Reversible homogeneous Finsler metrics with positive flag curvature

Abstract

Abstract In this work, we continue with the classification for positively curve homogeneous Finsler spaces ( G / H , F ) {(G/H,F)} . With the assumption that the homogeneous space G / H {G/H} is odd dimensional and the positively curved metric F is reversible, we only need to consider the most difficult case left, i.e. when the isotropy group H is regular in G. Applying the fixed point set technique and the homogeneous flag curvature formulas, we show that the classification of odd dimensional positively curved reversible homogeneous Finsler spaces coincides with that of L. Bérard Bergery in Riemannian geometry except for five additional possible candidates, i.e. SU ⁢ ( 4 ) / SU ⁢ ( 2 ) ( 1 , 2 ) ⁢ S ( 1 , 1 , 1 , - 3 ) 1 {\mathrm{SU}(4)/\mathrm{SU}(2)_{(1,2)}\mathrm{S}^{1}_{(1,1,1,-3)}} , Sp ⁢ ( 2 ) / S ( 1 , 1 ) 1 {\mathrm{Sp}(2)/\mathrm{S}^{1}_{(1,1)}} , Sp ⁢ ( 2 ) / S ( 1 , 3 ) 1 {\mathrm{Sp}(2)/\mathrm{S}^{1}_{(1,3)}} , Sp ⁢ ( 3 ) / Sp ⁢ ( 1 ) ( 3 ) ⁢ S ( 1 , 1 , 0 ) 1 {\mathrm{Sp}(3)/\mathrm{Sp}(1)_{(3)}\mathrm{S}^{1}_{(1,1,0)}} , and G 2 / SU ⁢ ( 2 ) {G_{2}/\mathrm{SU}(2)} with SU ⁢ ( 2 ) {\mathrm{SU}(2)} the normal subgroup of SO ⁢ ( 4 ) {\mathrm{SO}(4)} corresponding to the long root. Applying this classification to homogeneous positively curved reversible ( α , β ) {(\alpha,\beta)} metrics, the number of exceptional candidates can be reduced to only two, i.e. Sp ⁢ ( 2 ) / S ( 1 , 1 ) 1 {\mathrm{Sp}(2)/\mathrm{S}^{1}_{(1,1)}} and Sp ⁢ ( 3 ) / Sp ⁢ ( 1 ) ( 3 ) ⁢ S ( 1 , 1 , 0 ) 1 {\mathrm{Sp}(3)/\mathrm{Sp}(1)_{(3)}\mathrm{S}^{1}_{(1,1,0)}} .