nLab
Riemannian metric

Context

Riemannian geometry

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Definition

In terms of a 2-tensor

A Riemannian metric on a smooth manifold MM is defined as a covariant symmetric 2-tensor (.,.) p,pM(., .)_p, p \in M – a section of the symmetrized second tensor power of the tangent bundle – such that (v,v) p>0(v,v)_p \gt 0 for all vT p(M)v \in T_p(M). For convenience, we will write (v,w)(v,w) for (v,w) p(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)T_p(M) such that if X,YX,Y are (smooth) vector fields, the function (X,Y):M(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.

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

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

  3. Given an open covering U iU_i on MM, Riemannian metrics (,) i(\cdot, \cdot)_i on U iU_i, and a partition of unity ϕ i\phi_i subordinate to the covering U iU_i, we get a Riemannian metric on MM by

    (1)(v,w) p:= iϕ 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 vT p(M)v \in T_p(M), use the notation v:=(v,v)=(v,v) p\left \Vert{v} \right \Vert := (v,v) = (v,v)_p. If c:IMc: I \to M is a smooth curve for II an interval in \mathbb{R}, we define

(2)l(c):= Ic(t)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 MM into a metric space: for p,qMp,q \in M, let

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

The metric on MM induces the standard topology on MM. To see this, first note that it is a local question, so we can reduce to the case of MM an open ball in euclidean space n\mathbb{R}^n. Each tangent vector vT p(M)v \in T_p(M) can be viewed as an element of n\mathbb{R}^n in a natural way. Now let n\left \Vert{\cdot}\right \Vert_{\mathbb{R}^n} be the standard norm on n\mathbb{R}^n. By continuity, we can find δ>0\delta \gt 0 by shrinking MM if necessary such that for all vT p(M),pKv \in T_p(M), p \in K,

(4)δv nv pδ 1v 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 MM are necessarily comparable to the usual lengths in n\mathbb{R}^n. The result now follows.

References

An introduction in terms of synthetic differential geometry is in

Revised on August 16, 2011 00:11:20 by Urs Schreiber (194.81.173.201)