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%version 1.0,\ \   09.04.2018
%version 2.0,\ \   14.04.2018, lots of serious errors
%                              about Jacobi fields (Ivan Frolov)



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\newcommand{\firstdate}{11.04.2018}

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\pagestyle{fancy} 
\lhead{\tiny Hogde theory, HSE} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

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\listok{11}{Hodge theory 11: exponential map and Frobenius theorem}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\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.
}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Exponential map}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let $\nabla:\; TM \arrow TM \otimes \Lambda^1 M$ be a connection.
{\bf Geodesic} with respect to $\nabla$ is a map
$\gamma:\; [a, b]\arrow M$ which satisfies
$\nabla_{\dot \gamma}(\dot \gamma)=0$,
where $\dot \gamma(t)=\frac{\gamma(u)}{du}\restrict {u=t}
\in TM\restrict {\im \gamma}$ is the 
derivative of $\gamma$, and the equation 
$\nabla_{\dot \gamma}(\dot \gamma)$ is understood
as parallelism of $\dot \gamma$, considered as
a section of the vector bundle $TM\restrict {\im \gamma}$ 
with connection $\nabla$. 
\ed

\exercise
Let $v\in T_x M$. 
\enum 
\ite[!] Prove that for $\epsilon$
sufficiently small, there exists a geodesic
 $\gamma:\; [0, \epsilon]\arrow M$ such that
$\gamma(0)=x$ and $\dot\gamma(0)=v$.
\ite[!] Prove that geodesic is unique.
\ee
\ez

\definition
Let $U\subset T_x M$ be the set of tangent
vectors such that for any $v\in U$ there exists a geodesic
$\gamma_v:\; [0, 1]\arrow M$, such that
$\gamma_v(0)=x$ and $\dot\gamma_v(0)=v$.
The map $v \arrow \gamma(1)$ is called
{\bf exponential map}.
\ed

\exercise[!]
Prove that exponential map defines a diffeomorphism
from a neighbourhood of 0 in $U$ to $M$.
\ez

\definition
The corresponding coordinates are called {\bf geodesic
coordinates}, or {\bf normal coordinates}.
\ed

\exercise
Suppose that $M$ is a Riemannian manifold,
and $\nabla$ an orthogonal connection. Prove that the
exponential map $\exp$ is defined on all $T_xM$, that is,
for any vector $v\in T_x M$ there exists a geodesic
map $\gamma:\; [0, \infty[$ tangent to $v$
\enum
\ite[!] when $M$ is compact
\ite[*] when it is complete as a metric space.
\ite[**] Suppose that $\exp$ is defined on all $T_xM$.
Prove that $M$ is complete as a metric space.
\ee
\ez

\exercise[*]
Find a compact manifold and a connection
such that the exponential map
is not defined on the whole $T_x M$
(that is, for some vector $v\in T_x M$ there is no geodesic
map $\gamma:\; [0, 1]$ tangent to $v$).
\ez


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Jacobi fields}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let $M$ be a manifold with connection, and $\exp:\; T_x M \arrow M$
the exponential map (which can be defined on the whole $T_x M$
or locally in a neighbourhood $U$ of 0). Let $A\in \End(T_x M)$
be a matrix, and $a\in T(T_x M)$ 
a linear vector field $x\arrow A(x)$   associated
with $A$. {\bf  Jacobi field} is a vector field on $\exp(U)$
obtained as an image of $a$ under the exponential map. 
\ed

\exercise[!]
Let $\gamma_0$ be a geodesic passing through $x\in M$, and
$J\in TM \restrict{\gamma_0}$ a Jacobi field.
Prove that there exists a smooth family $\gamma_t$ of
geodesics passing through $x$ such that
$J\restrict{\gamma_0(r)}= \frac{\gamma_t(r)}{dt}$.
\ez

\hint
A flow of diffeomorphisms associated
with a Jacobi field maps a geodesic passing
through x to another geodesic passing through x.
\eh

\remark
Usually, one defines Jacobi field as a vector field
$J\in TM \restrict{\gamma_0}$ on a geodesic such that
$J\restrict{\gamma_0(r)}= \frac{\gamma_t(r)}{dt}$
for some smooth family $\gamma_t$ of geodesics.
As shown above, this definition is equivalent to ours.
\er

