Narrow Results

In this paper, we study a class of Finsler metrics defined by a Riemannian metric and a 1-form. We classify those projectively flat with constant flag curvature.

In this paper, we study Randers metrics with quadratic Riemann curvature as in the Riemannian case. We find equations that characterize R-quadratic Randers metrics. In particular, we show that R-quadratic Randers metrics must have constant S-curvature.

Locally dually flat Finsler metrics arise from information geometry. Such metrics have special geometric properties. In this paper, we characterize locally dually flat and projectively flat Finsler metrics and study a special class of Finsler metrics called Randers metrics which are expressed as the sum of a Riemannian metric and a one-form. We find some equations that characterize locally dually flat Randers metrics and classify those with isotropic S-curvature.

In this paper, we study the non-Riemannian quantity H in Finsler geometry. We obtain some rigidity theorems of a compact Finsler manifold under some conditions related to H. We also prove that the S-curvature for a Randers metric is almost isotropic if and only if H almost vanishes. In particular, S-curvature is isotropic if and only if H = 0.

In this paper, we prove a structure theorem for projectively flat Finsler metrics of negative constant flag curvature. We show that for such a Finsler metric if the orthogonal group acts as isometries, then the Finsler metric is a slight generalization of Chern–Shen's construction Riemann–Finsler geometry, Nankai Tracts in Mathematics, Vol. 6 (World Scientific Publishing, Hackensack, NJ, 2005), x+192 pp.

In this paper, we study Finsler surfaces of constant (flag) curvature. We show that the space of those, with two-dimensional isometric group depends on two arbitrary constants. We also give a new technique to recover Finsler metrics from the specified two constants. Using this technique we obtain some new Finsler surfaces of constant flag curvature with two-dimensional isometry group.

In this paper, we study a class of Finsler metrics called general (α, β)-metrics, which are defined by a Riemannian metric α and a 1-form β. We find an equation which is necessary and sufficient condition for such Finsler metric to be a Douglas metric. By solving this equation, we obtain all of general (α, β)-metrics with vanishing Douglas curvature under certain condition. Many new non-trivial examples are explicitly constructed.

We define a Larotonda space as a quotient space ${𝒫}={{𝒰}}_{{𝒜}}/{{𝒰}}_{{ℬ}}$ of the unitary groups of $C^{\ast }$-algebras $1\in {ℬ}\subset {𝒜}$ with a faithful unital conditional expectation

Given a point $\rho \in {𝒫}$ and a tangent vector $X\in {(T{𝒫})}_{\rho }$, we consider the problem of whether the geodesic $\delta$ of the linear connection satisfying these initial data is (locally) minimal for the metric. We find a sufficient condition. Several examples are given, of locally minimal geodesics.

In this paper, we present two kinds of total Chern forms $c(E,G)$ and ${𝒞}(E,G)$ as well as a total Segre form $s(E,G)$ of a holomorphic Finsler vector bundle $\pi :(E,G)\to M$ expressed by the Finsler metric $G$, which answers a question of Faran [The equivalence problem for complex Finsler Hamiltonians, in Finsler Geometry, Contemporary Mathematics, Vol. 196 (American Mathematical Society, Providence, RI, 1996), pp. 133–144] to some extent. As some applications, we show that the signed Segre forms ${(-1)}^k{s}_k(E,G)$ are positive $(k,k)$-forms on $M$ when $G$ is of positive Kobayashi curvature; we prove, under an extra assumption, that a Finsler–Einstein vector bundle in the sense of Kobayashi is semi-stable; we introduce a new definition of a flat Finsler metric, which is weaker than Aikou’s one [Finsler geometry on complex vector bundles, in A Sampler of Riemann–Finsler Geometry, MSRI Publications, Vol. 50 (Cambridge University Press, 2004), pp. 83–105] and prove that a holomorphic vector bundle is Finsler flat in our sense if and only if it is Hermitian flat.

Locally projectively flat Finsler metrics compose an important group of metrics in Finsler geometry. The characterization of these metrics is the regular case of the Hilbert’s Fourth Problem. In this paper, we study a class of Finsler metrics composed by a Riemann metric $\alpha =\sqrt{a_{ij}(x)y^i{y}^j}$ and a $1$-form $\beta =b_i(x)y^i$ called general ($\alpha$, $\beta$)-metrics. We classify those locally projectively flat when $\alpha$ is projectively flat. By solving the corresponding nonlinear PDEs, the metrics in this class are totally determined. Then a new group of locally projectively flat Finsler metrics is found.

In this paper, a new notion of isotropic curvature for sprays is introduced. We show that for a spray of scalar curvature, it is of isotropic curvature if and only if the non-Riemannian quantity $\chi$ vanishes. In fact, it is the first geometric quantity to show the spray of isotropic curvature even in the Finslerian case. How to determine a spray is induced by a Finsler metric or not is an interesting inverse problem. We study this problem when the spray is of isotropic curvature and show that a spray of zero curvature can be induced by a group of Finsler metrics. Further, an efficient way is given to construct a family of sprays of isotropic curvature which cannot be induced by any Finsler metric.

