  Let $M$ be a smooth Riemannian manifold which is the union of a compact part
and a finite number of Euclidean ends, $\RR^n \setminus B(0,R)$ for some $R >
0$, each of which carries the standard metric. Our main result is that the
Riesz transform on $M$ is bounded from $L^p(M) \to L^p(M; T^*M)$ for $1 < p <
n$ and unbounded for $p \geq n$ if there is more than one end. It follows from
known results that in such a case the Riesz transform on $M$ is bounded for $1
< p \leq 2$ and unbounded for $p > n$; the result is new for $2 < p \leq n$. We
also give some heat kernel estimates on such manifolds. We then consider the
implications of boundedness of the Riesz transform in $L^p$ for some $p > 2$
for a more general class of manifolds. Assume that $M$ is a $n$-dimensional
complete manifold satisfying the Nash inequality and with an $O(r^n)$ upper
bound on the volume growth of geodesic balls. We show that boundedness of the
Riesz transform on $L^p$ for some $p > 2$ implies a Hodge-de Rham
interpretation of the $L^p$ cohomology in degree 1, and that the map from $L^2$
to $L^p$ cohomology in this degree is injective.
