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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Complex geometry\\[15mm]
\small lecture 8: Levi-Civita connection}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\tiny\bf HSE, room 306, 16:20,
\\[2mm] 
October 17, 2020
}
\end{center}


\newpage


{\bf \blue Connections}


\definition
Recall that {\bf \blue a connection} on a bundle $B$
is an operator $\nabla:\; B \arrow B \otimes \Lambda^1 M$
satisfying $\nabla(fb) = b \otimes df + f \nabla(b)$,
where $f \arrow df$ is de Rham differential.
When $X$ is a vector field, we denote by
$\nabla_X(b)\in B$ the term $\langle \nabla(b), X\rangle$.

\remark {\bf \purple A connection $\nabla$ on $B$ gives
a connection $B^* \stackrel {\nabla^*} \arrow \Lambda^1 M \otimes B^*$
on the dual bundle,} by the formula
\[
d(\langle b, \beta\rangle) = \langle \nabla b, \beta\rangle+
\langle b, \nabla^*\beta\rangle
\]
These connections are usually denoted 
{\bf \red by the same letter $\nabla$.}

\remark
For any tensor bundle 
${\cal B}_1:=
B^*\otimes B^* \otimes ... \otimes B^* \otimes B\otimes B \otimes ... \otimes B$
{\bf \purple a connection on $B$ defines a connection on ${\cal B}_1$}
using the Leibniz formula:
\[
\nabla(b_1 \otimes b_2) = \nabla(b_1) \otimes b_2 + b_1 \otimes \nabla(b_2).
\]

\newpage

{\bf \blue Parallel transport along the connection}

\theorem
Let $B$ be a vector bundle with connection over $\R$.
Then for each $x\in \R$ and each vector $b_x \in B\restrict x$
{\bf \red there exists a unique section $b\in B$ such that
$\nabla b=0$, $b\restrict x= b_x$.}

\proof This is existence and uniqueness of solutions
of an ODE $\frac {db}{dt} + A(b)=0$. \endproof

\definition
Let $\gamma:\; [0, 1] \arrow M$ be a smooth path
in $M$ connecting $x$ and $y$, and $(B, \nabla)$ 
a vector bundle with connection. Restricting 
$(B, \nabla)$ to $\gamma([0,1])$, we obtain a 
bundle with connection on an interval.
Solve an equation  $\nabla(b)=0$
for $b\in B\restrict{\gamma([0,1])}$
and initial condition $b\restrict x= b_x$.
This process is called {\bf\blue parallel transport}
along the path via the connection.
The vector $b_y:= b\restrict y$
is called {\bf\blue vector obtained by parallel
transport of $b_x$ along $\gamma$}.
{\bf\blue Holonomy group} of $\gamma$ is the group
of endomorphisms of the fiber $B_x$ obtained
from parallel transports along all paths 
starting and ending in $x\in M$

\newpage

{\bf \blue Parallel tensors}

\definition
Let $B$ be a vector bundle, and 
$\Psi\in B^{\otimes i} \otimes (B^*)^{\otimes j}$ a tensor on $B$.
We say that {\bf \blue connection $\nabla$ preserves $\Psi$}
if $\nabla(\Psi)=0$. In this case we also say that $\Psi$
is {\bf \blue parallel} with respect to the connection.

\remark
$\nabla(\Psi)=0$ is equivalent to $\Psi$ being a solution of
$\nabla(\Psi)=0$ on each path $\gamma$. This means that 
{\bf \purple parallel transport preserves $\Psi$.} 

We obtained

\corollary
{\bf \red 
A tensor is parallel if and only if it is holonomy invariant.}


\example 
{\bf \blue Orthogonal connection}: given a positive definite form
$h\in \Sym^2 B^*$ on $B$, a connection $\nabla$ such that $\nabla(h)=0$
is called {\bf \blue orthogonal}.

\example
Suppose that $(B,I)$ is a complex vector bundle
equipped with a Hermitian metric $h$.
A connection $\nabla$ such that $\nabla(I)=\nabla(h)=0$
is called {\bf \blue unitary}, or {\bf \blue Hermitian}.


