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




  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • 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

            (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

          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

          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,

          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

          Last revised on May 23, 2015 at 06:48:02. See the history of this page for a list of all contributions to it.