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          \scriptsize {\it \small Geometry of manifolds,
            lecture 8 \hfil
  \tiny M. Verbitsky }}
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Geometry of manifolds \\[15mm]
\small Lecture 8: Vector bundles and locally trivial fibrations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf Math in Moscow and HSE
\\[2mm]  April 1, 2013
}
\end{center}

\newpage

{\bf \blue Locally trivial fibrations}

\definition
A smooth map $f:\; X \arrow Y$ is called {\bf \blue a locally trivial
fibration} if each point $y\in Y$ has a neighbourhood
$U\ni y$ such that $f^{-1}(U)$ is diffeomorphic to
$U\times F$, and the map
$f:\; f^{-1}(U)= U\times F \arrow U$ is a projection.
In such situation, $F$ is called {\bf \blue the fiber}
of a locally trivial fibration.

\definition {\bf \blue A trivial fibration}
is a map $X \times Y \arrow Y$.

\definition
{\bf \blue A vector bundle} on $Y$ is a locally trivial fibration
$f:\; X\arrow Y$ with fiber $\R^n$, with each fiber 
$V:= f^{-1}(y)$ equipped with a structure of a vector
space, smoothly depending on $y\in Y$.

\remark This definition {\bf \green is not very precise or rigorous,}
because ``smoothly depending on $y\in Y$'' {\bf \red needs to be explained.}


\remark {\bf \purple
This definition is compatible with the one we used previously}
(``a vector bundle is a locally free sheaf of $C^\infty M$-modules'').
This will be explained later today.


\newpage

{\bf \blue Fiber product}

\definition
Fix a topological space $M$, and let $\pi_1:\; X_1 \arrow M$ and
$\pi_2:\; X_2 \arrow M$ be continuous maps.
The {\bf \blue fiber product} is defined as
\[ X_1\times_M X_2:= \{(x_1,x_2)\in X_1\times
   X_2\ \ |\ \ \pi_1(x_1)=\pi_2(x_2)\}.
\]

\remark
Consider the projection $X_1\times_M X_2\stackrel \pi\arrow M$.
Then
\[ \pi^{-1}(m)= \pi_1^{-1}(m)\times \pi_2^{-1}(m).
\]

\exercise
Let $\pi_i:\; X_i \arrow M$, $i=1,2$ be trivial
fibrations, $X_i= M\times F_i$. {\bf \purple Prove that
$X_1 \times_M X_2 = M\times F_1\times F_2$.}

\remark
If $X_i$ are locally trivial fibrations over
$M$ with fiber $F_i$, {\bf \purple the fiber product
$X_1\times_M X_2$ is a locally trivial
fibration over $M$ with fiber $F_1 \times F_2$.}

\newpage

{\bf \blue Fiber product: universal property}

\remark The fiber product satisfies the following
{\bf \blue universal property}. 
Let $X_1\times_M
X_2\stackrel {\tilde \pi_i} \arrow X_i$
be the natural projection. Then any commutative square
\[\begin{CD} Y@>{f_1}>> X_1\\
@V{f_2}VV @VV{\pi_1}V\\
X_2 @>{\pi_2}>> M
\end{CD}
\]
induces a unique continuous map $f:\; Y \arrow X_1\times_M X_2$
such that $f\circ {\tilde \pi_i}=f_i$.
Moreover, {\bf \red any space satisfying the universal
property is homeomorphic to $X_1\times_M X_2$.}

\remark This statement is awkward, because I avoided the
language of categories. 

\exercise {\bf \purple
Translate this property to the language of categories.}

\newpage

{\bf \blue Topological groups over a base}

\definition
Fix a topological space $M$.
{\bf \blue A space over $M$}  $(X, f:\;
X \arrow M)$ is a topological space
equipped with a continuous map to $M$. {\bf A morphisms}
$\phi:\; (X_1, f_1) \arrow (X_2, f_2)$ is a continuous
map $\phi:\; X_1 \arrow X_2$ commuting with projections to $M$.