\newpage

{\bf \blue Torsors and affine spaces}

\definition
{\bf \blue A torsor} over a group $G$ is a space $X$
equipped with a free and transitive action of $G$,
$g,x \arrow \rho(g,x).$

\definition
{\bf \blue Morphism} of torsors 
$(X,G,\rho) \stackrel \Psi \arrow (X',G',\rho')$
is a pair $\Psi_X:\; X\arrow X', \Psi_G:\; G \arrow G'$,
where $\Psi_G$ is a group homomorphism satisfying
$\Psi_X(\rho(g,x)) = \rho'(\Psi_G(g),\Psi_X(x))$
(that is, compatible with the map $\Psi_X$).

\remark
{\bf \red This defines the category of torsors.}

\definition
{\bf \blue Affine space} is a torsor over a vector space $V$,
which is called {\bf \blue linearization}. 
The action of $V$ on $A$ is denoted 
 $a,v \arrow a +v$.

\example
Given two connections $\nabla$ and $\nabla_1$ on $B$,
the difference $\nabla-\nabla_1$ is an $\End(B)$-valued 1-form.
Converse is also true: for any $\End(B)$-valued 1-form 
$A\in \Lambda^1M \otimes \End(B)$, the operator $\nabla+A$ is a connection.
In other words, {\bf \red the space of connections is an
affine space over $\Lambda^1M \otimes \End(B)$}.


\newpage

{\bf \blue Affine space of orthogonal connections}

\claim
Let $B$ be a bundle with a scalar product. Then
{\bf \red the space of orthogonal connections on $B$ 
an affine space over $\Lambda^1 M \otimes \goth{so}(B)$.}

\proof
Let $s\in B^*\otimes B^*$ be a 2-form on $B$.
The action of $A:= \nabla-\nabla_1$ on $B^*\otimes B^*$
is given by $A(s) (x,y)= - s(A(x), y) - s(x, A(y))$.
Therefore, a difference $A$ of orthogonal connections satisfies
$h(A(x), y) = - h(x, A(y))$ for all $x, y\in B$. This is the same
as $A\in \Lambda^1 M \otimes \goth{so}(B)$. \endproof

Similarly one proves

\claim
Let $B$ be a bundle with a Hermitian structure and a
tensor $\Phi$, and ${\goth g}\subset \End(B)$ the 
Lie algebra of endomorphisms preserving $\Phi$. Then
{\bf \red the space of connections on $B$ preserving
  $\Phi$ is
an affine space over $\Lambda^1 M \otimes \g$.}


\newpage

{\bf \blue REMINDER: de Rham algebra}

\definition
Let $\Lambda^* M$ denote the vector bundle
with the fiber $\Lambda^*T^*_xM$ at $x\in M$
($\Lambda^*T^*M$ is the Grassman algebra of the cotangent
space $T^*_x M$). The sections of $\Lambda^i M$
are called {\bf \blue differential $i$-forms}.
 The algebraic operation ``wedge product'' defined
on differential forms is $C^\infty M$-linear; 
the space $\Lambda^* M$ of all differential forms
is called {\bf \blue the de Rham algebra}.

\remark $\Lambda^0 M = C^\infty M$.


\theorem
{\bf \red There exists a unique operator 
$C^\infty M\stackrel d \arrow \Lambda^1 M\stackrel d
\arrow \Lambda^2 M \stackrel d \arrow \Lambda^3 M \stackrel d \arrow ...$
satisfying the following properties}

1. On functions, $d$ is equal to the differential.\\
2. $d^2=0$ \\
3. $d(\eta \wedge \xi) = 
d(\eta) \wedge \xi + (-1)^{\tilde \eta}\eta \wedge
d(\xi)$,
where $\tilde \eta=0$ where $\eta\in \lambda^{2i}M$ is
{\bf \blue an
even form,} and $\eta\in \lambda^{2i+1}M$ is {\bf \blue odd.}


\definition
The operator $d$ is called {\bf \blue de Rham differential}.


