# Congruence (manifolds)

In the theory of smooth manifolds, a **congruence** is the set of integral curves defined by a nonvanishing vector field defined on the manifold.

Congruences are an important concept in general relativity, and are also important in parts of Riemannian geometry.

## A motivational example

The idea of a congruence is probably better explained by giving an example than by a definition. Consider the smooth manifold **R**². 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. The solutions of such systems are *families of parameterized curves*, in this case

This family is what is often called a *congruence of curves*, or just *congruence* for short.

This particular example happens to have two *singularities*, where the vector field vanishes. These are fixed points of the *flow*. (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.) These two singularities correspond to two *points*, rather than two curves. In this example, the other integral curves are all simple closed curves. Many flows are considerably more complicated than this. To avoid complications arising from the presence of singularities, usually one requires the vector field to be *nonvanishing*.