\definition
Let $B \stackrel \pi \arrow M$ be a continuous map, 
and $B\times_M B \stackrel \Psi \arrow M$ -
a morphism over $M$. This morphism is called {\bf\blue
associative multiplication} if it is associative on the fibers of
 $\pi$, that is, satisfies
$\Psi(a, \Psi(b,c))= \Psi(\Psi(a,b),c)$ for every
triple $a,b,c$ in the same fiber. \\
\phantom{X} A section $M \stackrel e \arrow B$ is called
{\bf\blue the unit} if the maps
$B \stackrel {\Id_B \times e}\arrow B\times_M B \stackrel \Psi \arrow B$
and $B \stackrel {e\times \Id_B}\arrow B\times_M B \stackrel \Psi \arrow B$
are equal to $Id_B$.  \\
\phantom{X} A morphism $\nu:\; B \arrow B$ over $M$ 
is called {\bf \blue a group inverse} if each of the maps
$B\stackrel \Delta \arrow B\times_M B \stackrel
{\Id_B\times \nu}\arrow B\times_M B \stackrel \Psi \arrow B$
and $B\stackrel \Delta \arrow B\times_M B \stackrel
{\nu\times \Id_B}\arrow B\times_M B \stackrel \Psi \arrow B$
 is a constant map, mapping $b$ to $e(\pi(b))$. 

A map $B \stackrel \pi \arrow M$
equipped with associative multiplication, unit and 
group inverse is called {\bf\blue a topological group over $M$}. 


\newpage

{\bf \blue Vector spaces over a base}

\definition
Let $G$ be an abelian group, and $k$ a field.
Suppose that for each non-zero $\lambda\in k$
there exists an automorphism $\phi_\lambda:\; G \arrow G$,
such that  $\phi_\lambda\circ\phi_{\lambda'}=\phi_{\lambda\lambda'}$,
and $\phi_{\lambda+\lambda'}(g)= \phi_{\lambda}(g)+\phi_{\lambda'}(g).$
Then $G$ is called {\bf \blue a vector space over $k$}.

\definition
Let $k=\R$ or $\C$.
An abelian topological group $B\stackrel \pi \arrow M$ over $M$
is called {\bf \blue a vector space over a base $M$},
or {\bf \blue a relative vector space over $M$}
if for each non-zero $\lambda\in k$ there exists a
continuous automorphism
$\phi_\lambda:\; B \arrow B$ of a group $B$ over $M$
satisfying assumptions of the above definition.

\remark
Let  $B\stackrel \pi \arrow M$ be a relative vector space
over $M$, $U\subset M$ an open subset,
and ${\cal B}(U)$ the space of sections of a map 
$\pi^{-1}(U) \stackrel \pi \arrow U$. Then
{\bf \red ${\cal B}(U)$ defines a sheaf of modules
over a sheaf $C^0(M)$ of continuous functions.}

\example
Let  $S\subset \R^n$ be a subset (not necessarily 
a smooth submanifold), $s\in S$ a point, and $v\in
T_s\R^n$ a vector. We sat that $v$ belongs to a {\bf\blue tangent cone} 
$C_sS$ if the distance from $S$ to a point $s+tv$
converges to 0 as $t\rightarrow 0$ faster than linearly:
$\lim\limits_{t \rightarrow 0}\frac{d(S, s+tv)}{t} = 0.$
{\bf \purple Then the set $CS$ of all pairs $(s, v), s\in S, v\in C_s S$
is a relative vector space over $S$}.

\newpage

{\bf \blue Total space of a vector bundle}


\definition
Let $B \arrow M$ be a smooth locally trivial
fibration with fiber $\R^n$.
Assume that  $B$ is equipped with a structure
of relative vector space over $M$, and all the maps
used in the definition of a relative vector space
are smooth. Then $B$
is called {\bf\blue a total space of a vector bundle.}

\remark
Let $\pi:\; B \arrow M$ be a total space of a vector bundle, 
$U\subset M$ open subset, and ${\cal B}(U)$ the space of all smooth sections
of $\pi^{-1}(U)\stackrel \pi \arrow U$. {\bf \purple Then
  ${\cal B}$ is a locally free sheaf of $C^\infty M$-modules}.

\theorem
{\bf \red Every locally free sheaf $C^\infty M$-modules is defined
from a total space of a vector bundle,} which
is determined uniquely by a sheaf.

The proof will be a couple of slides below.

