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Journals and Conferences
Introduction 198 1. Flag and Ricci Curvatures 199 1.1. Finsler metrics 199 1.2. Flag curvature 207 1.3. Ricci curvature 214 2. Randers Metrics in Their Defining Form 218 2.1. Basics 218 2.2. Characterising Einstein–Randers metrics 220 2.3. Characterising constant flag curvature Randers metrics 228 3. Randers Metrics Through Zermelo Navigation 230 3.1.… (More)
In this paper, we study Zermelo navigation on Riemannian manifolds and use that to solve a long standing problem in Finsler geometry. Namely, the complete classification of strongly convex Randers metrics of constant flag curvature.
We show that the evolution equations for a perfect fluid coupled to general relativity in a general lapse and shift,, are Hamiltonian relative to a certain Poisson structure. For the fluid variables, a Lie-Poisson structure associated to the dual of a semi-direct product Lie algebra is used, while the bracket for the gravitational variables has the usual… (More)
Guided by the Hopf fibration, we single out a family (indexed by a positive constant K) of right invariant Riemannian metrics on the Lie group S. Using the Yasuda–Shimada theorem as an inspiration, we determine for each K > 1 a privileged right invariant Killing field of constant length. Each such Riemannian metric pairs with the corresponding Killing field… (More)
This article uses the Berwald connection exclusively, together with its two curvatures, to cut an efficient path across the landscape of Finsler geometry. Its goal is to initiate differential geometers into two key research areas in the field: the search for unblemished “unicorns” and the study of Ricci flow. The exposition is almost self-contained.
By unicorns, I am referring to those mythical single-horned horse-like creatures for which there are only rumoured sightings by a privileged few. A similar situation exists in Finsler differential geometry. There, one has the hierarchy Euclidean ⊂ Minkowskian & Riemannian ⊂ Berwald ⊂ Landsberg among five families of metrics, in which the first two… (More)
For an n-dimensional complete connected Riemannian manifold M with sectional curvature KM ≥ 1 and diameter diam(M) > π2 , and a closed connected totally geodesic submanifold N of M , if there exist points x ∈ N and y ∈ M satisfying the distance d(x, y) > π 2 , then N is homeomorphic to a sphere. We also give a counterexample in 2-dimensional case to the… (More)
A Finsler metric is of sectional flag curvature if its flag curvature depends only on the section. In this article, we characterize Randers metrics of sectional flag curvature. It is proved that any non-Riemannian Randers metric of sectional flag curvature must have constant flag curvature if the dimension is greater than two. 0. Introduction Finsler… (More)