In algebraic geometry, p-curvature is an invariant of a connection on a coherent sheaf for schemes of characteristic p > 0.
It is a construction similar to a usual curvature, but only exists in finite characteristic.
Suppose X/S is a smooth morphism of schemes of finite characteristic p > 0, E a vector bundle on X, and
Moreover, the expression is p-linear in D. By the definition p-curvature measures the failure of the map
to be a homomorphism of restricted Lie algebras, just like the usual curvature in differential geometry measures how far this map is from being a homomorphism of Lie algebras.