Levi-Civita connection
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{{Authors|Jost Eschenburg}}{{Stub}} | {{Authors|Jost Eschenburg}}{{Stub}} | ||
− | == Definition == | + | == Definition== |
<wikitex>; | <wikitex>; | ||
Let $M$ be a smooth manifold with tangent bundle $TM$. | Let $M$ be a smooth manifold with tangent bundle $TM$. | ||
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for every $u\in\Rr^n_o$. | for every $u\in\Rr^n_o$. | ||
− | A | + | A ''semi-Riemannian metric'' on a $M$ is a linear bundle map |
$g : S^2TM \to \Rr$, sometimes called ''metric tensor'', | $g : S^2TM \to \Rr$, sometimes called ''metric tensor'', | ||
which is nondegenerate, that is $\det g_{ij} \neq 0$ | which is nondegenerate, that is $\det g_{ij} \neq 0$ | ||
where $g_{ij} = g(\phi_i,\phi_j)$. If $g$ is positive definite, | where $g_{ij} = g(\phi_i,\phi_j)$. If $g$ is positive definite, | ||
$g(X,X) > 0$ for every nonzero vector field $X$ on $M$, it is | $g(X,X) > 0$ for every nonzero vector field $X$ on $M$, it is | ||
− | called a | + | called a ''Riemannian metric''. A (''semi-'') ''Riemannian manifold'' is |
a smooth manifold $M$ together with a (semi-)Riemannian metric $g$. | a smooth manifold $M$ together with a (semi-)Riemannian metric $g$. | ||
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$(g_{kl})$. | $(g_{kl})$. | ||
− | For further information, see \cite{Milnor1963}, \cite{Kobayashi | + | For further information, see \cite{Milnor1963}, \cite{Kobayashi&Nomizu1963} and \cite{O'Neill1983}. |
</wikitex> | </wikitex> | ||
== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
− | + | == External links == | |
+ | *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]] | ||
[[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
Tex syntax errorbe 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 aTex syntax erroris a linear bundle map
, sometimes called metric tensor, which is nondegenerate, that is where . If is positive definite,
for every nonzero vector field onTex syntax error, it is
called a Riemannian metric. A (semi-) Riemannian manifold is
a smooth manifoldTex syntax errortogether 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