ${\begin{aligned}(\phi _{\upsilon }\times \phi _{\mathrm {o} })(O_{\upsilon }\times O_{\mathrm {o} })&=\{(\phi _{\upsilon }(p),\phi _{\mathrm {o} }(q))\in \mathbb {R} ^{2d}{\big |}p\in O_{\upsilon },q\in O_{\mathrm {o} }\}\\&=\phi _{\upsilon }(O_{\upsilon })\times \phi _{\mathrm {o} }(O_{\mathrm {o} })\\\end{aligned}}$
This set is open in $\mathbb {R} ^{2d}$ due to lemma 10.4, since it is the cartesian product of two open sets.
2. We prove that for all $\upsilon ,\mathrm {o} \in \Upsilon$, the function $\phi _{\upsilon }\times \phi _{\mathrm {o} }$ is a homeomorphism.
2.1. For bijectivity, see exercise 1.
2.2. We prove continuity.
Let $O\subseteq (\phi _{\upsilon }\times \phi _{\mathrm {o} })(O_{\upsilon }\times O_{\mathrm {o} })$ be open. Due to the definition of the subspace topology,
Lemma 10.4 implies that we have
    $O=\bigcup _{(x,y)\in O}B_{\epsilon _{(x,y)}}(x)\times B_{\epsilon _{(x,y)}}(y)$
(this equation can be proven by showing '$\subseteq$' and '$\supseteq$') and thus follows with lemma 10.5, that:
    ${\begin{aligned}(\phi _{\upsilon }\times \phi _{\mathrm {o} })^{-1}(O)&=(\phi _{\upsilon }\times \phi _{\mathrm {o} })^{-1}\left(\bigcup _{(x,y)\in O}B_{\epsilon _{(x,y)}}(x)\times B_{\epsilon _{(x,y)}}(y)\right)\\&=\bigcup _{(x,y)\in O}(\phi _{\upsilon }\times \phi _{\mathrm {o} })^{-1}(B_{\epsilon _{(x,y)}}(x)\times B_{\epsilon _{(x,y)}}(y))\\&=\bigcup _{(x,y)\in O}\phi _{\upsilon }^{-1}(B_{\epsilon _{(x,y)}}(x))\times (\phi _{\mathrm {o} })^{-1}(B_{\epsilon _{(x,y)}}(y))\end{aligned}}$
, which is open as the union of open sets (since we had equipped $M\times M$ with the product topology).
2.3. We prove continuity of the inverse.
Let $O\subseteq M\times M$ be open.
3. We prove that

