Riemannian metric


Riemannian geometry

Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



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


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.


An introduction in terms of synthetic differential geometry is in

Revised on May 23, 2015 06:48:02 by Urs Schreiber (