# Lagrangian in pseudo-Riemannian manifold and geodesics

I'm trying to solve the following problem without success.

Let $V$ be a smooth function on a pseudo-Riemannian manifold $(M, g)$, which is either bounded from above or from below. Show that there exists a function f related to V and a pseudo Riemannian metric: $$g_N=g+fdu^2,\; N=M\times\mathbb{R}$$ Where $u$ is the coordinate on the $\mathbb{R}$-factor, such that the solutions of the Lagrangian system defined by $\mathcal{L}(v)=\frac{1}{2}g(v,v)-V(\pi v)$ and $v\in TM$, correspond precisely to the geodesics on the manifold $(N, g_N)$.

From my point of view the system should be decoupled because the metric is split in two terms. Probably I totally misunderstand the concept of geodesic equations.

• The metric is only split into two terms if $f$ is independent of the projection onto $M$. This is definitely not the case here. So there will be coupling. For your question it may help to just write down the Lagrangian (in energy form) for geodesics in $(N,g_N)$ and compare. (Also, I think there should be some additional properties on $V$, otherwise you are not guaranteed to get a pseudo-Riemannian manifold.) – Willie Wong Nov 9 '15 at 5:51
• I have tried to do the calculations but I cannot get the coupling terms. – dknew Jan 21 '16 at 16:13