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%version 1.1,\ \   15.10.2020: A couple of corrections from Nikita Klemyatin



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\lhead{\tiny Complex geometry, HSE} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

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\listok{5}{Complex geometry handout 5: connections and torsion}

%{\scriptsize
%  {\bf Rules:} You may choose to solve only 
%``hard'' exercises (marked with !, * and **) 
%or ``ordinary'' ones (marked with ! or unmarked),
%or both, if you want to have extra stuff to work.
%To have a perfect score, a student must obtain
%(in average) a score of 10 points per week.
%
%If you have got credit for 2/3 of ordinary problems
%or 2/3 of ``hard'' problems, you receive  
%$6t$ points, where $t$ is a number depending on the
%date when it is done. Passing all ``hard'' 
%or all ``ordinary'' problems brings you $10t$ points.
%Solving of ``**'' (extra hard) problems is not
%obligatory, but each such problem gives you a credit
%for 2 ``*'' or ``!'' problems in the ``hard'' set.
%
%The first 3 weeks after giving a handout, $t=1.5$,
%between 21 and 35 days, $t=1$, and afterwards, $t=0.7$.
%The scores are not cumulative, only the
%best score for each handout counts.
%}

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\exercise
Let $G$ be a connected Lie group, 
and  $\nabla$ a connection on $TG$
such that $\nabla(X)=0$ for all left-invariant vector fields.
Prove that such $\nabla$ exists and is unique.
Prove that $\nabla$ is torsion-free
if and only if $G$ is commutative.
\ez

\exercise
Let $M$ be a manifold
equipped with an action of quaternion algebra in $TM$.
Prove that there exists a connection $\nabla$ on $TM$ 
such that for any quaternion $h\in {\Bbb H}$ and any
$X, Y\in TM$ one has $\nabla_Y(h(X))= h(\nabla_Y(X))$.
\ez

\exercise
Let $B\subset TM$ be a sub-bundle.
\enum
\ite
Prove that there exists a connection
$\nabla:\; TM \arrow TM \otimes \Lambda^1(M)$ such that
$\nabla(B) \subset B\otimes \Lambda^1(M)$.
\ite Suppose that $\nabla$ is torsion-free.
Prove that $B$ is an integrable distribution,
that is, $[B,B]\subset B$.
\ite Assume that $[B,B]\subset B$.
Prove that then $\nabla$ can be chosen torsion-free.
\ee
\ez



\definition
Let $\a(M)\subset\End(TM)$  be a bundle of Lie algebras.
{\bf The space of intrinsic torsion}
of $\a(M)$ is 
${\cal T}_\a:=\frac{\Lambda^2(M)\otimes
  TM}{\Alt_{12}(\Lambda^1(M)\otimes \a(M))}$.
Let $\a(M)$ be the bundle of Lie algebras
leaving invariant a collection ${\cal A}$ of tensors
or subspaces in the tensor powers of $TM$.
Suppose that $\nabla$ is a connection preserving
${\cal A}$. {\bf The intrinsic torsion of ${\cal A}$}
is the class in ${\cal T}_\a$ represented by the torsion
of $\nabla$.
\ed

\exercise
Let $X$ be a nowhere vanishing vector field on $M$,
and $\a(M)$ is the bundle of Lie algebras preserving $X$.
Prove that the intrinsic torsion space of $\a(M)$ is trivial.
\ez


\exercise
Let $\nu$ be a nowhere vanishing volume form on $M$,
and $\a(M)$ is the bundle of Lie algebras preserving $\nu$.
Prove that the intrinsic torsion space of $\a(M)$ is trivial.
Prove that there exists a torsion-free connection preserving $\nu$.
\ez

\exercise
Let  $B\subset TM$ be a sub-bundle,
and $\a(M)\subset\End(TM)$ the Lie algebra
of all $v\in \End(B)$ such that $v(B)\subset B$.
\enum
\ite Prove that the intrinsic torsion space
is isomorphic to $\Lambda^2 (B^*)\otimes (TM/B)$.
\ite
Prove that the intrinsic torsion of $B$
is its Frobenius form.
\ee
\ez

\definition
{\bf Yano structure} on a manifold $M$ is
$F\in \End(TM)$ of constant rank satisfying $F^3=-F$.
\ed

\exercise
Let $F$ be a Yano structure on $M$.
\enum
\ite Prove that there exists a connection $\nabla$ preserving $F$.
\ite Suppose that $\nabla$ is torsion-free. Prove that the bundle
$\ker F$ and the eigenbundle $\{x\in TM\otimes \C\ \ |\ \ F(x)=\1x\}$
are integrable.
\ee
\ez


\end{document}