\definition
{\bf Radial vector field} on a vector space $V=\R^n$ is
the linear vector field associated with the map 
$x\arrow x$. In standard coordinates it
can be written as $\sum x_i \frac{d}{dx_i}$.
\ed

\exercise
Let $X$ be a vector field defined
in a star-shaped neighbourhood of 0 in $\R^n$.
Prove that $X$ is a linear vector field,
if 
\enum
\ite $X$ commutes with the radial vector field $R$
\ite $[[X, R], R]=0$.
\ee
\ez

\hint
Use the Taylor series for $X$ and the following observation:
for any homogeneous vector field $R_d$ of degree $d$,
one has $[R, X]=dX$.
\eh 

\exercise
Let $\exp:\; T_x M \arrow M$ be the exponential
map, defining a diffeomorphism in a star-shaped 
neighbourhood  $U\subset T_xM$, and $\theta\in TM$
the image of the radial vector field under
$\exp$. Prove that $J$ is a Jacobi vector field
if and only if 
\enum
\ite[!] $[J,\theta]=0$
\ite[!] $[[J, \theta], \theta]=0$.
\ee
\ez


\remark
Abusing the language, we call
the vector field $\theta\in T\exp(U)$ 
defined above ``the radial vector field''
as well.
\er

%\exercise
%Let $\theta\in TM$ be the radial vector field
%on a manifold with connection $\nabla$.
%\enum
%\ite
%Prove that $\nabla_\theta \theta=0$.
%\ite Let $\theta'$ be a vector field 
%in a neighbourhood of $x\in M$ which satisfies
%$\nabla_{\theta'} {\theta}=0$. Suppose that
%$\theta'\restrict x=0$. Prove that 
%$\theta'=\const\theta$.
%\ee
%\ez

\definition
Let $\nabla:\; TM \arrow TM \otimes \Lambda^1 M$ be a connection.
{\bf Torsion} of $\nabla$ is a map 
$T_\nabla:\; \Lambda^2 TM \arrow TM$ mapping
vector fields $X, Y\in TM$ to $\nabla_X Y -\nabla_Y X-[X, Y]$.
A connection is called {\bf torsion-free} if its torsion vanishes.
{\bf Curvature} of $\nabla$ is a map
$\Theta_\nabla:\;  \Lambda^2 TM \arrow\End(TM)$
mapping vector fields $X, Y\in TM$ to 
$[\nabla_X,\nabla_Y]-\nabla_{[X,Y]}$ considered as 
a map from $TM$ to $TM$.
\ed

\exercise
Prove that torsion and curvature are $C^\infty(M)$-linear.
\ez

\exercise
Let $U$ be an image of a star-shaped domain
$W \subset T_x M$ under the exponential map $\exp:\; T_x M \arrow M$
which is a diffeomorphism on $W$. Denote by $\theta\in TU$
the radial vector field, and let $J\in TU$ be another vector field.
Assume that the connection $\nabla$ is torsion-free.
\enum
\ite Prove that $J$ is Jacobi if and only if 
$\nabla_J \theta= \nabla_\theta J$.
Prove that $J$ is Jacobi if and only if 
\begin{equation}\label{_Jacobi_via_3rd_commu_Equation_} 
\nabla_{[\theta, J]} \theta= 
   \nabla_\theta(\nabla_\theta J-\nabla_J\theta).
\end{equation}
\ite
Prove that $\theta$ is a radial vector field if and only
if $\nabla_\theta\theta=\theta$.
\ite
Prove that 
\[ -\nabla_{[\theta, J]} = 
   \Theta_\nabla(\theta, J)- \nabla_\theta\nabla_J + \nabla_J
   \nabla_\theta
\]
and
\[
-\nabla_{[\theta, J]} \theta = 
\Theta_\nabla(\theta, J)(\theta)-
\nabla_\theta\nabla_J\theta + \nabla_J \theta.
\]
Use this to rewrite
\eqref{_Jacobi_via_3rd_commu_Equation_} 
as 
\begin{equation}\label{_Jacobi_via_theta_Equation_}
\Theta_\nabla(\theta,
J)(\theta)=\nabla_\theta\nabla_\theta(J)-\nabla_J \theta.
\end{equation}
\ee
\ez

