Geodesic mapping

projective mapping

A mapping $f$ that transforms the geodesic lines of a space $U$ into the geodesic lines of a space $V$. A geodesic mapping $f : U \rightarrow V$, where $U$ and $V$ are spaces in which geodesics are defined, is a local homeomorphism (diffeomorphism if $U$ and $V$ are smooth manifolds).

A space that locally permits a geodesic mapping into a Euclidean space is called projectively flat. Geodesic mappings of one Riemannian space into another exist in exceptional cases. Among the Riemannian spaces only those of constant curvature are projectively flat [1]. The description of all Riemannian projectively-flat metric spaces constitutes Hilbert's fourth problem [2].

In the theory of spaces with an affine connection one does not speak of geodesic mappings but rather of geodesic transformations of a connection, which means a transition to another connection on the same manifold with preservation of the geodesics. The transition from a connection $\Gamma^i_{jk}$ to a connection $\bar\Gamma^i_{jk}$ is a geodesic mapping under the condition $\bar\Gamma^i_{jk} = \Gamma^i_{jk} + A^i_k\psi_j + A^i_j\psi_k$, where $\psi$ is a covector field. A space with an affine connection is projectively flat if and only if the projective curvature tensor vanishes.

References

[1]	J.A. Schouten, "Ricci-calculus. An introduction to tensor analysis and its geometrical applications" , Springer (1954) (Translated from German)
[2]	A.V. Pogorelov, "Hilbert's fourth problem" , Winston & Wiley (1979) (In Russian)