# Levi-Civita connection

## 1 Definition

Let $M$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M$ be a smooth manifold with tangent bundle $TM$$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:

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

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

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

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:

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

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].