A curvature collineation (often abbreviated to CC) is
vector field which preserves the
Riemann tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
in the sense that,
:
where
are the components of the Riemann tensor. The
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of all
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
curvature collineations forms a
Lie algebra under the
Lie bracket
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
operation (if the smoothness condition is dropped, the set of all curvature collineations need not form a Lie algebra). The Lie algebra is denoted by
and may be
infinite
Infinite may refer to:
Mathematics
* Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (group), a South Korean boy band
*''Infinite'' (EP), debut EP of American m ...
-
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
al. Every
affine vector field
An affine vector field (sometimes affine collineation or affine) is a projective vector field preserving geodesics and preserving the affine parameter. Mathematically, this is expressed by the following condition:
:(\mathcal_X g_)_=0
See also
...
is a curvature collineation.
See also
*
Conformal vector field
In conformal geometry, a conformal Killing vector field on a manifold of dimension ''n'' with (pseudo) Riemannian metric g (also called a conformal Killing vector, CKV, or conformal colineation), is a vector field X whose (locally defined) flo ...
*
Homothetic vector field In physics, a homothetic vector field (sometimes homothetic collineation or homothety) is a projective vector field which satisfies the condition:
:\mathcal_X g_=2c g_
where c is a real constant. Homothetic vector fields find application in the s ...
*
Killing vector field
*
Matter collineation
A matter collineation (sometimes matter symmetry and abbreviated to MC) is a vector field that satisfies the condition,
:\mathcal_X T_=0
where T_ are the energy–momentum tensor components. The intimate relation between geometry and physics ma ...
*
Spacetime symmetries
Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems. Spacetime symmetries are used in the study of exact ...
Mathematical methods in general relativity
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