## Lie groups

Definition 10.7:
Let $G$ be a $d$-dimensional manifold of class ${\mathcal {C}}^{n}$, which is simultaneously a group, i. e. we have a group operation $*:G\times G\to G$ regarding to which there exist an identity, which we in the following shall denote by $e$, and inverses in $G$, and which is associative. We call $G$ a $d$-dimensional Lie group of class ${\mathcal {C}}^{n}$ iff
* the function $\psi _{*}:G\times G\to G,\psi _{*}((g,h)):=g*h$ is differentiable of class ${\mathcal {C}}^{n}$, where $G\times G$ has the product manifold atlas, AND
* the function $G\to G,g\mapsto g^{-1}$ is also of class ${\mathcal {C}}^{n}$.
Definition 10.8:
Let $M$ be a manifold, and let $N\subseteq M$ be a subset of $M$ which is also a manifold. The inclusion of $N$ into $M$ is defined to be the function
    $\iota :N\to M,\iota (q):=q$
Definition 10.9:
Let $G$ be a Lie group of class ${\mathcal {C}}^{n}$, and let $H\subseteq G$ be a subset of $G$. We call $H$ a Lie subgroup of $G$ iff
* $H$ is a Lie group of class, but not necessarily with the subspace topology induced by $G$
* the inclusion of $H$ into $G$ is differentiable of class ${\mathcal {C}}^{n}$ and its differential is injective at every point.
Definitions 10.10:
Let $G$ be a Lie group with group operation $*$, and let $g\in G$. The left multiplication function with respect to $g$, denoted by $L_{g}$, is defined to be the function
    $L_{g}:G\to G,L_{g}(h):=g*h$
The right multiplication function with respect to $g$, denoted by $R_{g}$, is defined to be the function
    $R_{g}:G\to G,R_{g}(h):=h*g$
Theorem 10.11: Let $G$ be a $d$-dimensional Lie group of class ${\mathcal {C}}^{n}$. Then for each $g\in G$ the respective left multiplication function and the respective right multiplication are diffeomorphisms of class ${\mathcal {C}}^{n}$ from $G$ to itself.
Proof:
We only prove the claim for the left multiplication. The proof for the right multiplication goes the same way.
In this proof, the group operation of $G$ is denoted by $*$.
1. We show that $L_{g}$ is differentiable of class ${\mathcal {C}}^{n}$.
Let $g\in G$ be arbitrary. Since $G$ is a Lie group, the function
    $\psi _{*}:G\times G\to G,\psi _{*}(g,h)=g*h$
is differentiable of class ${\mathcal {C}}^{n}$, where $G\times G$ is equipped with the product manifold structure.
Let now $(O,\phi )$ and $(U,\theta )$ be two arbitrary elements in the atlas of $G$. We choose $(V,\chi )$ in the atlas of $G$ such that $g\in V$. As $\psi _{*}$ is differentiable of class ${\mathcal {C}}^{n}$, the function
    $\phi |_{*(\psi _{*}^{-1}(O)\cap V\times U)}\circ \psi _{*}|_{\psi _{*}^{-1}(O)\cap V\times U}\circ (\chi \times \theta )|_{\psi _{*}^{-1}(O)\cap V\times U}^{-1}$
is contained in ${\mathcal {C}}^{n}(\mathbb {R} ^{2d},\mathbb {R} ^{d})$. Therefore, also the function
    $f:\mathbb {R} ^{d}\to \mathbb {R} ^{d},f(x)=\phi |_{\psi _{*}(\psi _{*}^{-1}(O)\cap V\times U)}\circ \psi _{*}|_{\psi _{*}^{-1}(O)\cap V\times U}\circ (\chi \times \theta )|_{\psi _{*}^{-1}(O)\cap V\times U}^{-1}(\chi ^{-1}(g),x)$
is contained in ${\mathcal {C}}^{n}(\mathbb {R} ^{2d},\mathbb {R} ^{d})$; the partial derivatives exist and are equal to the partial derivatives of the last $d$ variables of the function
    $\phi |_{\psi _{*}(\psi _{*}^{-1}(O)\cap V\times U)}\circ \psi _{*}|_{\psi _{*}^{-1}(O)\cap V\times U}\circ (\chi \times \theta )|_{\psi _{*}^{-1}(O)\cap V\times U}^{-1}$
. But we have for all $x\in \theta (L_{g}^{-1}(O)\cap U)$:
    $f(x)=(\phi |_{L_{g}(|_{L_{g}^{-1}(O)\cap U})}\circ L_{g}|_{L_{g}^{-1}(O)\cap U}\circ \theta |_{L_{g}^{-1}(O)\cap U}^{-1})(x)$
and therefore the function
    $\phi |_{L_{g}(|_{L_{g}^{-1}(O)\cap U})}\circ L_{g}|_{L_{g}^{-1}(O)\cap U}\circ \theta |_{L_{g}^{-1}(O)\cap U}^{-1}$
is contained in ${\mathcal {C}}^{n}(\mathbb {R} ^{d},\mathbb {R} ^{d}$, which means the definition of differentiability of class ${\mathcal {C}}^{n}$ is fulfilled.
2. We show that $L_{g}$ is bijective.
We do so by noticing that an inverse function of $L_{g}$ is given by $L_{g^{-1}}$: For arbitrary $h\in G$, we have:
    $L_{g^{-1}}(L_{g}(h))=g*(g^{-1}*h)=(g*g^{-1})*h=e*h=h$
and
    $L_{g}(L_{g^{-1}}(h))=g^{-1}*(g*h)=(g^{-1}*g)*h=e*h=h$
3. We note that the inverse function is differentiable of class ${\mathcal {C}}^{n}$:
We use 1. with $g^{-1}$; $g^{-1}$ also is an element of $G$, and 1. proved that the left multiplication function is differentiable for every element of $G$, including $g^{-1}$.$\Box$

