Riemann curvature tensor
mathematicalfield of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor is the most standard way to express curvature of Riemannian manifolds. It is one of many things named after Bernhard Riemannand Elwin Bruno Christoffel. The curvature tensor is given in terms of a Levi-Civita connectionby the following formula:
Here is a linear transformation of the tangent space of the manifold; it is linear in each argument.
NB. Some authors define the curvature tensor with the opposite sign.
If and are coordinate vector fields then and therefore the formula simplifies to :i.e. the curvature tensor measures "noncommutativity of the covariant derivative".
The linear transformation is also called the curvature transformation or endomorphism.
The Riemann curvature tensor, especially in its coordinate expression (see below), is a central mathematical tool of
general relativity, the modern theory of gravity.
In local coordinates the Riemann curvature tensor is given by:where are the coordinate vector fields. The above expression can be written using
(see also the
list of formulas in Riemannian geometry).
The transformation of a vector after circling an infinitesimal rectangle is:.
Also define the purely covariant version by:
ymmetries and identities
The Riemann curvature tensor has the following symmetries::::
The last identity was discovered by Ricci, but is often called the first Bianchi identity or algebraic Bianchi identity, because it looks similar to the Bianchi identity below. These three identities form a complete list of symmetries of the curvature tensor, i.e. given any tensor which satisfies the identities above, one can find a Riemannian manifold with such a curvature tensor at some point. Simple calculations show that such a tensor has independent components.
Yet another useful identity follows from these three::
The Bianchi identity (often called the second Bianchi identity or differential Bianchi identity)involves the covariant derivative:
coordinate chartabout some point on the manifold, the above identities may be written in terms of the components of the Riemann tensor at this point as:
: (first Bianchi identity)
: (second Bianchi identity)
where the square brackets denote cyclic symmetrisation over the indices and the semi-colon is a covariant derivative.
For a two-dimensional
surface, the Bianchi identities imply that the Riemann tensor can be expressed as:where is the metric tensorand is a function called the Gaussian curvatureand "a", "b", "c" and "d" take values either "1" or "2". As expected we see that the Riemann curvature tensor only has one independent component.
The Gaussian curvature coincides with the
sectional curvatureof the surface. It is also exactly half the scalar curvatureof the 2-manifold, while the Ricci curvaturetensor of the surfaceis simply given by :
Curvature of Riemannian manifolds
Basic introduction to the mathematics of curved spacetime
Wikimedia Foundation. 2010.
Look at other dictionaries:
Curvature tensor — In differential geometry, the term curvature tensor may refer to: the Riemann curvature tensor of a Riemannian manifold see also Curvature of Riemannian manifolds; the curvature of an affine connection or covariant derivative (on tensors); the… … Wikipedia
Curvature of Riemannian manifolds — In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous… … Wikipedia
Tensor — For other uses, see Tensor (disambiguation). Note that in common usage, the term tensor is also used to refer to a tensor field. Stress, a second order tensor. The tensor s components, in a three dimensional Cartesian coordinate system, form the… … Wikipedia
Curvature — In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this … Wikipedia
Curvature form — In differential geometry, the curvature form describes curvature of a connection on a principal bundle. It can be considered as an alternative to or generalization of curvature tensor in Riemannian geometry. Contents 1 Definition 1.1 Curvature… … Wikipedia
Tensor-Diagramm-Notation — Die penrosesche graphische Notation – auch als penrosesche diagrammatische Notation, Tensor Diagramm Notation oder auch einfach Penrose Notation bezeichnet – ist eine von Roger Penrose vorgeschlagene Notation in der Physik und Mathematik, um eine … Deutsch Wikipedia
Tensor contraction — In multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the natural pairing of a finite dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components… … Wikipedia
Riemann tensor (general relativity) — The Riemann tensor (general relativity) is a mathematical object that describes gravitation and its effects in Einstein s theory of general relativity. Curvature and geodesic deviationThe Riemann tensor can be used to express the idea of… … Wikipedia
Tensor de curvatura — En geometría diferencial, el tensor de curvatura de Riemann , o simplemente tensor de curvatura o tensor de Riemann, supone una generalización del concepto de curvatura de Gauss, definido para superficies, a variedades de dimensiones arbitrarias … Wikipedia Español
Curvature invariant (general relativity) — Curvature invariants in general relativity are a set of scalars called curvature invariants that arise in general relativity. They are formed from the Riemann, Weyl and Ricci tensors which represent curvature and possibly operations on them such… … Wikipedia