Levi-Civita connection

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 $M$${{Authors|Jost Eschenburg}}{{Stub}} == Definition== <wikitex>; Let $M$ be a smooth manifold with tangent bundle $TM$. Let $\phi : \Rr^n_o \to M$ be a local parametrization, defined on some open domain $\Rr^n_o\subset\Rr^n$, and let $\phi_i = \partial_i\phi$ be the partial derivatives; the vectors $\phi_i(u)$, $i=1,\dots,n$ form a basis of $T_{\phi(u)}M$ for every $u\in\Rr^n_o$. A ''semi-Riemannian metric'' on a $M$ is a linear bundle map $g : S^2TM \to \Rr$, sometimes called ''metric tensor'', which is nondegenerate, that is $\det g_{ij} \neq 0$ where $g_{ij} = g(\phi_i,\phi_j)$. If $g$ is positive definite, $g(X,X) > 0$ for every nonzero vector field $X$ on $M$, it is called a ''Riemannian metric''. A (''semi-'') ''Riemannian manifold'' is a smooth manifold $M$ together with a (semi-)Riemannian metric $g$. Every semi-Riemannian manifold $(M,g)$ carries a particular affine [[Connections#Connections_on_the_tangent_bundle|connection]], the ''Levi-Civita connection''. This is a [[Covariant derivative|covariant derivative]] $\nabla$ on the tangent bundle $TM$ with the following two properties: # it has no [[Torsion tensor|torsion]], $T = 0$, # the metric is parallel, $\nabla g = 0$. The second equation involves the covariant derivative of the metric tensor $g : S^2TM \to \Rr$ which is defined in such a way that applications of tensors to vector fields satisfy the Leibniz product rule: $$ \partial_X g(Y,Z) = (\nabla_Xg)(Y,Z) + g(\nabla_XY,Z) + g(Y,\nabla_XZ) $$ for all $X,Y,Z \in \Gamma TM$. Thus the rules (1) and (2) can be rephrased as # $\nabla_XY - \nabla_YX = [X,Y]$, # $\partial_Xg(Y,Z) = g(\nabla_XY,Z) + g(Y,\nabla_XZ)$. for all $X,Y,Z \in \Gamma TM$. If the semi-Riemannian metric $g$ is fixed, we often write $$\langle X,Y \rangle = g(X,Y).$$ The Levi-Civita connection is uniquely determined by properties (1) and (2) which imply {{equation|$\begin{matrix} 2\langle\nabla_XY,Z\rangle &=& X\langle Y, Z\rangle + Y\langle Z,X\rangle - Z\langle X,Y\rangle \ && -\langle X,[Y,Z]\rangle + \langle Y,[Z,X]\rangle + \langle Z,[X,Y]\rangle \end{matrix}$|1}} for all $X,Y,Z \in \Gamma TM$ (Koszul formula). If we specialize to the coordinate vector fields $\phi_i$, the Lie bracket terms vanish: $$ 2\langle\nabla_i\phi_j,\phi_k\rangle = \partial_i g_{jk} + \partial_jg_{ki} - \partial_kg_{ij}. $$ Denoting the coefficents of $\nabla_i\phi_j$, the so called ''Christoffel symbols'' by $\Gamma_{ij}^k$, {{equation|$\nabla_i\phi_j = \sum_l \Gamma_{ij}^l\phi_l\, ,$|2}} we obtain the Levi-Civita formula {{equation|$\Gamma_{ij}^l = {1\over2}\sum_{k}g^{kl}\left(\partial_ig_{jk} + \partial_jg_{ki}- \partial_kg_{ij}\right)$|3}} where $(g^{kl})$ denotes the inverse matrix of the metric coefficient matrix $(g_{kl})$. For further information, see \cite{Milnor1963}, \cite{Kobayashi&Nomizu1963} and \cite{O'Neill1983}. </wikitex> == References == {{#RefList:}} == External links == *The Encyclopedia of Mathematics article on the [http://www.encyclopediaofmath.org/index.php/Levi-Civita_connection Levi-Civita connection] *The Wikipedia page on the [[Wikipedia:Levi-Civita connetion|Levi-Civita connection]] [[Category:Definitions]] [[Category:Connections and curvature]]M$ be a smooth manifold with tangent bundle $TM$$TM$. Let $\phi : \Rr^n_o \to M$$\phi : \Rr^n_o \to M$ be a local parametrization, defined on some open domain $\Rr^n_o\subset\Rr^n$$\Rr^n_o\subset\Rr^n$, and let $\phi_i = \partial_i\phi$$\phi_i = \partial_i\phi$ be the partial derivatives; the vectors $\phi_i(u)$$\phi_i(u)$, $i=1,\dots,n$$i=1,\dots,n$ form a basis of $T_{\phi(u)}M$$T_{\phi(u)}M$ for every $u\in\Rr^n_o$$u\in\Rr^n_o$.

