Riemannian metric (changes) in nLab

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Definition

In terms of a 2-tensor

A Riemannian metric on a smooth manifold MM is defined as a covariant symmetric 2-tensor (.,.) p,p∈M(., .)_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 v∈T 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→ℝ(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 ℝ n\mathbb{R}^n, there is a standard Riemannian metric coming from the usual inner product. More generally, if g ij:ℝ n→ℝg_{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∈ℝ nx \in \mathbb{R}^n, we get a Riemannian metric ∑ i,jg ijdx i⊗dx j\sum_{i,j} g_{i j} d x^i \otimes d x^j on ℝ n\mathbb{R}^n, where the sum is to be interpreted as a covariant tensor.

2. Given an immersion N→MN \to M, a Riemannian metric on MM induces one on NN in the natural way, simply by pulling back. For instance, any surface in ℝ 3\mathbb{R}^3 has a Riemannian structure based upon the standard Riemannian structure on ℝ 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 v∈T 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:I→Mc: I \to M is a smooth curve for II an interval in ℝ\mathbb{R}, we define

(2)l(c):=∫ I‖c′(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,q∈Mp,q \in M, let

(3)d(p,q):=inf c∣c(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 ℝ n\mathbb{R}^n. Each tangent vector v∈T p(M)v \in T_p(M) can be viewed as an element of ℝ n\mathbb{R}^n in a natural way. Now let ‖⋅‖ ℝ n\left \Vert{\cdot}\right \Vert_{\mathbb{R}^n} be the standard norm on ℝ n\mathbb{R}^n. By continuity, we can find δ>0\delta \gt 0 by shrinking MM if necessary such that for all v∈T p(M),p∈Kv \in T_p(M), p \in K,

(4)δ‖v‖ ℝ n≤‖v‖ p≤δ −1‖v‖ 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 MM are necessarily comparable to the usual lengths in ℝ n\mathbb{R}^n. The result now follows.

• signature of a metric

• invariant Riemannian metric

• 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)