Congruence (manifolds)
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High Quality Content by WIKIPEDIA articles! 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. 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 s...
High Quality Content by WIKIPEDIA articles! 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. 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. If we add more mathematical structure, our congruence may acquire new significance.
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