  We define and study multiplicative connections in the tangent bundle of a Lie
groupoid. Multiplicative connections are linear connections satisfying an
appropriate compatibility with the groupoid structure. Our definition is
natural in the sense that a linear connection on a Lie groupoid is
multiplicative if and only if its torsion is a multiplicative tensor in the
sense of Bursztyn-Drummond [5] and its geodesic spray is a multiplicative
vector field. We identify the obstruction to the existence of a multiplicative
connection. We also discuss the infinitesimal version of multiplicative
connections in the tangent bundle, that we call infinitesimally multiplicative
(IM) connections and we prove an integration theorem for IM connections.
Finally, we present a few toy examples.