In Finsler geometry, the projective Ricci curvature is an important projective invariant. In this paper, we characterize projective Ricci flat spherically symmetric Finsler metrics. Under a certain condition, we prove that a projective Ricci flat spherically symmetric Finsler metric must be a Douglas metric. Moreover, we construct a class of new nontrivial examples on projective Ricci flat Finsler metrics.

Every Finsler metric on a differential manifold induces a spray. The converse is not true. Therefore, it is one of the most fundamental problems in spray geometry to determine whether a spray is induced by a Finsler metric which is regular, but not necessary positive definite. This problem is called inverse problem. This paper discuss inverse problem of sprays with scalar curvature. In particular, we show that if such a spray $G$ on a manifold is of vanishing $H$-curvature, but $G$ has not isotropic curvature, then $G$ is not induced by any (not necessary positive definite) Finsler metric. We also find infinitely many sprays on an open domain $U\subset {ℝ}^n$ with scalar curvature and vanishing $H$-curvature, but these sprays have no isotropic curvature. This contrasts sharply with the situation in Finsler geometry.

In this paper, we discuss inverse problem in spray geometry. We find infinitely many sprays with non-diagonalizable Riemann curvature on a Lie group, these sprays are not induced by Finsler metrics. We also study left invariant sprays with non-vanishing spray vectors on Lie groups. We prove that if such a spray $S$ on a Lie group $G$ satisfies that $G$ is commutative or $S$ is projective, then $S$ is not induced by any (not necessary positive definite) left invariant Finsler metric. Finally, we construct an abundance of the left invariant sprays on Lie groups which satisfy the conditions in above result.

In Finsler geometry, the projective Ricci curvature is an important projective invariant. In this paper, we investigate the projective Ricci curvature of a class of general $(\alpha ,\beta )$-metrics satisfying a certain condition, which is invariant under the change of volume form. Moreover, we construct a class of new nontrivial examples on such Finsler metrics.

Ongoing searches for a quantum theory of gravity have repeatedly led to the suggestion that space–time might ultimately be anisotropic (Finsler-like) and/or exhibit multirefringence (multiple signal cones). Multiple (and even anisotropic) signal cones can be easily dealt with in a unified manner, by writing down a single Fresnel equation to simultaneously encode all signal cones in an even-handed manner. Once one gets off the signal cone and attempts to construct a full multirefringent space–time metric the situation becomes more problematic. In the multirefringent case we shall report a significant no-go result: in multirefringent models there is no simple or compelling way to construct any unifying notion of pseudo-Finsler space–time metric, different from a monorefringenent model, where the signal cone structure plus a conformal factor completely specifies the full pseudo-Riemannian metric.

To throw some light on this situation we use an analog model where both anisotropy and multirefringence occur simultaneously: biaxial birefringent crystal. But the significance of our results extends beyond the optical framework in which (purely for pedagogical reasons) we are working, and has implications for any attempt at introducing multirefringence and intrinsic anisotropies to any model of quantum gravity that has a low energy manifold-like limit.

Given any symplectomorphism on $D^{2n}(n\ge 1)$ which is $C^{\infty }$ close to the identity, and any completely integrable Hamiltonian system ${\mathrm{\Phi }}_H^t$ in the proper dimension, we construct a $C^{\infty }$ perturbation of $H$ such that the resulting Hamiltonian flow contains a “local Poincaré section” that “realizes” the symplectomorphism. As a (motivating) application, we show that there are arbitrarily small perturbations of any completely integrable Hamiltonian system which are entropy non-expansive (and, in particular, exhibit hyperbolic behavior on a set of positive measure). We use some results in Berger–Turaev [On Herman’s positive entropy conjecture, Adv. Math. 349 (2019) 1234—1288], though in higher dimensions we could simply apply a construction from [D. Burago and S. Ivanov, Boundary distance, lens maps and entropy of geodesic ows of Finsler metrics, Geom. & Topol. 20 (2016) 469–490].

A Finsler metric F is said to be spherically symmetric if the orthogonal group O(n) acts as isometries of F. In this paper, we show that every spherically symmetric Finsler metric of isotropic Berwald curvature is a Randers metric. We also construct explicitly a lot of new isotropic Berwald spherically symmetric Finsler metrics.

Here, a nonlinear analysis method is applied rather than classical one to study projective changes of Finsler metrics. More intuitively, a projectively invariant pseudo-distance is introduced and characterized with respect to the Ricci tensor and its covariant derivatives.

In this paper, we study a class of Finsler metrics, which are defined by a Riemannian metric $\alpha$ and a one-form $\beta$. They are called general $(\alpha ,\beta )$-metrics. We have proven that, every Landsberg general $(\alpha ,\beta )$-metric is a Berwald metric, under a certain condition. This shows that the hunting for an unicorn, one of the longest standing open problem in Finsler geometry, cannot be successful in the class of general $(\alpha ,\beta )$-metrics.