# Canonical Metrics on 3- and 4-Manifolds

From the Uniformization Theorem, it is known that every conformal class of metrics on a genus-$g$ Riemann surface with $n$ boundaries/punctures, subject to the condition $2g+n\ge 3$, contains a unique hyperbolic metric which can be considered as a canonical metric on the surface. I am wondering if similar results exist in the case of 3- or 4-manifolds where the manifolds admit Lorentzian metrics, metrics with signature $(-,+,+)$ or $(-,+,+,+)$ in the case of 3- or 4-manifold, respectively.

Let $M$ be a compact 3- or 4-manifolds, possibly with smooth boundaries, which admits Lorentzian metrics, and consider a diffeomorphism $f:M\longrightarrow M$. This will relate Lorentzian metrics $g_1$ and $g_2$ which are compatible with a given smooth structure on $M$:

$$f^*g_1=g_2$$

Where $f^*g_1$ denotes the pull-back of the metric $g_1$. If I denote the equivalence class of such Lorentzian metrics under diffeomorphism as $[g]$, there are two questions:

• Is there a canonical Lorentzian metric in $[g]$ for the general 3- or 4-manifolds?

• For the special case that the 4-manifold $M$ can smoothly be written as $M=\mathbb{R}\times X$, where $X$ is a compact 3-manifold possibly with boundaries, is there a canonical Lorentzian metric in $[g]$? By "smoothly", I mean there is a global diffeomorphism between $M$ and $\mathbb{R}\times X$;

A good reference on 3- or 4-manifolds containing related results is highly appreciated.

• You should have a look at Besse’s Einstein manifolds – Thomas Rot May 4 '18 at 13:30
• @ThomasRot Thanks for the reference. Actually, I am interested in more general Lorentzian metrics not only those that are solutions of the Einstein equations. – QGravity May 4 '18 at 18:58
• For the 3-dimensional case, you might like to look into the Perelman's geometrization theorem, which asserts the existence of (somewhat) canonical Riemannian metrics on 3-manifolds. (See here: en.wikipedia.org/wiki/Geometrization_conjecture .) I'm not sure what's known in the Lorentzian setting, but there has been plenty of work on it; see, for instance, here -- math.univ-lille1.fr/~kassel/flat-lorentzian.pdf . – HJRW May 11 '18 at 10:22
• I've replaced the '3-manifolds' tag by 'gt.geometric-topology', since I think most researchers in 3-manifolds will follow that tag. Please feel free to switch it back if you want. – HJRW May 11 '18 at 10:35
• @HJRW Thank you for the references. Also, there is a related article arxiv.org/pdf/math/0511034.pdf As I can understand, and I am not a math student, Thurston's conjecture describes the conditions under which a closed 3-manifolds admits a locally homogeneous metric. Not every 3-manifolds admit such a metric. However, I am interested in the case that the 3-manifold is an arbitrary Riemannian manifold. – QGravity May 16 '18 at 13:06

There is no canonical Lorentzian metric in $[g]$, because that would be a diffeomorphism invariant Lorentzian metric. The diffeomorphism group of any manifold has infinite dimension, and infinite dimensional stabilizer, so does not preserve any affine connection (by Bochner's lemma), or any rigid geometric structure. Recall that Bochner's lemma proves that the stabilizer of a point in the symmetry group of any affine connection is expressed as linear transformations in geodesic coordinates, so has dimension bounded by the square of the dimension of the manifold.
Edit: I can't seem to find a nice proof of what I called Bochner's lemma (which is perhaps not due to Bochner). The OP asked for one. Here is a proof. Take a $C^{\infty}$ manifold $M$ with a $C^{\infty}$ connection $\nabla$ on its tangent bundle. The connection determines an exponential map. Any $C^{\infty}$ diffeomorphism $\phi \colon M \to M$ fixing the connection fixes the exponential map: $\phi(\exp_p(tv))=\exp_{\phi(p)}(t\phi'(p)v)$. Suppose now that $p$ is a fixed point of $\phi$. Then $\phi'(p) \colon T_p M \to T_p M$ is related to $\phi \colon M \to M$ by the equation $\phi(e^{tv})=e^{t\phi'(v)}$, where we write $e^v$ to mean $\exp_p v$. Note that $v \mapsto e^v$ is a local diffeomorphism near $v=0$, with local inverse which we denote $\log$. Take linear coordinates $v=v^i e_i$ on $T_p M$ (i.e. take a basis $\{e_i\}$ for $T_p M$). Define coordinates $x^1,\dots,x^n$ on $M$ near $p$ by $x^i(q)=v^i$ if $v^ie_i = \log q$. So in these coordinates, $e^v$ is expressed as the identity map, and our equation $\phi(\exp_p(tv))=\exp_p(t\phi'(p)v)$ becomes in coordinates $\phi(x)=\phi'(0)x$, i.e. $\phi$ is linear in these coordinates.
If in additional $\phi$ preserves a pseudo-Riemannian metric (for example one for which $\nabla$ is the Levi--Civita connection), then $\phi'(p)$ preserves the quadratic form in $T_p M$ given by that connection.
• To in addition preserve a pseudo-Riemannian metric, of any signature, the lemma puts the stabilizer dimension at, at most, $n(n-1)/2$, and so the symmetry group of a pseudo-Riemannian metric is of dimension at most $n+n(n-1)/2$, equality only for space forms. See Kobayashi, Transformation Groups in Differential Geometry, for lots of rigidity theorems. – Ben McKay May 4 '18 at 5:59
• ($n$ being the dimension of the manifold) – Ben McKay May 4 '18 at 6:26
• Thank you for the answer. However, I am confused. Why should it be a diffeomorphism-invariant metric? Also, it seems to me that if we consider the space of all Lorentzian metrics on a fixed 4-manifold $M$, denoted as $\mathtt{Lor}(M)$, and its quotient by the diffeomorphism group $\mathtt{Dif}(M)$, denoted as $\mathtt{Lor}_{\mathtt{Dif}}(M)$ (all [g]), the projection $\mathtt{Lor}(M)\longrightarrow\mathtt{Lor}_{\mathtt{Dif}}(M)$ is not even a fibration because there is no continuous way to define a continuous section of this projection if there is no canonical or distiguished metric in $[g]$. – QGravity May 7 '18 at 15:54
• @QGravity: what did you mean by "canonical"? I assumed that you meant up to isomorphism of the data $(M.[g])$, i.e. diffeomorphism of the manifold and of the choice of Lorentz metric equivalence class. – Ben McKay May 7 '18 at 17:33