On a Class of Quasi-Einstein Finsler Metrics

Abstract

In this paper, we introduce the notion of quasi-Einstein Finsler metric, which is a natural generalization of quasi-Einstein metric in Riemannian geometry. This is also a generalization of Einstein Finsler metrics. Then we study and characterize quasi-Einstein square metrics. Furthermore, we determine quasi-Einstein square metrics. Moreover, we prove that locally projectively flat quasi-Einstein square metrics on a manifold of dimension $n\ge 3$ must be locally Minkowskian.

A Appendix

Proposition A.1

[19] For an $(\alpha ,\beta )$-metric $F=\alpha \phi (s)$, $s=\frac{\beta }{\alpha }$, the Ricci curvature of F is related to the Ricci curvature ${}^{\alpha }Ric$ of $\alpha$ by

$$\begin{aligned} Ric={}^{\alpha }Ric+T^{m}{}_{m}, \end{aligned}$$

(A.1)

where

$$\begin{aligned} T^{m}{}_{m}= & {} [(n-1)c_{1}+c_{2}]\frac{r_{00}^{2}}{\alpha ^{2}}+[(n-1)c_{3}+c_{4}]\frac{r_{00}s_{0}}{\alpha }+[(n-1)c_{5}+c_{6}]\frac{r_{00}r_{0}}{\alpha }\\&+[(n-1)c_{7}+c_{8}]\frac{r_{00|0}}{\alpha }+[(n-1)c_{9}+c_{10}]s_{0}^{2}+(r_{00}r-r_{0}^{2})c_{11}\\&+[(n-1)c_{12}+c_{13}]r_{0}s_{0}\\&+(r^{i}{}_{i}r_{00}+r_{00|b}-b^{i}r_{i0|0}-r_{0i}r^{i}{}_{0})c_{14}+[(n-1)c_{15}+c_{16}]r_{0i}s^{i}{}_{0}\\&+[(n-1)c_{17}+c_{18}]s_{0|0}\\&+c_{19}s_{0i}s^{i}{}_{0}+\alpha c_{20}r s_{0}+[(n-1)c_{21}+c_{22}]\alpha s_{i}s^{i}{}_{0}+\alpha (2r_{i}s^{i}{}_{0}+b^{i}s_{i|0}-2s_{0|b}\\&-2s_{0}r^{i}_{i}+3s_{i}r^{i}{}_{0}) c_{23}\\&+2\alpha Q s^{i}{}_{0|i}-4\alpha ^{2}Q^{2}\Psi s_{i}s^{i}-\alpha ^{2}Q^{2}s^{i}{}_{j}s^{j}{}_{i}, \end{aligned}$$

where

$$\begin{aligned} c_{1}= & {} 2\Psi \Theta _{s}(B-s^{2})-2s\Psi \Theta +\Theta ^{2}-\Theta _{s},\\ c_{2}= & {} (2\Psi \Psi _{ss}-\Psi _{s}^{2})(B-s^{2})^{2}-(6s\Psi \Psi _{s}+\Psi _{ss})(B-s^{2})+2s\Psi _{s},\\ c_{3}= & {} -4\Psi (2Q\Theta _{s}+Q_{s}\Theta )(B-s^{2})+2(Q_{s}\Theta +2Q\Theta _{s})+4Q\Theta (s\Psi -\Theta )-2\Theta _{B},\\ c_{4}= & {} 4[-\Psi (2Q\Psi _{ss}+Q_{s}\Psi _{s}+Q_{ss}\Psi )+Q\Psi _{s}^{2}](B-s^{2})^{2}\\&+2[-2\Psi ^{2}(Q-sQ_{s})+2Q_{ss}\Psi +Q_{s}\Psi _{s}+2Q\Psi _{ss}-\Psi _{sB}\\&+10sQ\Psi \Psi _{s}](B-s^{2})+2\Psi (Q-sQ_{s})-4\Psi _{s}-Q_{ss}-10sQ\Psi _{s},\\ c_{5}= & {} 2(2\Psi \Theta -\Theta _{B}),\\ c_{6}= & {} 2(2\Psi \Psi _{s}-\Psi _{sB})(B-s^{2})-2\Psi _{s},\\ c_{7}= & {} -\Theta ,\\ c_{8}= & {} -\Psi _{s}(B-s^{2}),\\ c_{9}= & {} 8Q\Psi (Q\Theta _{s}+\Theta Q_{s})(B-s^{2})+4Q^{2}(\Theta ^{2}-\Theta _{s})-4Q(\Theta Q_{s}-\Theta _{B}),\\ c_{10}= & {} 4[\Psi ^{2}(2QQ_{ss}-Q_{s}^{2})+2Q\Psi (Q\Psi _{ss}+Q_{s}\Psi _{s})-Q^{2}\Psi _{s}^{2}](B-s^{2})^{2}\\&+4[-4sQ\Psi (\Psi Q_{s}+Q\Psi _{s})-\Psi (2QQ_{ss}-Q_{s}^{2})-Q(Q\Psi _{ss}+Q_{s}\Psi _{s})+Q\Psi _{Bs}\\&+2\Psi ^{2}Q^{2}+Q_{s}\Psi _{B}](B-s^{2})-4s^{2}Q^{2}\Psi ^{2}+4(2+3sQ)(\Psi Q_{s}+Q\Psi _{s})\\&-8Q^{2}\Psi +2QQ_{ss}-Q_{s}^{2}+4sQ\Psi _{B},\\ c_{11}= & {} 4(\Psi ^{2}+\Psi _{B}),\\ c_{12}= & {} 4Q(-2\Theta \Psi +\Theta _{B}),\\ c_{13}= & {} 4[2\Psi (Q_{s}\Psi -Q\Psi _{s})+Q\Psi _{Bs}+Q_{s}\Psi _{B}](B-s^{2})+8sQ\Psi ^{2}+4Q\Psi _{s}-4(1-sQ)\Psi _{B},\\ c_{14}= & {} 2\Psi ,\\ c_{15}= & {} 4Q\Theta ,\\ c_{16}= & {} 4(B-s^{2})(Q \Psi _{s}-Q_{s}\Psi )+2Q_{s}-2(1+2sQ)\Psi ,\\ c_{17}= & {} 2Q\Theta ,\\ c_{18}= & {} 2(Q\Psi _{s}+Q_{s}\Psi )(b^{2}-s^{2})-Q_{s}+2sQ\Psi ,\\ c_{19}= & {} -2Q^{2}+2(1+sQ)Q_{s},\\ c_{20}= & {} -8Q(\Psi ^{2}+\Psi _{B}),\\ c_{21}= & {} -4Q^{2}\Theta ,\\ c_{22}= & {} 2[-2Q^{2}\Psi _{s}(B-s^{2})+Q\Psi ],\\ c_{23}= & {} 2Q\Psi . \end{aligned}$$