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Symmetric spaces and Lie groups

Table of Contents

The PDF version of this page can be downloaded by replacing html in the its address by pdf. For example /html/sheaf-cohomology.html should become /pdf/sheaf-cohomology.pdf.

This post is a part of the memoire of my M1 internship at I2M. The memoire contains, needless to say, less errors than this page.

1 Symmetric space

By de Rham decomposition, we now focus more on the building blocks: Riemannian manifolds with irreducible holonomy. The theory of Lie groups allows us to understand a block if it is symmetric.

A Riemannian manifold \(M\) is called symmetric if for every \(x\in M\), there exists an isometry \(s_x\) of \(M\) such that \(x\) is an isolated fixed point and \(s_x^2=Id\).

Let \(x\in M\) and \(v\in T_xM\), we note by \(\exp_x(v)\) the point of distance \(|v|\) in the geodesic starting in \(x\) with velocity \(v/|v|\). We remark that any isometry \(s_x\) with \(s_x^2=Id\) and \(x\) as isolated fixed point satisfies

\begin{equation} \label{eq:sxreverse} s_x(\exp_x(v)) = \exp_x(-v) \end{equation}

In fact the eigenvalues of \(T_xs_x\) have to be \(1\) or \(-1\), but as \(x\) is an isolated fixed point one has \(T_xs_x = -Id\). Then \(s_x\) as an isometry sends the geodesic starting at \(x\) with velocity \(v\) to one starting at \(s_x(x)=x\) with velocity \((s_x)_* v = -v\) and we have \eqref{eq:sxreverse}.

Equation \eqref{eq:sxreverse} tells us that \(s_x\) is a reflection of center \(x\) on every geodesic passing by \(x\). We can compose two reflections \(s_x,s_y\) to form a translation on the geodesic connecting \(x\) and \(y\). This shows that a symmetric space is complete and the group of isometries of the form \(s_x\circ s_y\) acts transitively on \(M\).

Let \(M\) be a symmetric Riemannian manifold then

  1. \(M\) is complete.
  2. Fix \(x_0\in M\), let \(G\) be the group generated by the isometries of form \(s_x\circ s_y,\ x,y\in M\) and \(H\) is the subgroup containing elements of \(G\) that fix \(x_0\), then \(G\) is Lie subgroup of \(Isom(M)\) connected by arc, \(H\) is a closed Lie subgroup of \(G\) and \(M\) is isometric to \(G/H\). Moreover the holonomy group of \(M\) is \(H\).

In general, for a Lie group \(G\) and a closed Lie subgroup \(H\), if \(G\) has a metric left-invariant by \(G\) and right-invariant by \(H\) (i.e. the metric on \(\frak{g}\) is invariant by action of \(H\) by adjoint) then \[ \frak{g} = \frak{h} \oplus^\perp \frak{m},\quad [\frak{h},\frak{m}]\subset \frak{m} \] But if \(G/H\) is symmetric then one has the following extra information \[ [\frak{m},\frak{m}]\subset \frak{h} \] It turns out that this condition is quite strong and allowed E. Cartan to classify all such pairs \((\frak{g},\frak{h})\).

2 Locally symmetric space

The previous results can be extended to locally symmetric spaces.

Let \(M\) is a Riemannian manifold, the followings are equivalent

  1. For every \(x\in M\), there exists a neighborhood \(U\) of \(x\) and an isometry \(s_x:\ U\longrightarrow U\) such that \(s_x^2=Id\) and \(x\) is the unique fixed point of \(s_x\).
  2. The curvature tensor \(R\) satisfies

\[ \nabla R = 0 \] If they are satisfied, \(M\) is called locally symmetric.

Let \(M\) be a locally symmetric Riemannian manifold, then there exists a unique symmetric simply connected Riemannian manifold \(N\) such that \(M\) and \(N\) are locally isometric, i.e. for every \(x\in M\) and \(y\in N\), there exists neighborhoods \(U\) of \(x\) and \(V\) of \(y\) that are isometric.

As a result, the reduced holonomy of \(M\) is the same as the holonomy of \(N\).

3 Annex: Group of isometries as Lie group

We explain in this annex some subtle details: how can a group of isometries be a manifold. We state, with Montgomery-Zippin, Transformation groups as reference, the following general result:

Let \(G\) be a group acting faithfully on a connected manifold \(M\) of class \(C^k\) such that each action is \(C^1\) and \(G\) is locally compact. Then \(G\) is a Lie group and the map \(G\times M\longrightarrow M\) is \(C^1\).

Note that we equip a group of isometries with the compact-open topology, as \(M\) is locally compact and therefore second-countable (i.e. the topology admits a countable base), we see that a group of isometries is also second-countable. It suffices to prove the local compactness for the group of (all) isometries as this property is inherited by its closed subgroup. The detail can be found in Kobayashi-Nomizu's Foundations of differential geometry (Volume I, Theorem 4.7).

Let \(M\) be a connected, locally-compact metric space and \(G\) be the group of isometries of \(M\), then

  1. \(G\) is locally compact.
  2. \(G_a\) the subset of isometries fixing a point \(a\in M\) is compact.
  3. If, in addition, \(M\) is compact then \(G\) is also compact.
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Last updated 2017-08-31 Thu 02:25.