\exercise[!]
Let $J\in TM \restrict \gamma$ be a vector field tangent
to the geodesic. Denote by $\ddot J$ the second derivative
of $J$ along $\gamma$, with $\ddot J=\nabla_{\dot
  \gamma}\nabla_{\dot \gamma}(J)$.
Prove that $J$ is a Jacobi field if and only if
$\ddot J= \Theta_\nabla(\dot\gamma, J, \dot\gamma)$
\ez

\hint 
Let $t$ be a parametrization of a geodesic $\gamma(t)$.
Prove that $\theta$ is tangent to $\gamma$ and
$t\dot \gamma = \theta$ on $\gamma$, and deduce that
\[ 
\nabla_\theta\nabla_\theta = t^2 \nabla_{\dot
  \gamma}\nabla_{\dot \gamma} + t\nabla_{\dot \gamma}.
\]
using \eqref{_Jacobi_via_theta_Equation_}.
\eh

\remark
$\ddot J= \Theta_\nabla(\dot\gamma, J, \dot\gamma)$ is a second order ODE. \er





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Torsion and Frobenius theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let $B\subset TM$ be a sub-bundle of tangent bundle,
and $\Phi:\; \Lambda^2 B \arrow TM/B$ maps
vector fields $b_1, b _2\in B\subset TM$ to 
their commutator $[b_1, b_2]\mod B$.
Then $\Phi$ is called {\bf Frobenius form}.
\ed

\exercise
Prove that the Frobenius form is $C^\infty M$-linear.
\ez

\exercise
Let $B \subset TM$ be a sub-bundle. Prove that there
exists a connection $\nabla:\; TM \arrow TM \otimes \Lambda^1 M$ 
such that $\nabla(B) \subset B \otimes \Lambda^1 M$.
\ez

\remark In this case we say that {\bf connection $\nabla$
preserves $B$}.
\er

\exercise
Suppose that a connection $\nabla:\; TM \arrow TM \otimes \Lambda^1 M$
preserves a sub-bundle $B\subset TM$. 
\enum
\ite
Prove that $T_\nabla$ maps $\Lambda^2 B$ to $B$
if and only if its Frobenius form vanishes.
\ite[!] Suppose that the Frobenius form of $B$
vanishes. Prove that there exists a 
torsion-free connection preserving $B\subset TM$
\ee
\ez

\hint
Use the same argument as used in the proof
of existence of Levi-Civita connection.
\eh

\exercise
Let $B\subset TM$ be a sub-bundle, preserved
by a connection $\nabla$,
and $\Theta_\nabla$ its curvature. 
Prove that $\Theta_\nabla(X, Y, Z)\in B$
whenever $Z\in B$.
\ez

\exercise
Let $B\subset TM$ be a sub-bundle, preserved
by a torsion-free connection $\nabla$. 
\enum
\ite Denote by $\theta$ the radial
vector field, defined in a neighbourhood of $x\in M$.
Prove that for any Jacobi field $J$ such that $J\restrict
x \in B$ and $\nabla_\theta J\restrict x \in B$, 
one has $J\in B$.
\ite[!]
Use his to show that the exponential map maps
the subspace $B\restrict x \subset T_x M$ to a submanifold of $M$
tangent to $B$.
\ee
\ez

\exercise[!]
Prove Frobenius theorem:
given $B\subset TM$ such that $[B, B]\subset B$,
for any $x\in M$ there exists a neighbourhood 
$U \subset M$  and a family of submanifolds
$V_t\subset U$ parametrized by $t\in Z$ such that
$U$ is obtained as a disjoint union of $V_t$,
and $TV_t = B$ at each point of $V_t$ for each $t\in Z$.
\ez

\remark In these assumptions, submanifolds $V_t$ are called
{\bf leaves of the foliation} defined by $B$,
and $B$ is called  {\bf an involutive} or {\bf holonomic}
sub-bundle of $TM$.
\er


\end{document}

