In the theory of
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...
s, a congruence is the set of
integral curve
In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations.
Name
Integral curves are known by various other names, depending on the nature and interpre ...
s defined by a nonvanishing
vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
defined on the manifold. Congruences are an important concept in
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, and are also important in parts of
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
.
A motivational example
The idea of a congruence is probably better explained by giving an example than by a definition. Consider the smooth manifold R². 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
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.
Definit ...
s; a flow defines an
action
Action may refer to:
* Action (philosophy), something which is done by a person
* Action principles the heart of fundamental physics
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video gam ...
by the one-dimensional
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
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
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
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
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
, 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
geodesic
In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a conn ...
s on an ordinary round sphere (with the North pole excised). If we had added the standard Euclidean metric
instead, our curves would have become
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
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
Hopf bundle
In differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influe ...
or ''Hopf fibration''. The integral curves or fibers respectively are certain ''pairwise linked'' great circles, the
orbit
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
s in the space of unit norm
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s under left multiplication by a given unit quaternion of unit norm.
Congruences in Lorentzian manifolds
In a
Lorentzian manifold
In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere non-degenerate bilinear form, nondegenerate. This is a generalization of a Riema ...
, such as a
spacetime
In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
model in general relativity (which will usually be an
exact or approximate solution to the
Einstein field equation
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it.
The equations were published by Albert Einstein in 1915 in the ...
), 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
has vanishing
covariant derivative
In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to:
Statistics
* Covariance matrix, a matrix of covariances between a number of variables
* Covariance or cross-covariance between ...
,
.
See also
*
Congruence (general relativity) In general relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime. Often ...
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).
Differential topology