## Left invariant vector fields

Definition 10.12:
Let $G$ be a Lie group. We call a vector field $\mathbf {V} \in {\mathfrak {X}}(M)$ left invariant iff
    $\forall g,h\in G:(dL_{g})_{h}(\mathbf {V} (h))=\mathbf {V} (gh)$
The set of all left invariant vector fields of a lie group is denoted by the corresponding lower case fraktur letter; for example, the set of all left invariant vector fields of a Lie group $G$ one would denote by ${\mathfrak {g}}$, and the set of all left invariant vector fields of a Lie group $H$ would be denoted by ${\mathfrak {h}}$.
Let us repeat the definition of a Lie subalgebra:
Definition 6.2:
Let $L$ with $[\cdot ,\cdot ]$ be a Lie algebra. A subset of $L$ which is a Lie algebra with the restriction of $[\cdot ,\cdot ]$ on that subset is called a Lie subalgebra.
Theorem 10.13:
Let $G$ be a Lie group. Then ${\mathfrak {g}}$, together with pointwise addition and the vector field Lie bracket, is a Lie subalgebra of ${\mathfrak {X}}(M)$ with pointwise addition and the vector field Lie bracket.
Proof:
Let $\mathbf {V} ,\mathbf {W} \in {\mathfrak {g}}$. It suffices to show that $[\mathbf {V} ,\mathbf {W} ]\in {\mathfrak {g}}$, because then ${\mathfrak {g}}$ is a Lie algebra with the restriction of the vector field Lie bracket.
Indeed, we have for all $g,h\in G$ and $\varphi \in {\mathcal {C}}^{n}(G)$:
    ${\begin{aligned}(dL_{g})_{h}([\mathbf {V} ,\mathbf {W} ](h))(\varphi )&=[\mathbf {V} ,\mathbf {W} ](h)(L_{g}^{*}\varphi )\\&=\mathbf {V} (h)(\mathbf {W} (L_{g}^{*}\varphi ))-\mathbf {W} (h)(\mathbf {V} (L_{g}^{*}\varphi ))\\&=\mathbf {V} (h)((dL_{g})_{\cdot }(\mathbf {W} \varphi ))-\mathbf {W} (h)((dL_{g})_{\cdot }(\mathbf {V} \varphi ))\\&=\mathbf {V} (h)(\mathbf {W} (g\cdot )(\varphi ))-\mathbf {W} (h)(\mathbf {V} (g\cdot )(\varphi ))\\&=\mathbf {V} (h)(\mathbf {W} \varphi \circ L_{g})-\mathbf {W} (h)(\mathbf {V} \varphi \circ L_{g})\\&=\mathbf {V} (h)(L_{g}^{*}(\mathbf {W} \varphi ))-\mathbf {W} (h)(L_{g}^{*}(\mathbf {V} \varphi \circ L_{g}))\\&=(dL_{g})_{h}(\mathbf {V} (h))(\mathbf {W} \varphi )-(dL_{g})_{h}(\mathbf {W} (h))(\mathbf {V} \varphi )\\&=\mathbf {V} (gh)(\mathbf {W} \varphi )-\mathbf {W} (gh)(\mathbf {V} \varphi )\\&=[\mathbf {V} ,\mathbf {W} ](gh)(\varphi )\end{aligned}}$
, where by $(dL_{g})_{\cdot }(\mathbf {W} \varphi )$ the function
    $p\mapsto (dL_{g})_{p}(\mathbf {W} \varphi )$
is meant and by $\mathbf {W} (g\cdot )(\varphi )$ the function
    $p\mapsto \mathbf {W} (gp)(\varphi )$
is meant (as both are equal to $\mathbf {W} (L_{g}^{*}\varphi )$, both are differentiable of class ${\mathcal {C}}^{n}$).$\Box$
Theorem 10.14:
Let $G$ be a Lie group of class ${\mathcal {C}}^{n}$. Then there exists a vector space isomorphism between ${\mathfrak {g}}$ and $T_{e}G$.
Proof:
We choose the function
    $\Theta :{\mathfrak {g}}\to T_{e}G,\Theta (\mathbf {V} ):=\mathbf {V} (e)$
We will now show that this function is the desired isomorphism.
1. We prove linearity: Let $\mathbf {V} ,\mathbf {W} \in {\mathfrak {g}}$ and $c\in \mathbb {R}$. We have:
    ${\begin{aligned}\Theta (\mathbf {V} +c\mathbf {W} )&=(\mathbf {V} +c\mathbf {W} )(e)\\&=\mathbf {V} (e)+c\mathbf {W} (e)\\&=\Theta (\mathbf {V} )+c\Theta (\mathbf {W} )\end{aligned}}$
2. We prove bijectivity.
2.1. We prove injectivity.
Let $\Theta (\mathbf {V} )=\Theta (\mathbf {W} )$ (i. e. $\mathbf {V} (e)=\mathbf {W} (e)$) for $\mathbf {V} ,\mathbf {W} \in {\mathfrak {g}}$. Since $\mathbf {V} ,\mathbf {W}$ are left invariant, it follows for all $g\in G$ and $\varphi \in {\mathcal {C}}^{n}$, that
    ${\begin{aligned}\mathbf {V} (g)(\varphi )&=\mathbf {V} (ge)(\varphi )\\&=(dL_{g})_{e}(\mathbf {V} )(\varphi )\\&=\mathbf {V} (e)(L_{g}^{*}(\varphi ))\\&=\mathbf {W} (e)(L_{g}^{*}(\varphi ))\\&=(dL_{g})_{e}(\mathbf {W} )(\varphi )\\&=\mathbf {W} (ge)(\varphi )\\&=\mathbf {W} (g)(\varphi )\end{aligned}}$
2.2. We prove surjectivity.
Let $\mathbf {V} _{e}\in T_{e}G$ be arbitrary. We define
    $\mathbf {U} _{\mathbf {V} _{e}}(g):=(dL_{g})_{e}\mathbf {V} _{e}$
Due to theorem 2.19, this is a vector field. It is also left invariant because for all $g,h\in G$ and $\varphi \in {\mathcal {C}}^{n}(M)$, we have:
    ${\begin{aligned}(dL_{g})_{h}(\mathbf {U} _{\mathbf {V} _{e}}(h))(\varphi )&=\mathbf {U} _{\mathbf {V} _{e}}(h)(L_{g}^{*}(\varphi ))\\&=(dL_{h})_{e}\mathbf {V} _{e}(L_{g}^{*}(\varphi ))\\&=\mathbf {V} _{e}(L_{h}^{*}(L_{g}^{*}(\varphi )))\\&=\mathbf {V} _{e}(\varphi \circ L_{g}\circ L_{h})\\&=\mathbf {V} _{e}(\varphi \circ L_{gh})\\&=(dL_{gh})_{e}\mathbf {V} _{e}(\varphi )\\&=\mathbf {U} _{\mathbf {V} _{e}}(gh)(\varphi )\end{aligned}}$
Further, we have for all $\varphi \in {\mathcal {C}}^{n}(M)$:
    ${\begin{aligned}\Theta (\mathbf {U} _{\mathbf {V} _{e}})(\varphi )=\mathbf {U} _{\mathbf {V} _{e}}(e)(\varphi )=(dL_{e})_{e}\mathbf {V} _{e}(\varphi )=\mathbf {V} _{e}(\varphi )\end{aligned}}$
3. We note that the inverse of $\Theta$ is linear since the inverse of a linear bijective function is always linear.
$\Box$
The next theorem shows that in a Lie group, all left invariant vector fields are complete.
Lemma 10.15:
Let $G$ be a Lie group, let $\mathbf {V} \in {\mathfrak {g}}$ and let $\Phi _{\mathbf {V} }$ be the flow of $\mathbf {V}$. Then for all $g\in G$ and all $(x,h)$ in the domain of $\Phi {\mathbf {V} }$, we have:
    $\Phi _{\mathbf {V} }(x,g*h)=h*\Phi _{\mathbf {V} }(x,g)$
Proof:
Let $h\in G$ be arbitrary, and let <mat>I_h</math> be the unique largest interval such that $0\in I_{h}$ and there exists a unique integral curve $\gamma _{h}:I_{h}\to M$ such that
    $\gamma _{h}'$
Theorem 10.16:
Let $G$ be a Lie group and let $\mathbf {V} \in {\mathfrak {g}}$. Then $\mathbf {V}$ is complete.
Proof:
Let $g\in G$ be arbitrary and let $\gamma _{g}$ be an integral curve at $g$.

