24. Riemannian Manifolds

In this notebook, we recap some basic definitions of Riemannian manifolds, especially the metric tensor, Christoffel symbols, and several curvature quantities.

Let \((M,g)\) be an \(N\)-dimensional Riemannian manifold with metric tensor \(g\). That means \(M\) is a smooth manifold, and \(g\) is a symmetric and positive definite bilinear form on the tangent space of \(M\). If \(g\) has signature \(-+\dots+\), then \((M,g)\) is called a pseudo-Riemannian manifold. For instance, in general relativity, the metric has the signature \(-+++\), where the associated negative coordinate corresponds to time. For numerical experiments and simulations we will approximate the exact metric by Regge finite elements.

For simplicity we assume that \(M\) can be covered by a single chart, inducing local coordinates \(x^1,\dots, x^N\) with basis vectors \(\partial_1,\dots,\partial_N\) (a frame of vector fields \(\partial_i\in \mathfrak{X}(M)\) ), and their corresponding dual basis of 1-forms \(dx^1,\dots, dx^N\), \(dx^i\in \Lambda^1(M)\). That means \(dx^i(\partial_j)=\delta^i_{j}\). Then, we can write the metric \(g\) and its inverse in terms of the coordinates via

\[\begin{align*} g_{ij} = g(\partial_i,\partial_j),\qquad g^{ij}= g^{-1}(dx^i,dx^j),\qquad g= g_{ij}dx^i\otimes dx^j,\qquad g^{-1} = g^{ij}\,\partial_i\otimes \partial_j. \end{align*}\]

Note that both \(g_{ij}\) and \(g^{ij}\) are symmetric. We utilized Einstein’s summation convention of repeated indices. That means if an index appears as a subscript and superscript, we implicitly sum over it. The volume form \(\omega\in \Lambda^N(M)\) used for integration on the manifold can be expressed in coordinates as

\[\begin{align*} \omega = \sqrt{\det (g_{ij})}\,dx^1\wedge\dots\wedge dx^N, \end{align*}\]

where \(\wedge:\Lambda^i(M)\times\Lambda^j(M)\to\Lambda^{i+j}(M)\) denotes the wedge-product of two forms.

The Levi-Civita connection \(\nabla\) is the unique connection that is metric-compatible (Leibnitz rule) and torsion-free, i.e. for all \(X,Y,Z\in\mathfrak{X}(M)\)

\[\begin{align*} & X(g(Y,Z)) = g(\nabla_XY,Z)+ g(Y,\nabla_XZ)\\ & \nabla_XY -\nabla_YX = [X,Y], \end{align*}\]

where \([X,Y]\in \mathfrak{X}(M)\) denotes the commutator (Lie-bracket) defined by

\[\begin{align*} [X,Y]f = X(Y(f))-Y(X(f)),\qquad \forall f\in C^{\infty}(M). \end{align*}\]

The Christoffel symbols of the first and second kinds are defined via

\[\begin{align*} \Gamma_{ijk}= g(\nabla_{\partial_i}\partial_j,\partial_k),\qquad \nabla_{\partial_i}\partial_j=\Gamma_{ij}^k\partial_k \end{align*}\]

and in terms of the metric tensor

\[\begin{align*} \Gamma_{ijk}=\frac{1}{2}\left(\partial_ig_{jk}+\partial_jg_{ik}-\partial_kg_{ij}\right),\qquad \Gamma_{ij}^k = g^{kl}\Gamma_{ijl}. \end{align*}\]

The full fourth-order Riemann curvature tensor, encoding the complete curvature information of the Riemannian manifold \(M\), is given by

\[\begin{align*} \mathfrak{R}(X,Y,Z,W) = g(\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z,W),\qquad \mathfrak{R}_{ijkl} = \partial_i \Gamma_{jkl}-\partial_j \Gamma_{ikl} + \Gamma_{ik}^p\Gamma_{jlp}- \Gamma_{jk}^p\Gamma_{ilp}. \end{align*}\]

The Riemann curvature tensor is skew-symmetric in its first two and second two components and symmetric when switching the first two with the second two indices. Further, there holds the first (algebraic) and second (differential) Bianchi identity

\[\begin{align*} &\mathfrak{R}(X,Y,Z,W)=-\mathfrak{R}(Y,X,Z,W)=-\mathfrak{R}(X,Y,W,Z)= \mathfrak{R}(Z,W,X,Y),\\ &\mathfrak{R}(X,Y,Z,W)+\mathfrak{R}(X,Z,W,Y)+\mathfrak{R}(X,W,Y,Z)=0,\\ &\nabla_V\mathfrak{R}(X,Y,Z,W)+\nabla_Z\mathfrak{R}(X,Y,W,V)+\nabla_W\mathfrak{R}(X,Y,V,Z)=0. \end{align*}\]

Note that there exist two different sign conventions for the Riemann curvature tensor! The Ricci curvature tensor coincides in both conventions and reads (in our convention)

\[\begin{align*} \mathrm{Ric}_{ij} = \mathfrak{R}_{kijl}g^{kl}, \end{align*}\]

i.e., the first and last indices are contracted (in the other convention, the first and third indices get contracted). A second contraction gives the scalar curvature

\[\begin{align*} S = \mathrm{Ric}_{ij} g^{ij} = \mathfrak{R}_{kijl}g^{kl}g^{ij}. \end{align*}\]

In two dimensions, the Gauss curvature is given for any linear independent \(X,Y\) by

\[\begin{align*} K = \frac{\mathcal{R}(X,Y,Y,X)}{g(X,X)g(Y,Y)-g(X,Y)^2}, \end{align*}\]

which is independent of the choice of \(X,Y\). Further, it is related to the scalar curvature by \(S=2K\).

The notation of Gauss curvature can be extended to arbitrary dimensions considering a plane spanned by two independent vectors \(X,Y\). The result is denoted as the sectional curvature \(K(X,Y)\).

The Einstein tensor, which is zero in two dimensions but in general non-zero for dimensions greater than 2, is defined by

\[\begin{align*} G_{ij}=\mathrm{Ric}_{ij}-\frac{1}{2}S\,g_{ij}. \end{align*}\]

On a codimension 1 manifold \(F\) (also called hyper-surface), the second fundamental form \(I\!I\) acting on vector fields on the hyper-surface is defined by

\[\begin{align*} I\!I^n(X,Y)= g(n,\nabla_XY)=-g(\nabla_Xn,Y),\qquad X,Y\in \mathfrak{X}(F), \end{align*}\]

where \(n\) denotes a \(g\)-normalized normal vector on \(F\). Again, the sign may depend on the convention (changing the normal vector’s direction changes the sign). The second fundamental form is symmetric. In two dimensions, the second fundamental form reduces to the geodesic curvature of a curve

\[\begin{align*} \kappa^n_g = g(\nabla_tt,n)=-g(\nabla_tn,t), \end{align*}\]

where \(t\) is a \(g\)-normalized tangent vector.