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WPCleaner v1.14 - fix link

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In the theory of [[smooth manifold]]s, a '''congruence''' is the set of [[integral curve]]s 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
:<math>\vec{X} = ( x^2 - y^2 ) \, \partial_x + 2 \, x y \, \partial_y</math>
These correspond to a system of ''first order linear ordinary differential equations'', in this case
:<math>\dot{x} = x^2 - y^2,\; \dot{y} = 2 \, x y</math>
where dot denotes a derivative with respect to some (dummy) parameter. The solutions of such systems are ''families of parameterized curves'', in this case
:<math> x(\lambda) = \frac{x_0 - (x_0^2+y_0^2) \, \lambda}{1 - 2 \, x_0 \, \lambda + (x_0^2 + y_0^2) \, \lambda^2} </math>
:<math> y(\lambda) = \frac{y_0}{1 - 2 \, x_0 \, \lambda + (x_0^2 + y_0^2) \, \lambda^2} </math>
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 point (mathematics)|fixed point]]s of the ''flow''. (A flow is a one dimensional group of [[diffeomorphism]]s; a flow defines an [[group action|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 curve]]s. 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.

==Congruences in Riemannian manifolds==

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
:<math>ds^2 = \left( \frac{2}{1 + x^2 + y^2} \right)^2 \, \left( dx^2 + dy^2 \right) </math>
our congruence might become a ''geodesic congruence''. Indeed, in the example from the preceding section, our curves become [[geodesic]]s on an ordinary round sphere (with the North pole excised). If we had added the standard Euclidean metric <math>ds^2 = dx^2 + dy^2</math> instead, our curves would have become [[circle]]s, but not geodesics.

An interesting example of a Riemannian geodesic congruence, related to our first example, is the [[Clifford congruence]] on P³, which is also known at the [[Hop pi bundle]] or ''Hop pi fibration''. The integral curves or fibers respectively are certain ''pairwise linked'' great circles, the [[orbit]]s in the space of unit norm [[quaternion]]s under left multiplication by a given unit quaternion of unit norm.

==Congruences in Lorentzian manifolds==

In a [[Lorentzian manifold]], such as a [[spacetime]] model in general relativity (which will usually be an [[exact solutions in general relativity|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. A congruence is called a ''geodesic congruence'' if the tangent vector field <math>\vec{X}</math> has vanishing [[covariant derivative]], <math>\nabla_{\vec{X}} \vec{X} = 0</math>.

==See also==

*[[congruence (general relativity)]]

==References==

*{{cite book | author=Lee, John M. | title=Introduction to smooth manifolds | location=New York | publisher=Springer | year=2003 | isbn=0-387-95448-1}} A textbook on manifold theory. See also the same author's textbooks on topological manifolds (a lower level of structure) and Riemannian geometry (a higher level of structure).

[[Category:Differential topology]]

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en>Niceguyedc: WPCleaner v1.14 - fix link