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Register a new userRigidity of Busemann convex Finsler metrics
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We prove that a Finsler metric is nonpositively curved in the sense of Busemann if and only if it is affinely equivalent to a Riemannian metric of nonpositive sectional curvature. In other terms, such Finsler metrics are precisely Berwald metrics of nonpositive flag curvature. In particular in dimension 2 every such metric is Riemannian or locally isometric to that of a normed plane. In the course of the proof we obtain new characterizations of Berwald metrics in terms of the so-called linear parallel transport.
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We generalize the notion of Zermelo navigation to arbitrary pseudo-Finsler metrics possibly defined in conic subsets. The translation of a pseudo-Finsler metric $F$ is a new pseudo-Finsler metric whose indicatrix is the translation of the indicatrix of $F$ by a vector field $W$ at each point, where $W$ is an arbitrary vector field. Then we show that the Matsumoto tensor of a pseudo-Finsler metric is equal to zero if and only if it is the translation of a semi-Riemannian metric, and when $W$ is homothetic, the flag curvature of the translation coincides with the one of the original one up to the addition of a non-positive constant. In this case, we also give a description of the geodesic flow of the translation.
In this work, we consider a class of Finsler metrics using the warped product notion introduced by Chen, S. and Zhao (2018), with another warping, one that is consistent with static spacetimes. We will give the PDE characterization for the proposed metrics to be Ricci-flat and explicitly construct two non-Riemannian examples.
In this paper, we consider a special class of singular Finsler metrics: $m$-Kropina metrics which are defined by a Riemannian metric and a $1$-form. We show that an $m$-Kropina metric ($m e -1$) of scalar flag curvature must be locally Minkowskian in dimension $nge 3$. We characterize by some PDEs a Kropina metric ($m=-1$) which is respectively of scalar flag curvature and locally projectively flat in dimension $nge 3$, and obtain some principles and approaches of constructing non-trivial examples of Kropina metrics of scalar flag curvature.
In this paper we have studied the class of Finsler metrics, called C3-like metrics which satisfy the un-normal and normal Ricci flow equation and proved that such metrics are Einstein.
If the flag curvature of a Finsler manifold reduces to sectional curvature, then locally either the Finsler metric is Riemannian, or the flag curvature is isotropic.
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