LEFT INVARIANT $(\alpha,\beta)$-METRICS ON 4-DIMENSIONAL LIE GROUPS

Let $G$ be a 4-dimensional Lie group with an invariant para-hypercomplex structure and let $F= \beta+ a\alpha+\beta^2/{\alpha}$ be a left invariant $(\alpha,\beta)$-metric, where $\alpha$ is a Riemannian metric and $\beta$ is a 1-form on $G$, and $a$ is a real number. We prove that the flag curvature of $F$ with parallel 1-form $\beta$ is non-positive, except in Case 2, in which $F$ admits both negative and positive flag curvature. Then, we determine all geodesic vectors of $(G,F)$.

Higher rank rigidity for Berwald spaces

We generalize the higher rank rigidity theorem to a class of Finsler spaces, i.e. Berwald spaces. More precisely, we prove that a complete connected Berwald space of finite volume and bounded non-positive flag curvature with rank at least two whose universal cover is irreducible is a locally symmetric space or a locally Minkowski space.

Homogeneous Finsler spaces and the flag-wise positively curved condition

In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) condition), which means that in each tangent plane, there exists a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact coset spaces admitting non-negatively curved homogeneous Finsler metrics satisfying the (FP) condition. Using a crucial technique we developed previously, we prove that most of these coset spaces cannot be endowed with positively curved homogeneous Finsler metrics. We also prove that any Lie group whose Lie algebra is a rank 2 non-Abelian compact Lie algebra admits a left invariant Finsler metric satisfying the (FP) condition. As by-products, we find the first example of non-compact coset space {S^{3}\times\mathbb{R}} which admits homogeneous flag-wise positively curved Finsler metrics. Moreover, we find some non-negatively curved Finsler metrics on {S^{2}\times S^{3}} and {S^{6}\times S^{7}} which satisfy the (FP) condition, as well as some flag-wise positively curved Finsler metrics on {S^{3}\times S^{3}} , shedding some light on the long standing general Hopf conjecture.

Reversible homogeneous Finsler metrics with positive flag curvature

In this work, we continue with the classification for positively curve homogeneous Finsler spaces {(G/H,F)}. With the assumption that the homogeneous space {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. {\mathrm{SU}(4)/\mathrm{SU}(2)_{(1,2)}\mathrm{S}^{1}_{(1,1,1,-3)}}, {\mathrm{Sp}(2)/\mathrm{S}^{1}_{(1,1)}}, {\mathrm{Sp}(2)/\mathrm{S}^{1}_{(1,3)}}, {\mathrm{Sp}(3)/\mathrm{Sp}(1)_{(3)}\mathrm{S}^{1}_{(1,1,0)}}, and {G_{2}/\mathrm{SU}(2)} with {\mathrm{SU}(2)} the normal subgroup of {\mathrm{SO}(4)} corresponding to the long root. Applying this classification to homogeneous positively curved reversible {(\alpha,\beta)} metrics, the number of exceptional candidates can be reduced to only two, i.e. {\mathrm{Sp}(2)/\mathrm{S}^{1}_{(1,1)}} and {\mathrm{Sp}(3)/\mathrm{Sp}(1)_{(3)}\mathrm{S}^{1}_{(1,1,0)}}.