Levi-Civita connection
(→References) |
m |
||
Line 52: | Line 52: | ||
For further information, see \cite{Milnor1963}, \cite{Kobayashi&Nomizu1963} and \cite{O'Neill1983}. | For further information, see \cite{Milnor1963}, \cite{Kobayashi&Nomizu1963} and \cite{O'Neill1983}. | ||
</wikitex> | </wikitex> | ||
− | |||
== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
− | |||
== External links == | == External links == | ||
*The Encyclopedia of Mathematics article on the [http://www.encyclopediaofmath.org/index.php/Levi-Civita_connection Levi-Civita connection] | *The Encyclopedia of Mathematics article on the [http://www.encyclopediaofmath.org/index.php/Levi-Civita_connection Levi-Civita connection] | ||
*The Wikipedia page on the [[Wikipedia:Levi-Civita connetion|Levi-Civita connection]] | *The Wikipedia page on the [[Wikipedia:Levi-Civita connetion|Levi-Civita connection]] | ||
[[Category:Definitions]] | [[Category:Definitions]] | ||
+ | [[Category:Connections and curvature]] |
Latest revision as of 11:21, 21 May 2013
The user responsible for this page is Jost Eschenburg. No other user may edit this page at present. |
This page has not been refereed. The information given here might be incomplete or provisional. |
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 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 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] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR1393940 (97c:53001a) Zbl 0508.53002
- [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
3 External links
- The Encyclopedia of Mathematics article on the Levi-Civita connection
- The Wikipedia page on the Levi-Civita connection