PROJECTIVELY RELATED EINSTEIN FINSLER SPACES

The main objective of this paper is to find the necessary and sufficient condition of a given Finsler metric to be Einstein in order to classify the Einstein Finsler metrics on a compact manifold. The considered Einstein Finsler metric in the study describes all different kinds of Einstein metrics which are pointwise projective to the given one. This study has resulted in the following theorem that needs the proof of three prepositions. Let F be a Finsler metric (n 2) projectively related to an Einstein non-projectively flat Finsler metric F , then F is Einstein if and only if F = lambda F where lambda is a constant. A Schur type lemma is also proved.