\newpage

{\bf \blue Cartan formula}

\claim
For any $\eta \in \Lambda^1 M$, and $X,Y\in TM$
one has
{\bf \red\[
d\eta(X,Y) = \eta([X,Y])- \Lie_X(\eta(Y))+ \Lie_Y(\eta(X)).
\]}
\phantom{x} \hspace{-14mm} \proof Two sides of this 
equation define two operators $d, d_1\Lambda^1 M \arrow
\Lambda^2 M$. Both operators
satisfy the Leibniz rule
$d(f\eta)= df\wedge d\eta + f d\eta$.
When $\eta=df$ is exact, one has
 \begin{multline*}
 \eta([X,Y])- \Lie_X(\eta(Y))+ \Lie_Y(\eta(X)) = \\
= \Lie_{[X,Y]}(f)-\Lie_X\Lie_Y(f)+\Lie_Y\Lie_X(f)=0
\end{multline*}
hence $d_1(\alpha)=0$ on all closed forms.
A map $\delta:\; \Lambda^1(M)\arrow \Lambda^2(M)$
which vanishes on closed forms and satisfies the
Leibniz rule is de Rham differential, which can be seen
from the axiomatic
definition of $d$.
\endproof

\newpage

{\bf \blue Torsion}

\definition
Let $\nabla$ be a connection on $\Lambda^1 M$,
\[ \Lambda^1 \stackrel \nabla \arrow \Lambda^1 M \otimes \Lambda^1M.\]
{\bf \blue Torsion of $\nabla$} $T_\nabla:\; \Lambda^1 M
\arrow \Lambda^2 M$ is a map
$\nabla \circ \Alt  - d$, where 
$\Alt:\;  \Lambda^1 M \otimes \Lambda^1M\arrow \Lambda^2 M$
is exterior multiplication. 


\remark
\begin{align*}
T_\nabla(f\eta) = & \Alt(f\nabla\eta + df\otimes \eta) - d(f\eta)\\
= &f\bigg [\Alt(\nabla\eta) - d\eta\bigg] + df\wedge \eta - df\wedge \eta=
f T_\nabla(\eta).
\end{align*}
{\bf \purple Therefore $T_\nabla$ is linear}.

\definition 
Let $(M,g)$ be a Riemannian manifold.
A connection $\nabla$ on $TM$ is called {\bf \blue orthogonal}
if  $\nabla(g)=0$, and {\bf \blue Levi-Civita connection}
if it is orthogonal and has zero torsion.

\theorem {\bf \blue (``the fundamental theorem of Riemannian
geometry'')} {\bf \red  Every Riemannian manifold admits a Levi-Civita
connection, and it is unique.}

{\bf \green Will be proven later today}.

\newpage

{\bf \blue Gregorio Ricci-Curbastro, Tullio Levi-Civita}

{\setlength\tabcolsep{20mm}
\begin{tabular}{cc}
\epsfig{file=Ricci-Curbastro.jpeg,width=0.32\linewidth} &
\epsfig{file=Levi-Civita_3.jpeg,width=0.35\linewidth}\\
Gregorio Ricci-Curbastro, & Tullio Levi-Civita, \\
 1853-1925 & 1873-1941
\end{tabular}
}

{\green \em
...With his former student Tullio Levi-Civita, he wrote his
most famous single publication, a pioneering work on the
calculus of tensors, signing it as Gregorio Ricci. This
appears to be the only time that Ricci-Curbastro used the
shortened form of his name in a publication, and continues
to cause confusion.}

\newpage

{\bf \blue Torsion and commutator of vector fields}

\remark
Cartan formula gives
\begin{align*}
T_\nabla(\eta)(X,Y) = &\nabla_X(\eta)(Y) - \nabla_Y(\eta)(X)-
d\eta(X,Y) \\ =& \nabla_X(\eta)(Y) - \nabla_Y(\eta)(X)
-\eta([X,Y])- \Lie_X(\eta(Y))+ \Lie_Y(\eta(X)).
\end{align*}
On the other hand, 
$\nabla_X(\eta)(Y)= \Lie_X(\eta(Y)) - \eta(\nabla_X(Y))$.
Comparing the equations,  we obtain
\[
T_\nabla(\eta)(X,Y)=\eta\bigg(\nabla_X(Y)- \nabla_Y(X) - [X,Y]\bigg).
\]
{\bf \blue Torsion is often defined as a map
$\Lambda^2 TM \arrow TM$ using the formula
$\nabla_X(Y)- \nabla_Y(X) - [X,Y]$.}