\newpage

{\bf \blue Fiber of a locally free sheaf}


\definition
Let ${\cal B}$ be an $n$-dimensional locally free
sheaf of $C^\infty$-modules on $M$,
$x\in M$ a point,  ${\cal B}_x$ the space of germs of ${\cal B}$
in $x$, and ${\goth m}_x\subset C^\infty_x M$
the maximal ideal in the ring of germs $C^\infty_x M$
of smooth functions. Define {\bf\blue the fiber} of
${\cal B}$ in $x$ as a quotient
${\cal B}_x/{\goth m}_x{\cal B}_x$.
A fiber of ${\cal B}$ is denoted
${\cal B}\restrict x$.

\remark
{\bf \purple A  fiber of an $n$-dimensional bundle
is an $n$-dimensional vector space.}

\remark
Let  ${\cal B}= C^\infty M^n$, and  
$b\in {\cal B}\restrict x$ a point of a fiber, represented
by a germ $\phi \in {\cal B}_x=C^\infty_m M^n$,
$\phi=(f_1, ..., f_n)$. Consider a map $\Psi$ from the set
of all fibers ${\cal B}$ to $M \times \R^n$,
mapping $(x, \phi=(f_1, ..., f_n))$
to $(f_1(x), ..., f_n(x))$. {\bf \red Then $\Psi$ is bijective.}
Indeed, ${\cal B}\restrict x=\R^n$.


\newpage

{\bf \blue Total space of a vector bundle from its sheaf
  of sections}


\definition
Let ${\cal B}$ be an $n$-dimensional locally free sheaf
of $C^\infty$-modules. Denote the set of all
vectors in all fibers of  ${\cal B}$ over all points of $M$
by $\Tot {\cal B}$. Let  $U\subset M$ be an open subset of $M$,
with  ${\cal B}\restrict U$ a trivial bundle.
Using the local bijection $\Tot {\cal B}(U)=U \times \R^n$
we consider topology on $\Tot {\cal B}$
induced by open subsets in 
$\Tot {\cal B}(U)=U \times \R^n$ for all
open subsets $U\subset M$ and all trivializations of
${\cal B}\restrict U$. Then
$\Tot {\cal B}$ is called {\bf \blue a total space of a
  vector bundle ${\cal B}$.}

\claim 
The space $\Tot {\cal B}$ with this topology {\bf \purple is
a locally trivial fibration over $M$, with fiber $\R^n$.}
Moreover, it is a relative vector space over $M$, and
{\bf \purple the sheaf of smooth sections of $\Tot {\cal B}\arrow M$
is isomorphic to ${\cal B}$.}


\remark {\bf \red This gives an equivalence between locally free
sheaves of $\C^\infty$-modules and the 
total spaces of vector bundles,} defined abstractly in
terms of locally trivial fibrations.


\newpage

{\bf \blue Tensor product}


\definition
Let $V,V'$ be $R$-modules, $W$ a free abelian group
generated by $v\otimes v'$, with $v\in V, v'\in V'$,
and $W_1\subset W$ a subgroup generated by combinations
$rv \otimes v'-v\otimes rv'$, $(v_1+ v_2)\otimes v'-
v_1 \otimes v' - v_2 \otimes v'$ and $v\otimes (v'_1+ v'_2)-
v\otimes v'_1 - v\otimes v'_2$.
Define {\bf\blue the tensor product} $V \otimes_R V'$
as a quotient group $W/W_1$.

\exercise
Show that $r \cdot v\otimes v'\mapsto (rv)\otimes v'$
{\bf \purple defines an $R$-module structure on $V \otimes_R V'$.}

\remark
Let ${\cal F}$ be a sheaf of rings, and
${\cal B}_1$ and ${\cal B}_2$ be sheaves
of locally free $(M, {\cal F})$-modules.
{\bf \purple Then
\[ U \arrow {\cal B}_1(U)\otimes_{{\cal F}(U)} {\cal B}_2(U)
\]
is also a locally free sheaf of modules.}


\definition
{\bf \blue Tensor product} of vector bundles
is a tensor product of the corresponding sheaves of modules.

