Levi-Civita connection
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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] 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