Geodesics of a "diagonal" metric

Question

Are there any relations that exist to simplify Christoffel symbols/connection coefficients for a diagonal metric which has the same function of the coordinates at each entry? In other words, I have a metric

$g = f(x_1,x_2,\cdots) \; \begin{pmatrix} 1 & 0 & \cdots \\ 0 & 1 & & \\ \vdots & & \ddots \end{pmatrix}$

And I want to calculate geodesics. I think they'll be straight lines, i.e. they will take the same shape as they would if the space was Euclidean, but they will be traversed with some varying speed. (That was originally my motivation for looking at this metric, as I have some curves that are 'straight' but not traveled at uniform speed, and so I introduced this metric hoping they would become geodesics).

The geodesic equation is \begin{equation} \frac{d^2x^{\lambda}}{ds^2} + \Gamma^{\lambda}_{\mu \nu} \frac{dx^{\mu}}{ds}\frac{dx^{\nu}}{ds}=0 \end{equation}

I'm hoping that the second term will factor into a constant vector $y^{\lambda}$ multiplied by a scalar function of the coordinates $Y(x_1,x_2,\cdots)$. But it isn't obvious to me if this happens.

On a related note, has anyone ever looked at extending this machinery to spaces of infinite dimension?

Answer 1

As @Thomas correctly pointed the question is about geodesics in conformally flat manifolds.

A "simpler" example of a conformally flat manifold is the sphere $\mathbb{S}^n$ with the metric induced from its standard immersion into $\mathbb{R}^{n+1}$, the conformal equivalence of metrics is provided by the stereographic projection. The geodesics on the sphere are great circles, as known.

In fact, all "space forms", i.e. manifolds of constant sectional curvature, are conformally flat.

More advanced treatment one can find in a recent paper of Paul Tod "Some examples of the behaviour of conformal geodesics" here.

Conformal geodesics are also known under the name of conformal circles. Michael Eastwood recently showed us a calculation that explains some details that one needs when reading the paper Bailey T.N., Eastwood M.G. Conformal circles and parametrizations of curves in conformal manifolds.

With respect to the infinite-dimensional case I can suggest to take a look at this poster where, in particular, one can find a reference to the standard source: Kriegl A., Michor P.W. The Convenient Setting of Global Analysis.

Answer 2

The previous answers are perfectly good, but let me explain briefly why you should not expect geodesics to be straight lines in the first place.

The conformal factor $f$ can be interpreted as an optic index; for example if you take $f$ to be $1$ on a half space and $2$ on its complement, geodesics through the separating hyperplane will correspond to the travel of light through an interface between two materials of different indexes. Then you see refraction (which for example explains why a straw in a glass of water looks broken). There is a well known formula with sines relating the angles of the geodesic before and after the interface, and you probably know a few other examples of this phenomenon.

Of course, the above example is not smooth, but the same kind of thing happen with smooth $f$ (you get continuous refraction in a material with varying optial index). A way to see this easily is to consider the case where $f$ is equal to $1$ except near a given non straight curve $\gamma$, and $f$ is very small at $\gamma$. Then the length (according to $g$) of $\gamma$ can be much smaller than the length of the line between its endpoints, which therefore cannot be a (minimizing) geodesic.

Answer 3

A typical example of a metric like the one you are looking at is the upper half plane with it's hyperbolic metric $f(y)\delta_{ij}$. It's geodesics are circles, not (only) straight lines, so I'd say your hope that you'll get (geometrically) the same geodesics is not justified.

Edit: the kind of metric you are looking at is conformally equivalent to the Euclidean metric, which is a term which may allow you to google and find more results of interest to you. I'm quite familiar with the topic in two (real) dimensions only, so would not want to comment on this in general. (In two dimensions, these are basically holomorphic maps)