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\section*{Levi-Civita connection}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{riemannian_geometry}{}\paragraph*{{Riemannian geometry}}\label{riemannian_geometry}

[[!include Riemannian geometry - contents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_terms_of_christoffel_symbols}{In terms of Christoffel symbols}\dotfill \pageref*{in_terms_of_christoffel_symbols} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{in_physics}{In physics}\dotfill \pageref*{in_physics} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \textbf{Levi-Civita connection} is the unique symmetric [[connection on a bundle|connection]] on the [[tangent bundle]] of a [[Riemannian manifold]] or [[pseudo-Riemannian manifold]] that is compatible with the [[Riemannian metric|metric]] or [[pseudo-Riemannian metric|pseudo-metric]]. The [[curvature]] and [[geodesic]]s on a pseudo-Riemannian manifold are taken with respect to this connection. The existence and uniqueness of the Levi-Civita connection is called the \emph{[[fundamental theorem of Riemannian geometry]]}.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{definition_2}{}\paragraph*{{Definition}}\label{definition_2}

For $(X,g)$ a [[Riemannian manifold]], the \textbf{Levi-Civita connection} $\nabla_g$ on $X$ is the unique [[connection on a bundle|connection]] on the [[tangent bundle]] $T X$ that

\begin{enumerate}%
\item the [[covariant derivative]] of the metric vanishes, $\nabla_{g} g = 0$;


\item $\nabla_g$ has vanishing [[torsion of a metric connection|torsion]].



\end{enumerate}
We say in detail what this means in ``[[first order formulation of gravity|first order formalism]]''/[[Cartan geometry]].

\begin{defn}
\label{}\hypertarget{}{}
For $(X,g)$ a $(d-k,k)$-[[dimension]]al [[Riemannian manifold]] (for $k = 0$) or [[pseudo-Riemannian manifold]] (for $k = 1$), the \textbf{Levi-Civita connection} $\nabla_g$ on $X$ is the unique $ISO(d-k,k)$-[[connection on a bundle]] $\nabla$ (for $ISO(d-k)$ the [[Poincare group]]) such that

\begin{enumerate}%
\item \textbf{metric compatibility}: the [[vielbein]] $e$ gives the metric: $g = e \cdot e$;


\item \textbf{torsion freeness}: the curvature of the vielbein (the [[torsion]]) vanishes: $T = 0$.



\end{enumerate}
\end{defn}
More in detail, locally on a patch $U_i \subset X$ the $ISO(d-k,k)$-connection $\nabla$ is given by a [[Poincare Lie algebra]]-[[Lie algebra valued 1-form|valued 1-form]]

\begin{displaymath}
(e, \omega) : T U_i \to \mathfrak{iso}(d-k,k) \simeq \mathbb{R}^d \ltimes \mathfrak{so}(d-k,k)
\end{displaymath}
with

\begin{enumerate}%
\item $e_i$ an $\mathbb{R}^d$-valued form -- the [[vielbein]];


\item $\omega_i$ a [[special orthogonal Lie algebra]]-valued form -- the ``[[spin connection]]''.



\end{enumerate}
The [[curvature]] 2-form of this similarly decomposes into

\begin{enumerate}%
\item the [[torsion]] $T^a := F_{e}^a = d e^a + \omega^a{}_b \wedge e^b$,

(this equation is also called the \textbf{first Cartan structure equation})


\item the [[Riemann curvature]] $R_{g}{}^a{}_b := F_{\omega}^a{}_b = d \omega^a{}_b + \omega^a{}_c \wedge \omega^c{}_d$.

(this equation is also called the \textbf{second Cartan structure equation})



\end{enumerate}
The [[Bianchi identity]] satisfied by this curvature is

\begin{enumerate}%
\item $d F_e^a + \omega^a{}_b \wedge F_e^b = F_\omega^a{}_b \wedge e^b$;


\item $d F_\omega^a{}_b + \omega^a{}_c \wedge F_\omega^c{}_b - F_\omega^a{}_c \wedge \omega^c{}_b = 0$.



\end{enumerate}
The metric compatibility condition in the definition of Levi-Civita connection says that

\begin{displaymath}
g = e^a \otimes e_a
  \,.
\end{displaymath}
The torsion-freeness condition says that

\begin{displaymath}
F_e = 0
  \,.
\end{displaymath}
\hypertarget{in_terms_of_christoffel_symbols}{}\subsection*{{In terms of Christoffel symbols}}\label{in_terms_of_christoffel_symbols}

The Levi-Civita connection may be discussed in terms of its components -- called [[Christoffel symbol]]s -- given by the canonical local trivialization of the [[tangent bundle]] over a coordinate patch. This has been the historical route and is still widely used in the literature.

\textbf{Metric compatibility}

Here a metric $g$ is \textbf{compatible} with the connection $\nabla$ or \emph{preserved} by it (here thought of in its incarnation as a [[covariant derivative]]) if and only if $\nabla_X g = 0$ for all $X$, which is equivalent to the preservation of the metric inner product of tangent vectors under [[parallel transport|parallel translation]]. Since

\begin{displaymath}
X(g(X_1,X_2))  = (\nabla_X g)(X_1,X_2) + g(\nabla_X X_1, X_2) + g(X_1, \nabla_X X_2) ,
\end{displaymath}
by the fact that covariant differentiation commutes with contractions and satisfies the derviative identity, compatibility is equivalent to

\begin{displaymath}
X (g (X_1,X_2)) = g(\nabla_X X_1, X_2) + g(X_1, \nabla_X X_2),
\end{displaymath}
for all $X,X_1, X_2$.

