Levi-Civita connection
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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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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> | ||
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== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
[[Category:Definitions]] | [[Category:Definitions]] |
Revision as of 16:49, 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 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] 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