Geodesic in plane when metric depends on single variable?

[rhe]

Hello,

I have a p+1 dimensional manifold describing the parameter space of a family of probability densities. The p+1 dimensions are (beta, t1, t2, ..., tp), all reals, and beta restricted to the positive reals. The (fisher) metric on this manifold is a function of beta only, hence the submanifolds for a constant beta are flat, and the geodesics are straight lines in these submanifolds.

Consider two points P=(beta1,0,0,...,0) and Q=(beta2,t1,t2,...,tp), and the geodesic connecting these two points. I suspect that this geodesic lies in the 2D-plane spanned by the vector (1,0,0,...,0) and the unit vector pointing towards Q. Is this true?

If so, this could significantly speed up the numerical determination of geodesics and geodesic distances between probability densities for this type of models.

Thank you

 

[Reb]

Consider two points P=(beta1,0,0,...,0) and Q=(beta2,t1,t2,...,tp), and the geodesic connecting these two points. I suspect that this geodesic lies in the 2D-plane spanned by the vector (1,0,0,...,0) and the unit vector pointing towards Q. Is this true?

No, it's not, unless your manifold is a plane.