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\begin{document}

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\section*{Riemannian metric}

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_terms_of_a_2tensor}{In terms of a 2-tensor}\dotfill \pageref*{in_terms_of_a_2tensor} \linebreak
\noindent\hyperlink{in_terms_of_a_vielbein}{In terms of a Vielbein}\dotfill \pageref*{in_terms_of_a_vielbein} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{lengths_of_curves}{Lengths of Curves}\dotfill \pageref*{lengths_of_curves} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\hypertarget{in_terms_of_a_2tensor}{}\subsubsection*{{In terms of a 2-tensor}}\label{in_terms_of_a_2tensor}

A \textbf{Riemannian metric} on a [[smooth manifold]] $M$ is defined as a covariant symmetric 2-[[tensor]] $(., .)_p, p \in M$ -- a section of the symmetrized second tensor power of the [[tangent bundle]] -- such that $(v,v)_p \gt 0$ for all $v \in T_p(M)$. For convenience, we will write $(v,w)$ for $(v,w)_p$. In other words, a Riemannian metric is a collection of (positive) inner products on each of the [[tangent space]]s $T_p(M)$ such that if $X,Y$ are (smooth) [[vector field]]s, the function $(X,Y): M \to \mathbb{R}$ defined by taking the inner product at each point, is smooth. A [[manifold]] together with a Riemannian metric is called a [[Riemannian manifold]].

\hypertarget{in_terms_of_a_vielbein}{}\subsubsection*{{In terms of a Vielbein}}\label{in_terms_of_a_vielbein}

\begin{quote}%
for the moment see [[Poincare Lie algebra]] and [[first-order formulation of gravity]]


\end{quote}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

There are several ways to get Riemannian metrics:

\begin{enumerate}%
\item On $\mathbb{R}^n$, there is a standard Riemannian metric coming from the usual inner product. More generally, if $g_{i j}: \mathbb{R}^n \to \mathbb{R}$ are smooth functions such that the matrix $(g_{i j}(x))$ is symmetric and positive definite for all $x \in \mathbb{R}^n$, we get a Riemannian metric $\sum_{i,j} g_{i j} d x^i \otimes d x^j$ on $\mathbb{R}^n$, where the sum is to be interpreted as a covariant tensor.


\item Given an immersion $N \to M$, a Riemannian metric on $M$ induces one on $N$ in the natural way, simply by pulling back. For instance, any surface in $\mathbb{R}^3$ has a Riemannian structure based upon the standard Riemannian structure on $\mathbb{R}^3$---based simply on the usual inner product---and induced on the surface.


\item Given an open covering $U_i$ on $M$, Riemannian metrics $(\cdot, \cdot)_i$ on $U_i$, and a partition of unity $\phi_i$ subordinate to the covering $U_i$, we get a Riemannian metric on $M$ by

\begin{displaymath}
(v,w)_p := \sum_i \phi_i(p) (v,w)_{i,p}.
\end{displaymath}
Thus, using 1) above, any smooth manifold---which necessarily admits partitions of unity---can be given a Riemannian metric.



\end{enumerate}
\hypertarget{lengths_of_curves}{}\subsection*{{Lengths of Curves}}\label{lengths_of_curves}

A Riemannian metric allows us to take the length of a curve in a manner resembling the standard case. Given $v \in T_p(M)$, use the notation $\left \Vert{v} \right \Vert := (v,v) = (v,v)_p$. If $c: I \to M$ is a smooth curve for $I$ an interval in $\mathbb{R}$, we define

\begin{displaymath}
l(c) := \int_I \left \Vert{c'(t)}\right \Vert d t;
\end{displaymath}
this is easily checked to be independent of parametrization, just as in the usual case. Using this, we can make a Riemannian manifold $M$ into a metric space: for $p,q \in M$, let

\begin{displaymath}
d(p,q) := \inf_{c \mid c(a)=p,c(b)=q} l(c).
\end{displaymath}
The metric on $M$ induces the standard [[topology]] on $M$. To see this, first note that it is a local question, so we can reduce to the case of $M$ an open ball in euclidean space $\mathbb{R}^n$. Each tangent vector $v \in T_p(M)$ can be viewed as an element of $\mathbb{R}^n$ in a natural way. Now let $\left \Vert{\cdot}\right \Vert_{\mathbb{R}^n}$ be the standard norm on $\mathbb{R}^n$. By continuity, we can find $\delta \gt 0$ by shrinking $M$ if necessary such that for all $v \in T_p(M), p \in K$,

\begin{displaymath}
\delta \left \Vert{v}\right \Vert_{\mathbb{R}^n} \leq \left \Vert{v}\right \Vert_p \leq \delta^{-1}   \left \Vert{v}\right \Vert_{\mathbb{R}^n} ;
\end{displaymath}
in particular, the lengths of curves in $M$ are necessarily comparable to the usual lengths in $\mathbb{R}^n$. The result now follows.

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

\begin{itemize}%
\item [[orthogonal structure]]


\item [[Riemannian manifold]], [[pseudo-Riemannian manifold]]


\item [[Levi-Civita connection]]


\item [[moduli space of Riemannian metrics]]



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

An introduction in terms of [[synthetic differential geometry]] is in

\begin{itemize}%
\item [[Gonzalo Reyes]], \emph{Metrics, connections and curvature} (\href{http://po-start.com/reyes/wp-content/uploads/2007/01/metrics.pdf}{pdf})

\end{itemize}
[[!redirects Riemannian metrics]]

[[!redirects metric tensor]] [[!redirects metric tensors]]



\end{document}