family of metrics with same geodsics

Question

For every bi-invariant metric on a lie group we know geodesics are flow of left invariant vector fields, so this question naturally arise: are there family of metrics on manifolds that have same geodesics?

Answer 1

There are examples of metrics with the same geodesics on one manifold. You may find a discussion about them in Can one recover a metric from geodesics?

The short summery of the examples from the answer in the following: for any two metrics $g_1$ on $M_1$ and $g_2$ on $M_2$ the product metrics $g=g_1+ g_2$ and $\bar g= 2g_1 + g_2$ on $M_1 + M_2$ have the same geodesics considered as PARAMETERIZED curves.

If you consder geodesics as curves without preferable parameterisation on them, a local example of two 2-dim metrics that have the same geodesics is due to Dini 1869 and is (for arbitrary functions $X(x)$ and $Y(y)$ such that the next formulas do correspond to metrics) $$g= (X(x)- Y(y))(dx^2 + dy^2) \ \ and \ \ \bar g= \left(\tfrac{1}{Y(y)}- \tfrac{1}{X(x)}\right)\left(\tfrac{dx^2}{X(x)} + \tfrac{dy^2}{Y(y)}\right).$$ The example can be generalized for any dimension (Levi-Civita 1896), and the formulas of Dini and Levi-Civita are actually local normal forms of metrics having the same geodesics (see the discussion in http://arxiv.org/abs/1301.2492, where also metrics of indefinite signature are discussed).

If you are interested in metrics having the same geodesics with a biinvariant metric, then only the first example is possible. More precisly, if a bi-invariant Riemannian metric $g$ has the same geodesics considered as unparameterised curves with a metric $\bar g$, then parameterised geodesics of these two metrics are also the same. Moreover, the second metric is also bi-invariant and moreover the metric $g$ in the appropriate coordinate system has the form $g_0+ g_1 +...+g_k$ such that $g_0$ is $dx_1^2 +...+ dx^2_{k_0}$ and the metric $\bar g$ in the same coordinate system has the form $g'_0 + \lambda_1 g_1 + ...+ \lambda_kg_k$, where $\lambda_i$ are constants and $g'_0$ is $\sum_{i,j=1}^{k_0} C_{ij} dx_idx_j$, where $C_{ij}$ is a constant matrix.

This follows from the result of Sinjukov (Dokl. Akad. Nauk SSSR (N.S.) 98, (1954) 21--23) , see the discussion in Geodesic transformations of the complex projective plane

See also the question/answer in A property of bi-invarient Finsler metrics on SU(N) which is also related