## The exponential function

Definition 10.17:

## The adjoint function

For the next definition, we recall that the automorphism group of a group was given by the set of group isomorphisms from the group to itself with composition as the group operation. Indeed, this is a group (see exercise 3).
We further recall that for a group $G$, the automorphism group is denoted by ${\text{Aut}}(G)$.
Definition 10.18:
Let $G$ be a Lie group, whose group operation we shall denote by $*$. If for all $g\in G$ we define the function $\psi _{g}:G\to G$ by $\psi _{g}(h):=g*h*g^{-1}$, the adjoint function of $G$, denoted by ${\text{Ad}}(g)$ is given by
    ${\text{Ad}}:G\to {\text{Aut}}(G),{\text{Ad}}(g)(h):=g*h*g^{-1}$
Theorem 10.19:
Let $G$ be a manifold of class ${\mathcal {C}}^{n}$, where $n\in \mathbb {N} _{0}\cup \{\infty \}$. For each $g\in G$, ${\text{Ad}}(g)$ is of class ${\mathcal {C}}^{n}$.
Proof:
We have:
    ${\text{Ad}}(g)=L_{g}\circ R_{g^{-1}}$
Therefore, the claim follows from theorems 2.29 and 10.11.$\Box$
Theorem 10.20:
Let $G$ be a Lie group and $\mathbf {V} \in {\mathfrak {X}}(G)$. Then

## Exercises

* Let $X,Y,Z,W$ be sets $f:X\to Z$ and $g:Y\to W$ be two bijective functions. Prove that $f\times g$ is bijective.
* Prove lemma 10.5.
* Let $G$ be a group and ${\text{Aut}}(G)$ be the set of group isomorphisms from $G$ to $G$. Prove that ${\text{Aut}}(G)$ together with the composition as operation is a group.