Any two Riemannian metrics can easily be deformed into each other, only obtaining positive definite metrics in between.
However, for metrics of other signatures this might not be possible.


**Which Lorentzian metrics does the two-torus $\Bbb T^2$ admit, up to continuous deformations via Lorentzian metrics?** In other words, I'm interested in $\pi\_0(\mathbf{LMet}(\Bbb T^2))$.


Of course there is the standard metric $g:=dx^2-dy^2$, but there are also metrics whose lightcones turn around several times when one wanders along a non-nullhomotopic circle, explicitly given by $g\circ (M\otimes M)$ where $M\in C(\Bbb T^2, End(T\Bbb T^2))$, and hence can not be deformed into the standard metric, thus yielding an injection $\langle T^2, SO(2)\rangle \to \pi\_0(\mathbf{LMet}(\Bbb T^2))$. Are these already all different metrics?



Lorentzian metrics on the torus up to continuos deformations
Yes those are already all different metrics. Since the tangent bundle of the 2-torus $\mathbb{T}^2$ is trivial you have a correspondence between the set of homotopy classes of maps $\mathbb{T}^2\to\mathbb{RP}^1$ and $\pi\_0(\mathbf{LMet}(\mathbb{T}^2))$. Since $\mathbb{T}^2=S^1\times S^1$ and $\mathbb{RP}^1\simeq S^1$ are $K(\mathbb{Z}^2,1)$ and $K(\mathbb{Z},1)$ respectively, you get $\pi\_0(\mathbf{LMet}(\mathbb{T}^2))\simeq\hom(\mathbb{Z}^2,\mathbb{Z})$.


In general there is a bijection between the set of path components of metrics of signature $(p,q)$ on a given vector bundle $E\to X$ and the set of homotopy classes of rank $p$ subbundles of $E$. When $E$ is not trivial the classification is a difficult task. For $p=1$ interesting results are in this paper by Ulrich Koschorke, "Homotopy classification of line fields and of Lorentz metrics on closed manifolds", [Math. Proc. Camb. Phil. Soc. (2002), 132, issue 02, p.281-300](http://dx.doi.org/10.1017/S0305004101005606).


