  Let $M_1$, $\cdots$, $M_\ell$ be complete, connected and non-collapsed
manifolds of the same dimension, where $2\le \ell\in\mathbb{N}$, and suppose
that each $M_i$ satisfies a doubling condition and a Gaussian upper bound for
the heat kernel. If each manifold $M_i$ has volume growth either bigger than
two or equal to two, then we show that the Riesz transform $\nabla \L^{-1/2}$
is bounded on $L^p(M)$ for each $1<p<2$ on the gluing manifold
$M=M_1\#M_2\#\cdots \# M_\ell$.
