On Projectively Related Einstein Metrics

Abstract

We study the following problem: given an Einstein metric on a manifold, characterize and study all Einstein metrics which are pointwise projective to the given one. By definition, two metrics are said to be pointwise projectively related if they have the same geodesics as point sets. This is closely related to Hilbert's Fourth Problem. In this paper, we show that pointwise projectively related Einstein metrics satisfy a simple equation along geodesics. In particular, we prove the following rigidity theorem: if two pointwise projectively related Einstein metrics are complete with negative Einstein constants, then one is a multiple of another. A Riemann metric has two features: length and angle, while a Finsler metric has only one feature: length. In projective metric geometry, the second feature of Riemann metrics is generally not used. Thus, we do not restrict our attention to Riemannian metrics. Our method is ``Finslerian''.