Given a surjective submersion, the following are equivalent: (i) it is locally trivial, (ii) it admits a complete connection, (iii) it admits a fibered complete metric. Surjective submersions are examples of Lie groupoids, and conversely, any Lie groupoid can be regarded as a surjective submersion over a stack. In this talk I will discuss the classic result and its groupoid analogue, relating linearization of Lie groupoids and complete Riemannian groupoids.

Joint work with M. de Melo (USP).