Let $(M,g)$ be a compact pseudo-Riemannian manifold (closed or with boundary). A geodesic $\gamma:(a,b)\to M$ is called null if $g_{ij}\dot\gamma^i\dot\gamma^j=0$. The geodesic flow can be seen as a dynamical system on $TM$ (or $T^*M$ if you prefer). A geodesic is null everywhere or nowhere, so we can define null geodesic flow on the subbundle (light cone bundle) $$ CM = \{(x,v)\in TM;g_{ij}v^iv^j=0\}. $$ Is there a natural way to somehow quotient out the scaling freedom in the parametrization of geodesics to make the null geodesic flow live on a compact bundle? (A more detailed question can be found at the end.)

Let me now try to explain what I mean by natural. Consider now a compact Riemannian manifold $(M,g)$ and the geodesic flow on it. We can scale all geodesics to have unit speed; the rest of the solutions to the geodesic equation are obtained by scaling. Let $$ SM = \{(x,v)\in TM;g_{ij}v^iv^j=1\} $$ denote the unit sphere bundle. We can think of the geodesic flow as a dynamical system on $SM$, generated by the geodesic vector field $X$. If we equip $SM$ with the Sasaki metric and the corresponding volume form, then $X$ preserves the volume form. In particular, for $f,h\in C^\infty(SM)$ we have $$ \langle Xf,h\rangle = -\langle f,Xh\rangle +\text{boundary terms} $$ in the $L^2$ inner product corresponding to this volume form. I consider the compact unit sphere bundle $SM$ together with the Sasaki metric and the geodesic vector field $X$ to be a natural setting for the geodesic flow on $M$.

Let $(M_1,g_1)$ and $(M_2,g_2)$ be two compact Riemannian manifolds. Equip the product $M=M_1\times M_2$ with the pseudo-Riemannian product metric $g_1-g_2$ (whereas the Riemannian product metric would be $g_1+g_2$). A geodesic on $M$ is a product of a geodesic on $M_1$ and one on $M_2$. Such a geodesic is null iff the two geodesics have the same speed. We can scale these speeds to one. We can therefore make the null geodesic flow live on the compact bundle $$ LM=SM_1\times SM_2\subset CM, $$ whose fibers are essentially $S^{\dim(M_1)-1}\times S^{\dim(M_2)-1}$. If $X_1$ and $X_2$ are the geodesic vector fields on $SM_1$ and $SM_2$, then $X=X_1+X_2$ generates the null geodesic flow on $LM$. If we equip $SM_1$ and $SM_2$ with the natural metric, then $X$ preserves the volume form corresponding to the product metric. Now we have managed to make the null geodesic flow live on the compact bundle $LM$ with a natural metric and a generating vector field preserving the metric.

If the pseudo-Riemannian manifold is not a product of two Riemannian manifolds, is there a way to construct a compact subbundle of $CM$ that comes with a volume form and a geodesic vector field preserving it? The construction should, ideally, be such that in the product case it simply reproduces the bundle I described above (product of the two unit sphere bundles).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.