Levi-Civita connection
(Created page with "{{Authors|Jost Eschenburg}}{{Stub}} == Definition == <wikitex>; Let $M$ be a smooth manifold with tangent bundle $TM$. Let $\phi : \Rr^n_o \to M$ be a local parametrization, d...") |
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a smooth manifold $M$ together with a (semi-)Riemannian metric $g$. | a smooth manifold $M$ together with a (semi-)Riemannian metric $g$. | ||
− | Every semi-Riemannian manifold $(M,g)$ carries a particular affine connection, | + | Every semi-Riemannian manifold $(M,g)$ carries a particular affine |
− | the ''Levi-Civita connection''. This is a covariant derivative | + | [[Connections#Connections_on_the_tangent_bundle|connection]], |
+ | the ''Levi-Civita connection''. This is a [[Covariant dertivative|covariant derivative]] | ||
$\nabla$ on the tangent bundle $TM$ with the following two properties: | $\nabla$ on the tangent bundle $TM$ with the following two properties: | ||
− | # it has no torsion, $T = 0$, | + | # it has no [[Torsion tensor|torsion]], $T = 0$, |
# the metric is parallel, $\nabla g = 0$. | # the metric is parallel, $\nabla g = 0$. | ||
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where $(g^{kl})$ denotes the inverse matrix of the metric coefficient matrix | where $(g^{kl})$ denotes the inverse matrix of the metric coefficient matrix | ||
$(g_{kl})$. | $(g_{kl})$. | ||
+ | |||
+ | For further information, see \cite{Milnor1963}, \cite{Kobayashi-Nomizu1963} and \cite{O'Neill1983} | ||
</wikitex> | </wikitex> | ||
== References == | == References == |
Revision as of 16:47, 15 March 2013
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1 Definition
Let be a smooth manifold with tangent bundle . Let be a local parametrization, defined on some open domain , and let be the partial derivatives; the vectors , form a basis of for every .
A {\it semi-Riemannian metric} on a is a linear bundle map , sometimes called metric tensor, which is nondegenerate, that is where . If is positive definite, for every nonzero vector field on , it is called a {\it Riemannian metric}. A (semi-) Riemannian manifold is a smooth manifold together with a (semi-)Riemannian metric .
Every semi-Riemannian manifold carries a particular affine connection, the Levi-Civita connection. This is a covariant derivative on the tangent bundle with the following two properties:
- it has no torsion, ,
- the metric is parallel, .
The second equation involves the covariant derivative of the metric tensor which is defined in such a way that applications of tensors to vector fields satisfy the Leibniz product rule:
for all . Thus the rules (1) and (2) can be rephrased as
- .
for all . If the semi-Riemannian metric is fixed, we often write
The Levi-Civita connection is uniquely determined by properties (1) and (2) which imply
for all (Koszul formula). If we specialize to the coordinate vector fields , the Lie bracket terms vanish:
Denoting the coefficents of , the so called Christoffel symbols by ,
we obtain the Levi-Civita formula
where denotes the inverse matrix of the metric coefficient matrix .
For further information, see [Milnor1963], [Kobayashi-Nomizu1963] and [O'Neill1983]
2 References
- [Kobayashi-Nomizu1963] Template:Kobayashi-Nomizu1963
- [Milnor1963] J. Milnor, Morse theory, Princeton University Press, 1963. MR0163331 (29 #634) Zbl 0108.10401
- [O'Neill1983] B. O'Neill, Semi-Riemannian geometry, Academic Press Inc., 1983. MR719023 (85f:53002) Zbl 0531.53051