We have just proved

\claim
{\bf \red The torsion tensor 
$\nabla_X(Y)- \nabla_Y(X) - [X,Y]$ is dual to the 
torsion $\nabla \circ \Alt  - d:\; 
\Lambda^1 M \arrow \Lambda^2 M$ defined above}.
\endproof

\newpage

{\bf \blue Linearization of the torsion}

\remark
Consider the space ${\cal A}(\Lambda^1 M)$ of connections
on $\Lambda^1 M$. The torsion defines an affine map
\[ {\cal A}(\Lambda^1 M) 
\arrow \Hom(\Lambda^1M, \Lambda^2 M)= TM \otimes\Lambda^2 M .
\]
because $T(\nabla + \alpha) = T(\nabla) +
\Alt_{12}(\alpha)$,
where $\Alt_{12}:\; \Lambda^1M \otimes \End(\Lambda^1M) \arrow
\Lambda^2M \otimes TM$ is antisymmetrization in the first two 
indices.

\definition
{\bf \blue Liearized torsion}
is a map 
\[
T_{lin}:\; \Lambda^1(M) \otimes \Lambda^1(M) \otimes TM
\arrow \Lambda^2 M  \otimes TM
\]
obtained as a linearization of the torsion map. {\bf
  \purple It is 
equal to $\Alt_{12}$.}

\newpage

{\bf \blue Existence of orthogonal connections}

\claim
Let $B$ be a vector bundle equipped with a scalar product.
{\bf \red Then $B$ admits an orthogonal connection}.

\proof
Chose a covering $\{U_i\}$, such that $B$ is trivial on
each $U_i$ and admits an orthonormal basis in each $U_i$.
On each $U_i$ we chose a connection $\nabla_i$
preserving this basis. Let $\psi_i$ be a partition of unit
subjugated to  $\{U_i\}$. Then 
{\bf \purple the formula $\nabla(b):= \sum \nabla_i(\psi_i
  b)$
defines an orthogonal connection}.

\theorem {\bf \blue (``the fundamental theorem of Riemannian
geometry'')} {\bf \red  Every Riemannian manifold admits a Levi-Civita
connection, and it is unique.}

\proof
See the next slide.

\newpage

{\bf \blue Levi-Civita connection (existence and uniqueness)}


\pstep Chose an orthogonal connection $\nabla_0$ on
$\Lambda^1 M$. The space ${\cal A}$ of orthogonal connections is
affine and {\bf \purple its linearization is  
$\Lambda^1 M \otimes {\goth{so}}(TM)$.}
We shall identify $\goth{so}(TM)$ and $\Lambda^2 M$.
Then {\bf \purple ${\cal A}$ is an affine space over
$\Lambda^1 M \otimes \Lambda^2 M$}.

{\bf \green Step 2:}  Then the linearized torsion map is
\[ T_{lin} :\; \Lambda^1 M \otimes {\goth {so}}(TM)=
\Lambda^1(M) \otimes \Lambda^2 M
\stackrel{\Alt_{12}} \arrow \Lambda^2 M \otimes \Lambda^1M =
\Lambda^2 M \otimes T M.
\]
{\bf \purple It is an isomorphism}. Indeed, on the right
and on the left there are bundles of the same rank, hence
it would suffice to show that $T_{lin}=\Alt_{12}$ is injective. However,
if $\eta \in \ker T_{lin}$, it is a form which is
symmetric on first two arguments and antisymmetric
on the second two, giving 
$\eta(x,y,z) = \eta(y,x,z) = - \eta (y,z, x).$
This gives $\sigma(\eta) =-\eta$, where $\sigma$ is
a cyclic permutation of the arguments. Since
$\sigma^3=1$, this implies $\eta=0$.

{\bf \green Step 3:} We have shown that {\bf \purple
an orthogonal connection is uniquely determined by its
torsion}. Indeed, torsion map is an isomorphism of affine
spaces.

 {\bf \green Step 4:} Let $\nabla:= \nabla_0
 -T_{lin}^{-1}(T_{\nabla_0})$.
Then $T_\nabla=
T_{\nabla_0}-T_{lin}(T_{lin}^{-1}(T_{\nabla_0}))=0$,
hence {\bf \red $\nabla$ is torsion-free.}
\endproof



\end{document}
