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Register a new userHilbert forms for a Finsler metrizable projective class of sprays
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The projective Finsler metrizability problem deals with the question whether a projective-equivalence class of sprays is the geodesic class of a (locally or globally defined) Finsler function. In this paper we use Hilbert-type forms to state a number of different ways of specifying necessary and sufficient conditions for this to be the case, and we show that they are equivalent. We also address several related issues of interest including path spaces, Jacobi fields, totally-geodesic submanifolds of a spray space, and the equivalence of path geometries and projective-equivalence classes of sprays.
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In this paper, we first give two fundamental principles under a technique to characterize conformal vector fields of $(alpha,beta)$ spaces to be homothetic and determine the local structure of those homothetic fields. Then we use the principles to study conformal vector fields of some classes of $(alpha,beta)$ spaces under certain curvature conditions. Besides, we construct a family of non-homothetic conformal vector fields on a family of locally projectively Randers spaces.
An $(alpha,beta)$-manifold $(M,F)$ is a Finsler manifold with the Finsler metric $F$ being defined by a Riemannian metric $alpha$ and $1$-form $beta$ on the manifold $M$. In this paper, we classify $n$-dimensional $(alpha,beta)$-manifolds (non-Randers type) which are positively complete and locally projectively flat. We show that the non-trivial class is that $M$ is homeomorphic to the $n$-sphere $S^n$ and $(S^n,F)$ is projectively related to a standard spherical Riemannian manifold, and then we obtain some special geometric properties on the geodesics and scalar flag curvature of $F$ on $S^n$, especially when $F$ is a metric of general square type.
An $(alpha,beta)$-metric is defined by a Riemannian metric and $1$-form. In this paper, we investigate the known characterization for $(alpha,beta)$-metrics of isotropic S-curvature. We show that such a characterization should hold in dimension $nge 3$, and for the 2-dimensional case, there is one more class of isotropic S-curvature than the higher dimensional ones. Further, we construct corresponding examples for every two-dimensional class, especially for the class that the norm of $beta$ with respect to $alpha$ is not a constant.
A systematic development of the so-called Palatini formalism is carried out for pseudo-Finsler metrics $L$ of any signature. Substituting in the classical Einstein-Hilbert-Palatini functional the scalar curvature by the Finslerian Ricci scalar constructed with an independent nonlinear connection $mathrm{N}$, the metric and affine equations for $(mathrm{N},L)$ are obtained. In Lorentzian signature with vanishing mean Landsberg tensor $mathrm{Lan}_i$, both the Finslerian Hilbert metric equation and the classical Palatini conclusions are recovered by means of a combination of techniques involving the (Riemannian) maximum principle and an original argument about divisibility and fiberwise analyticity. Some of these findings are also extended to (positive definite) Riemannian metrics by using the eigenvalues of the Laplacian. When $mathrm{Lan}_i eq 0$, the Palatini conclusions fail necessarily, however, a good number of properties of the solutions remain. The framework and proofs are built up in detail.
An $(alpha,beta)$-metric is defined by a Riemannian metric $alpha$ and $1$-form $beta$. In this paper, we study a known class of two-dimensional $(alpha,beta)$-metrics of vanishing S-curvature. We determine the local structure of those metrics and show that those metrics are Einsteinian (equivalently, isotropic flag curvature) but generally are not Ricci-flat.
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