An affine vector field (sometimes affine collineation or affine) is a
projective vector field A projective vector field (projective) is a smooth vector field on a semi Riemannian manifold (p.ex. spacetime) M whose flow preserves the geodesic structure of M without necessarily preserving the affine parameter of any geodesic. More intuitivel ...
preserving
geodesics and preserving the
affine parameter
In geometry, a geodesic () is a curve representing in some sense the 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 connection ...
. Mathematically, this is expressed by the following condition:
:
See also
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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 ...
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Curvature collineation
A curvature collineation (often abbreviated to CC) is vector field which preserves the Riemann tensor in the sense that,
:\mathcal_X R^a_=0
where R^a_ are the components of the Riemann tensor. The set of all smooth curvature collineations form ...
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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 ...
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Killing vector field
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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 ...
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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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