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\listok{3}{LCK manifolds 3: Homotheties on Riemannian manifolds}

\exercise
Let $\nabla$ be a torsion-free connection on a manifold $M$,
and $X$ a vector field which satisfies $\nabla(X)=\Id_{TM}$.
\enum
\item Prove that either the holonomy group of $\nabla$ is non-compact,
or $M$ is non-compact.
\item Find an example of such connection when $M$ is compact.
\ee
\ez

\exercise
Let $\nabla$ be a Levi-Civita connection on $M$,
$R\in \Lambda^2 M \otimes \End(TM)$ its curvature tensor, and
and $X\in TM$ a vector field which satisfies $\nabla(X)=\Id_{TM}$.
Prove that $R(X,Y)=0$ for any $Y\in TM$.
\ez

\hint
Compute $\nabla_Y\nabla_Z X - \nabla_Z\nabla_Y X$.
\eh

\exercise
Let $\nabla$ be the Levi-Civita connection on a Riemannian
manifold $(M,g)$
and $X$ a vector field satisfying $\nabla(X)=\Id_{TM}$.
\enum
\item Prove that $(M,g)$ is locally isometric to a Riemannian cone.
\item Let $R\in \Lambda^2 M \otimes \End(TM)$ be the curvature tensor.
Prove that $\Lie_X(R)=0$ and $\nabla_X (R)=-2R$.
\item Suppose that $(M,g)$ is Einstein: $\Ric(M)=c g$.
Prove that $c=0$.
%\item  Show that the zero set of $X$ is either one point
%or empty.
%\item Suppose that the sero set of $X$ is non-empty.
%Prove that $\nabla$ is flat.
\ee
\ez

\exercise
Let $X$ be a complete Riemannian manifold, 
$M= C(X)$ its Riemannian cone,
and $\phi:\; M \arrow M$ an isometry. Prove
that $\phi$ is induced by an isometry of $X$.
\ez

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