A semi-Riemannian metric on a $M$$M$ is a linear bundle map $g : S^2TM \to \Rr$$g : S^2TM \to \Rr$, sometimes called metric tensor, which is nondegenerate, that is $\det g_{ij} \neq 0$$\det g_{ij} \neq 0$ where $g_{ij} = g(\phi_i,\phi_j)$$g_{ij} = g(\phi_i,\phi_j)$. If $g$$g$ is positive definite, $g(X,X) > 0$$g(X,X) > 0$ for every nonzero vector field $X$$X$ on $M$$M$, it is called a Riemannian metric. A (semi-) Riemannian manifold is a smooth manifold $M$$M$ together with a (semi-)Riemannian metric $g$$g$.

Every semi-Riemannian manifold $(M,g)$$(M,g)$ carries a particular affine connection, the Levi-Civita connection. This is a covariant derivative $\nabla$$\nabla$ on the tangent bundle $TM$$TM$ with the following two properties:

1. it has no torsion, $T = 0$$T = 0$,
2. the metric is parallel, $\nabla g = 0$$\nabla g = 0$.

The second equation involves the covariant derivative of the metric tensor $g : S^2TM \to \Rr$$g : S^2TM \to \Rr$ which is defined in such a way that applications of tensors to vector fields satisfy the Leibniz product rule:

$\displaystyle \partial_X g(Y,Z) = (\nabla_Xg)(Y,Z) + g(\nabla_XY,Z) + g(Y,\nabla_XZ)$ $$\displaystyle \partial_X g(Y,Z) = (\nabla_Xg)(Y,Z) + g(\nabla_XY,Z) + g(Y,\nabla_XZ)$$

for all $X,Y,Z \in \Gamma TM$$X,Y,Z \in \Gamma TM$. Thus the rules (1) and (2) can be rephrased as

1. $\nabla_XY - \nabla_YX = [X,Y]$$\nabla_XY - \nabla_YX = [X,Y]$,
2. $\partial_Xg(Y,Z) = g(\nabla_XY,Z) + g(Y,\nabla_XZ)$$\partial_Xg(Y,Z) = g(\nabla_XY,Z) + g(Y,\nabla_XZ)$.

for all $X,Y,Z \in \Gamma TM$$X,Y,Z \in \Gamma TM$. If the semi-Riemannian metric $g$$g$ is fixed, we often write

$\displaystyle \langle X,Y \rangle = g(X,Y).$ $$\displaystyle \langle X,Y \rangle = g(X,Y).$$

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}$$\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}$(1)

for all $X,Y,Z \in \Gamma TM$$X,Y,Z \in \Gamma TM$ (Koszul formula). If we specialize to the coordinate vector fields $\phi_i$$\phi_i$, the Lie bracket terms vanish:

$\displaystyle 2\langle\nabla_i\phi_j,\phi_k\rangle = \partial_i g_{jk} + \partial_jg_{ki} - \partial_kg_{ij}.$ $$\displaystyle 2\langle\nabla_i\phi_j,\phi_k\rangle = \partial_i g_{jk} + \partial_jg_{ki} - \partial_kg_{ij}.$$

Denoting the coefficents of $\nabla_i\phi_j$$\nabla_i\phi_j$, the so called Christoffel symbols by $\Gamma_{ij}^k$$\Gamma_{ij}^k$,

$\nabla_i\phi_j = \sum_l \Gamma_{ij}^l\phi_l\, ,$$\nabla_i\phi_j = \sum_l \Gamma_{ij}^l\phi_l\, ,$(2)

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)$$\Gamma_{ij}^l = {1\over2}\sum_{k}g^{kl}\left(\partial_ig_{jk} + \partial_jg_{ki}- \partial_kg_{ij}\right)$(3)

where $(g^{kl})$$(g^{kl})$ denotes the inverse matrix of the metric coefficient matrix $(g_{kl})$$(g_{kl})$.

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