For a smooth morphism pp of smooth analytic spaces or of smooth schemes p:XSp\colon X \to S a pp-connection is an 𝒪 X\mathcal{O}_X-linear map S:p *T ST X\nabla_S\colon p^* T_S \to T_X such that dp S=id p *T S\mathrm{d}p \circ \nabla_S = id_{p^* T_S}. The “differentialdp\mathrm{d}p here is the map T Sp *T XT_S \to p^* T_X induced by the universality of the pullback and the differential. A pp-connection is flat/integrable if the corresponding (by adjunction) map T Sp *T XT_S \to p_* T_X commutes with brackets of vector fields.

Last revised on July 29, 2010 at 13:55:26. See the history of this page for a list of all contributions to it.