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\lhead{\tiny Topologia das Variedades, IMPA, 2018} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  IMPA,\ \ \sc\version}
\rhead{{\tiny  Misha Verbitsky}}

\begin{document}

\listok{3}{Topologia das Variedades: class exercise 3}

\exercise
Consider the product of two spheres $M:=S^n\times S^m$
equipped with the action of the group $G:=SO(n+1)\times SO(m+1)$.
Find the space of $G$-invariant differential forms on $M$.
\ez


\exercise
Consider the complex projective space $M=\C P^2$,
equipped with an action of the group $G=U(2)$.
\enum
\ite Find the space of $G$-invariant differential forms on $M$.
\ite Prove that the de Rham cohomology of $H^*(\C P^2)$ are isomorphic
to the algebra of $G$-invariant forms. Find this space and
describe its multiplicative structure.
\ite Prove that any diffeomorphism of $\C P^2$ preserves the orientation.
\ee
\ez

\hint Use the algebra structure on the de Rham cohomology.
\eh

\definition
{\bf Symmetric space} is a complete Riemannian manifold $M$
such that for each $x\in M$ there exists an isometry
$\tau_x$ fixing $x$ and acting on its tangent space $T_x M$ as $-\Id$.
\ed

\exercise
Prove that the following spaces admit a Riemannian structure
making them symmetric.
\enum
\ite Sphere $S^n$ and a real projective space $\R P^n$ (for all $n$).
\ite Complex projective space $\C P^n$.
\ite Grassmann space of all $k$-dimensional planes in $\R^n$.
\ee
\ez

\exercise
Prove that the space $SO(1,n)/SO(n)$
admits an $SO(n,1)$-invariant metric, which is unique up to a 
constant multiplier. Prove that it is symmetric.
\ez

\remark The space $SO(1,n)/SO(n)$
with this metric is called {\bf the hyperbolic space}.
\er


\exercise
Let $M$ be a connected symmetric space. Prove that its isometry
group acts on $M$ transitively.
\ez

\exercise
Let $M$ be a symmetric space, $M=G/H$, where $G$ is its isometry group
and $H$ the stabilizer group of a point $x\in M$. 
\enum
\ite Prove that all $G$-invariant
differential forms on $M$ are even. 
\ite Use this to show that the de Rham cohomology
of a symmetric space $G/H$, with $G$ compact, are isomorphic to the space
of $G$-invariant differential forms.
\ee
\ez





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