\textbf{Uniqueness and existence on $\mathbb{R}^n$}

Now assume $M \subset \mathbb{R}^n$ and we have such a connection associated to $g$.\newline Then the connection is uniquely determined by its [[Christoffel symbols]], which we can determine in terms of $g$ by a bit of elementary algebra. In other words, we just need to compute $\nabla_{\partial_i} \partial_j$. Now

\begin{displaymath}
\partial_k g( \partial_i, \partial_j) = g( \nabla_{\partial_k} \partial_i, \partial_j) + g(  \partial_i, \nabla_{\partial_k}\partial_j).
\end{displaymath}
We can get two other equations by cyclic permutation:

\begin{displaymath}
\partial_i g( \partial_j, \partial_k) = g( \nabla_{\partial_i} \partial_j, \partial_k) + g(  \partial_j, \nabla_{\partial_i}\partial_k)
\end{displaymath}
\begin{displaymath}
\partial_j g( \partial_k, \partial_i) = g( \nabla_{\partial_j} \partial_k, \partial_i) + g(  \partial_k, \nabla_{\partial_j}\partial_i)
\end{displaymath}
So let $S_{i j} := \nabla_{\partial_i} \partial_j = \nabla_{\partial_j} \partial_i$, by symmetry. Let $T_{i j k} := \partial_i g( \partial_j, \partial_k)$; these are smooth real functions. These equations can be written

\begin{displaymath}
T_{k i j} = g( S_{i k}, \partial_j) + g( S_{j k}, \partial_i)
\end{displaymath}
\begin{displaymath}
T_{i j k} = g( S_{i j}, \partial_k) + g( S_{i k}, \partial_j)
\end{displaymath}
\begin{displaymath}
T_{j k i} = g( S_{j k}, \partial_i) + g( S_{i j}, \partial_k)
\end{displaymath}
These are three linear equations in the unknowns $g( S_{i k}, \partial_j),   g( S_{j k}, \partial_i), g( S_{i j}, \partial_k)$. The system is nonsingular, so we get a unique solution, and consequently by nondegeneracy a unique possibility for the $S_{i j}$.

Incidentally, we have in fact shown the uniqueness assertion of the general theorem, since that is local.

We shall now prove existence in this restricted case. Choose $S_{i j}$ to satisfy the system of three equations outlined above where $i \lt j \lt k$. Then set $S_{j i} := S_{i j}$, and we have a connection $\nabla$ with $\nabla_{\partial_i} \partial_j := S_{i j}$ since the vector fields $\partial_i$ are a frame (i.e. a basis at each tangent space on $M$). It is symmetric, since the torsion $T$ vanishes (by $S_{i j}=S_{j i}$) on pairs $(\partial_i,\partial_j)$, and hence identically, since it is a tensor.

We must check for compatibility. The difference of the two terms in (1) vanishes when $X,X_1,X_2$ are of the form $\partial_i$. The vanishing holds generally because the difference of the two sides, which is $(\nabla_X g)(X_1,X_2)$, is a tensor. Hence compatibility follows.

\textbf{Uniqueness and existence in the general case}

We have already shown the uniqueness assertion, since that is local. Connections restrict to connections on open subsets.

We have proved the existence of $\nabla$ when $M$ is an open submanifold of $\mathbb{R}^n$ (though not necessarily with the canonical metric $\sum_{i=1}^n d x_i \otimes d x_i$). In general, cover $M$ by open subsets $U_i$ diffeomorphic to an open set in $\mathbb{R}^n$. We get connections $\nabla_i$ on $U_i$ compatible with $g|_{U_i}$.

We claim that $\nabla_i|_{U_i \cap U_j} = \nabla_j|_{U_i \cap U_j}$. This is an easy corollary of uniquness. So we can patch the connections together to get the one Levi-Civita connection on $M$.

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\hypertarget{in_physics}{}\subsubsection*{{In physics}}\label{in_physics}

In the [[physics]], the theory of [[general relativity]] models the field of [[gravity]] in terms of the Levi-Civita connection on a [[Lorentzian manifold]]. See there for more details.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Weitzenböck connection]]


\item [[connection on a bundle]]

\begin{itemize}%
\item [[parallel transport]], [[holonomy]]

\end{itemize}

\item [[principal connection]]

\begin{itemize}%
\item [[affine connection]], \textbf{Levi-Civita connection}, [[Cartan connection]]

\end{itemize}

\item [[connection on a 2-bundle]]


\item [[connection on an infinity-bundle]]

\begin{itemize}%
\item [[higher parallel transport]]

\end{itemize}


\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

A discussion in terms of [[synthetic differential geometry]] is in

\begin{itemize}%
\item [[Gonzalo Reyes]], \emph{General Relativity:}

\emph{Metrics, connections and curvature} (\href{http://po-start.com/reyes/wp-content/uploads/2007/01/metrics.pdf}{pdf})

\emph{The Riemann-Christoffel tensor} (\href{http://po-start.com/reyes/wp-content/uploads/2007/01/the-riemann-christoffel-tensor.pdf}{pdf})

\emph{Affine connections, parallel transport and sprays} (\href{http://po-start.com/reyes/wp-content/uploads/2009/01/affineconnections.pdf}{pdf})



\end{itemize}
[[!redirects Levi-Civita connections]] [[!redirects metric connecitons]]



\end{document}