2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/223910We study non-abelian differentiable gerbes over stacks using the theory of Lie groupoids. More precisely, we develop the theory of connections on Lie groupoid $G$-extensions, which we call "connections on gerbes", and study the induced connections on various associated bundles. We also prove analogues of the Bianchi identities. In particular, we develop a cohomology theory which measures the existence of connections and curvings for $G$-gerbes over stacks. We also introduce $G$-central extensions of groupoids, generalizing the standard groupoid $S^1$-central extensions. As an example, we apply our theory to study the differential geometry of $G$-gerbes over a manifold.67 pages, references added and updated, final version to appear in Adv. MathDifferential GeometryHigh Energy Physics - TheoryMathematical Physicstext