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:
![\displaystyle \partial_X g(Y,Z) = (\nabla_Xg)(Y,Z) + g(\nabla_XY,Z) + g(Y,\nabla_XZ)](/images/math/a/4/c/a4c9f639252271a9b85197d7e0001497.png)
for all . Thus the rules (1) and (2) can be rephrased as
-
,
-
.
for all . If the semi-Riemannian metric
is fixed, we often write
![\displaystyle \langle X,Y \rangle = g(X,Y).](/images/math/6/b/4/6b4989cd183662e4422ee23f73db0665.png)
The Levi-Civita connection is uniquely determined by properties (1) and (2) which imply
![\begin{matrix} 2\langle\nabla_XY,Z\rangle &=& X\langle Y, Z\rangle + Y\langle Z,X\rangle - Z\langle X,Y\rangle \\ && -\langle X,[Y,Z]\rangle + \langle Y,[Z,X]\rangle + \langle Z,[X,Y]\rangle \end{matrix}](/images/math/9/9/1/991fad38bc38324459e730b0090a0028.png)
for all (Koszul formula). If we specialize to the coordinate vector fields
,
the Lie bracket terms vanish:
![\displaystyle 2\langle\nabla_i\phi_j,\phi_k\rangle = \partial_i g_{jk} + \partial_jg_{ki} - \partial_kg_{ij}.](/images/math/a/a/8/aa80c5db96fe1b74fcf56d9656c53452.png)
Denoting the coefficents of , the so called Christoffel symbols by
,
![\nabla_i\phi_j = \sum_l \Gamma_{ij}^l\phi_l\, ,](/images/math/3/b/a/3ba6bbd4afe868f6ef497d3d3946fd9e.png)
we obtain the Levi-Civita formula
![\Gamma_{ij}^l = {1\over2}\sum_{k}g^{kl}\left(\partial_ig_{jk} + \partial_jg_{ki}- \partial_kg_{ij}\right)](/images/math/e/e/6/ee68d5ece74b4d270dddc766dd61859e.png)
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