nLab
Riemannian metric

Definition

In terms of a 2-tensor

A 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 $(v,v{)}_p>0$ 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 spaces $T_p(M)$ such that if $X,Y$ are (smooth) vector fields, the function $(X,Y):M\to ℝ$ defined by taking the inner product at each point, is smooth. A manifold together with a Riemannian metric is called a Riemannian manifold.

In terms of a Vielbein

for the moment see Poincare Lie algebra and first-order formulation of gravity

Examples

There are several ways to get Riemannian metrics:

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

2. 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 ${ℝ}^3$ has a Riemannian structure based upon the standard Riemannian structure on ${ℝ}^3$—based simply on the usual inner product—and induced on the surface.

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

   (1)$$(v,w{)}_p:=\sum _i{\varphi }_i(p)(v,w{)}_{i,p}.$$(v,w)_p := \sum_i \phi_i(p) (v,w)_{i,p}.

   Thus, using 1) above, any smooth manifold—which necessarily admits partitions of unity—can be given a Riemannian metric.

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 $\parallel v\parallel :=(v,v)=(v,v{)}_p$. If $c:I\to M$ is a smooth curve for $I$ an interval in $ℝ$, we define

(2)$$l(c):={\int }_I\parallel c\prime (t)\parallel dt;$$l(c) := \int_I \left \Vert{c'(t)}\right \Vert d t;

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

(3)$$d(p,q):=\underset{c\mid c(a)=p,c(b)=q}{\inf }l(c).$$d(p,q) := \inf_{c \mid c(a)=p,c(b)=q} l(c).

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 ${ℝ}^n$. Each tangent vector $v\in T_p(M)$ can be viewed as an element of ${ℝ}^n$ in a natural way. Now let ${\parallel \cdot \parallel }_{{ℝ}^n}$ be the standard norm on ${ℝ}^n$. By continuity, we can find $\delta >0$ by shrinking $M$ if necessary such that for all $v\in T_p(M),p\in K$,

(4)$$\delta {\parallel v\parallel }_{{ℝ}^n}\le {\parallel v\parallel }_p\le {\delta }^{-1}{\parallel v\parallel }_{{ℝ}^n};$$\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} ;

in particular, the lengths of curves in $M$ are necessarily comparable to the usual lengths in ${ℝ}^n$. The result now follows.

Related concepts

• orthogonal structure

• Riemannian manifold, pseudo-Riemannian manifold

• Levi-Civita connection

• moduli space of Riemannian metrics

References

An introduction in terms of synthetic differential geometry is in

• Gonzalo Reyes, Metrics, connections and curvature (pdf)