In the theory of smooth manifolds, a congruence is the set of integral curves defined by a nonvanishing vector field defined on the manifold.
Vector fields can be specified as first order linear partial differential operators, such as These correspond to a system of first order linear ordinary differential equations, in this case where dot denotes a derivative with respect to some (dummy) parameter.
(A flow is a one-dimensional group of diffeomorphisms; a flow defines an action by the one-dimensional Lie group R, having locally nice geometric properties.)
To avoid complications arising from the presence of singularities, usually one requires the vector field to be nonvanishing.
If we add more mathematical structure, our congruence may acquire new significance.
For example, if we make our smooth manifold into a Riemannian manifold by adding a Riemannian metric tensor, say the one defined by the line element our congruence might become a geodesic congruence.
Indeed, in the example from the preceding section, our curves become geodesics on an ordinary round sphere (with the North pole excised).
In a Lorentzian manifold, such as a spacetime model in general relativity (which will usually be an exact or approximate solution to the Einstein field equation), congruences are called timelike, null, or spacelike if the tangent vectors are everywhere timelike, null, or spacelike respectively.