\newpage

{\bf \blue Dual bundle and bilinear forms}

\definition
Let $V$ be an $R$-module. {\bf\blue A dual $R$-module}
$V^*$ is $\Hom_R(V, R)$ with the $R$-module structure
defined as follows: $r\cdot h(\dots) \mapsto rh(\dots)$.

\claim
Let ${\cal B}$ be a vector bundle, that is,
a locally free sheaf of  $C^\infty M$-modules, and
$\Tot{\cal B}\stackrel \pi \arrow M$
its total space. Define ${\cal B}^*(U)$
as a space of smooth functions on $\pi^{-1}(U)$ 
linear in the fibers of $\pi$. 
{\bf \purple
Then  ${\cal B}^*(U)$ is a locally free sheaf over
$C^\infty(M)$.}

\definition
This sheaf is called {\bf \blue the dual vector bundle},
denoted by $B^*$. Its fibers are dual to the fibers 
of $B$.

\definition
{\bf \blue Bilinear form} on a bundle ${\cal B}$ 
is a section of $({\cal B}\otimes {\cal B})^*$.
A symmetric bilinear form on a real bundle ${\cal B}$
is called {\bf\blue positive definite} if it gives
a positive definite form on all fibers of 
${\cal B}$. Symmetric positive definite
form is also called {\bf\blue a metric}.
A skew-symmetric bilinear form on
${\cal B}$ is called {\bf\blue non-degenerate}
if it is non-degenerate on all fibers of ${\cal B}$.



\newpage

{\bf \blue Subbundles}

\definition
{\bf\blue A subbundle} ${\cal B}_1\subset {\cal B}$ 
is a subsheaf of modules which is also a vector bundle.

\definition
{\bf\blue Direct sum} $\oplus$ of vector bundles is a direct
sum of corresponding sheaves.

\example
Let ${\cal B}$ be a vector bundle equipped with a metric
(that is, a positive definite symmetric form),
and ${\cal B}_1 \subset {\cal B}$ a subbundle.
Consider a subset $\Tot {\cal B}_1^\bot\subset \Tot {\cal B}$,
consisting of all $v\in {\cal B}\restrict x$
orthogonal to ${\cal B}_1\restrict x \subset {\cal
B}\restrict x$. {\bf \purple Then $\Tot {\cal B}_1^\bot$ 
is a total space of a subbundle, denoted as ${\cal
B}_1^\bot\subset {\cal B}$,} and we have an isomorphism
${\cal B}= {\cal B}_1\oplus {\cal B}_1^\bot$.

\remark
A total space of a direct sum of vector
bundles ${\cal B}\oplus {\cal B}'$ {\bf \purple is homeomorphic to 
$\Tot {\cal B}\times_M \Tot {\cal B}'$.}

\exercise
Let ${\cal B}$ be a real vector bundle.
{\bf \red Prove that ${\cal B}$ admits a metric.}



\end{document}



%
%\newpage
%
%{\bf \blue Morphisms of sheaves}
%
%\definition
%Let ${\cal B}, {\cal B}'$ be sheaves on
%$M$. {\bf\blue A sheaf morphism} from ${\cal B}$ to ${\cal B}'$
%is a collection of homomorphisms ${\cal B}(U)\arrow {\cal B}'(U)$,
%defined for each open subset $U\subset M$,
%and compatible with the restriction maps:
%\[
%\begin{CD}
%{\cal B}(U) @>>> {\cal B}'(U)\\
%@VVV@VVV\\
%{\cal B}(U_1) @>>> {\cal B}'(U_1)
%\end{CD}
%\]
%
%\remark
%Morphisms of sheaves of modules are defined in the same
%way, but in this case {\bf \purple the maps ${\cal B}(U)\arrow {\cal
%  B}'(U)$ should be compatible with the module structure.}
%
%
%\definition
%A sheaf morphism is called {\bf\blue injective}
%if it is injective on germs
%and {\bf\blue surjective}, if it is surjective on germs.
%
%
%\definition
%Let ${\cal B} \stackrel \phi \arrow {\cal B}'$ be a
%morphism of locally free sheaves of  $C^\infty M$-modules.
%It is called {\bf\blue a smooth morphism}, or
%{\bf\blue a morphism of vector bundles} if 
%$\phi$ has locally free kernel and locally 
%free cokernel.
%
