Abstract
The gauge equation is a generalization of the conjugacy relation for Koszul connection to 1bundle morphisms that are not isomorphisms. The existence of non-trivial solution to this equation, 2especially when duality is imposed upon related connections, gives important information about the 3geometry of the manifolds considered. In this article, we use the gauge equation to introduce spectral 4sequences that are further specialized to Hessian structures. 5