On the projective Ricci curvature

The characterizations on a class of weakly weighted Einstein-Finsler metrics

“…Then F is weakly weighted Einstein-Kropina metric satisfyingRic a,c = (n − 1) 3θ F + σ F 2with respect to some volume form dV = e −(n+1)f dV BH if and only if equations (5.3) and (5.5),(5.11) hold for some 1-form ζ and u = u(x) and σ are determined by (5.10) and (5.12) respectively.6 The weakly weighted Einstein-Kropina metrics with ν = 0 and κ = 0In this section we shall consider the weakly weighted Einstein-Kropina metrics with ν = 0 and κ = 0. In this case, the generalized weighted Ricci curvatures are just the projective Ricci curvature ([5][17])Ric a,c = PRic and a = n−1 n+1 , c = − n−1 (n+1) 2 .Then, the polynomials P i 's in (5.2) can be simplified as follows.P 1 = −2(n − 1)b 2 r 00 f 0 , P 2 = b 4 Ric α + b 2 b k r 00;k + (n − 2)(b 2 s 0;0 − s 2 0 + b 2 r 0;0 − 2r 0 s 0 − r 2 0 ) + b 2 r 00 r k k −nr 00 r + 2b 2 r 0k s k 0 + (n − 1)b 4 Hess F f (y) + (n − 1) 2b 2 r 0 f 0 + b 4 f 2 0 , P 3 = −ns 0 r + b 2 b k s 0;k + b 2 s 0 r k k − b 4 s k 0;k + (n − 0k s k + (n − 1)b 2 s k s k 0 − 3(n − 1)θb 4 ,At the same time, (5.3) is reduced equivalently toζα 2 = −2(n − 1)b 2 r 00 f 0 . (6.1)From (6.1), we find that there exists a scalar function η = η(x) on M such that r 00 = ηα 2 .…”