# Closed Semi-Riemannian manifolds with non-compact isometry group

Does anyone know a good reference for general results about closed Semi-Riemannian manifolds which have a non-compact isometry group?

Edit: My goal is to understand a bit better what the intuition behind compactness / non-compactness of the isometry of a closed Semi-Riemannian manifold is. For example are there topological restrictions on the underlying space? An example of the kind of statements I'm looking for is the following theorem (for Lorentzian manifolds):

Theorem: Let $(M,g)$ be a closed, simply connected Lorentz manifold and assume $M$ and $g$ are analytic. Then the isometry group of $M$ is compact.

• There are quite a few results about pseudo-Riemannian manifolds. Could you please be more specific about what is it that you want to get? In any case, here is a nice construction of a PR-manifold with a reach group of isometries. Let $G$ be a semisimple group and use left translations of the killing form on $T_eG$ to obtain a PR-structure on $G$. By the conjugation invariance of the form, this structure is also right invariant. Take a cocompact lattice $\Gamma<G$ and form the quotient structure on $G/\Gamma$. The left action of $G$ on this space will be by isometries. Commented Aug 14, 2018 at 14:08
• Thanks for your comment. I adjusted my question accordingly. To your construction: so whenever $G$ is non-compact this yields an example of a compact PS-manifold $G/\Gamma$ with non-compact isometry group (and if $G$ is compact itself, then the metric on $G$ would be Riemannian), right?
– JS.
Commented Aug 14, 2018 at 15:21
• take a look here: ihes.fr/~/gromov/PDF/1[74].pdf Commented Aug 14, 2018 at 19:46
• Thank you. I will take a look at it. I have a question to your construction: Can we say anything about the signature of the PR-metric we obtain from the Killing form? In the case of $SL(2,\mathbb{R})$ it has signature $(2,1)$. Do all signatures appear?
– JS.
Commented Aug 15, 2018 at 8:36
• take a look here: arxiv.org/pdf/1804.08695.pdf Commented Aug 15, 2018 at 11:56

Of course there is D’Ambra's 1988 paper "Isometry groups of Lorentz manifolds", from which the theorem you state is taken. A later paper, taking a more general perspective, is this one by D’Ambra and Gromov from 1991.

Zimmer's school had its impact then, with Kowalski's thesis from 1994 and further papers by Zimmer, Adams, Stuck, Witte-Morris and Iozzi.

Later contribution was made by Zeghib's school and in particular Charles Frances.

There are many other contributions. Take a look at the list of participants of this conference. Many of the later contributions are in the language of Cartan geometries or rigid geometric structures etc. so you might miss them by a careless literature briefing.

My advice it to look at some of Frances' late papers, such as this, and the references there. Maybe even better: make a contact with Frances who is actively working on the subject.

Let me add just a little bit of actual mathematics.

Let $G$ be a semisimple Lie group. Then its Killing form is an $\text{Ad}$-invariant non-degenerate symmetric form on its Lie algebra, which I identify here the tangent space at the identity. It is definite iff $G$ is compact. Applying left translations we obtain a pseudo-Riemannian structure on $G$ which is both left and right translation invariant. Picking a cocompact lattice $\Gamma<G$ we form a quotient manifold $G/\Gamma$ and endow it with the quotient pseudo-Riemannian structure (note that $G\to G/\Gamma$ is a covering map). The left action of G on this space will be by isometries. The particular example of $G=\text{SL}_2(\mathbb{R})$ will give us a 3-dimensional compact Lorentzian manifold endowed with a non-trivial continuous action of $\text{SL}_2(\mathbb{R})$ by isometries. In a sense, this is as large as a group of isometries of compact Lorentzian manifold can get: non-compact simple Lie groups other then (locally) $\text{SL}_2(\mathbb{R})$ have no non-trivial continuous action of on compact Lorentzian manifolds. Similar theorems (with larger sets of exceptions) could be proved for pseudo-Riemannain manifolds with higher signatures.