No Arabic abstract
Let $M$ be a differentiable manifold, $T_xM$ be its tangent space at $xin M$ and $TM={(x,y);xin M;y in T_xM}$ be its tangent bundle. A $C^0$-Finsler structure is a continuous function $F:TM rightarrow mathbb [0,infty)$ such that $F(x,cdot): T_xM rightarrow [0,infty)$ is an asymmetric norm. In this work we introduce the Pontryagin type $C^0$-Finsler structures, which are structures that satisfy the minimum requirements of Pontryagins maximum principle for the problem of minimizing paths. We define the extended geodesic field $mathcal E$ on the slit cotangent bundle $T^ast Mbackslash 0$ of $(M,F)$, which is a generalization of the geodesic spray of Finsler geometry. We study the case where $mathcal E$ is a locally Lipschitz vector field. We show some examples where the geodesics are more naturally represented by $mathcal E$ than by a similar structure on $TM$. Finally we show that the maximum of independent Finsler structures is a Pontryagin type $C^0$-Finsler structure where $mathcal E$ is a locally Lipschitz vector field.
In this paper, we use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on $S^3$ with ${rm Ric} = 2 F^2$, ${rm Ric}=0$ and ${rm Ric}=- 2 F^2$, respectively. This family of metrics provide an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant, but the flag curvature is not.
In this paper, we study the spectrums of faithful dimension pairs on a closed Finsler manifold and obtain a Gromov type and a Buser type lower bounds for eigenvalues. Furthermore, for the Lusternik-Schnirelmann spectrum, we not only obtain a better lower bound, but also estimate the multiplicity of each eigenvalue.
In this paper we investigate the spectral problem in Finsler geometry. Due to the nonlinearity of the Finsler-Laplacian operator, we introduce textit{faithful dimension pairs} by means of which the spectrum of a compact reversible Finsler metric measure manifold is defined. Various upper and lower bounds of such eigenvalues are provided in the spirit of Cheng, Buser and Gromov, which extend in several aspects the results of Hassannezhad, Kokarev and Polterovich. Moreover, we construct several faithful dimension pairs based on Lusternik-Schnirelmann category, Krasnoselskii genus and essential dimension, respectively; however, we also show that the Lebesgue covering dimension pair is not faithful. As an application, we show that the Bakry-Emery spectrum of a closed weighted Riemannian manifold can be characterized by the faithful Lusternik-Schnirelmann dimension pair.
We discuss new sufficient conditions under which an affine manifold $(M, abla)$ is geodesically connected. These conditions are shown to be essentially weaker than those discussed in groundbreaking work by Beem and Parker and in recent work by Alexander and Karr, with the added advantage that they yield an elementary proof of the main result.
In this paper, we study the spectral problem on a compact Finsler manifold with or without boundary. More precisely, given a certain collection of sets in Sobolev space $H^{1,2}(M)$ and a dimension-like function, we can define a corresponding spectrum. Such a spectrum satisfies nice properties. In particular, the eigenfunction corresponding to each eigenvalue always exists. And a Cheng type upper bound estimate for eigenvalues is obtained. Moreover, some interesting examples are constructed and investigated in this paper.