Zermelo navigation on Riemannian manifolds

By David Bao, C. Robles, Zhongmin Shen. 2003

442 citations recorded by the source corpus.

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.

Subfields: Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/727e76d184bfbd9f7fb899203392f3811d72444d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Astronomy and Astrophysics

Citations in the corpus

On two curvature-driven problems in Riemann–Finsler geometry (2007, background): “• Randers metrics with constant flag curvature have been classified in [Bao et al. 2004], with a historical account of the problem in [Bao 2004].”
RANDERS METRICS OF SCALAR FLAG CURVATURE (2009, background): “This leads to the classification of Randers metrics of constant flag curvature [3].”
On Some Non-Riemannian Quantities in Finsler Geometry (2013, background): “There are Randers metrics on S with positive constant curvature and Ξ = 0 [13] [3].”
Weak Finsler structures and the Funk weak metric (2008, background): “hen, p = √ g + µω is the Lagrangian of a Finsler structure on M. Such a Finlser structure is usually called a Randers metric on M, and it has applications on physics (cf. e.g. [3] §11.3, and see also [4] for the relation of this metric with the Zermelo navigation problem.) Lemma 5.5. Let Ωe be a wea”
Finsler metrics of scalar flag curvature with special non-Riemannian curvature properties (2008, background): “In particular, Randers metrics of constantflag curvature have been classified [ 3 ].”
A Finsler geodesic spray paradigm for wildfire spread modelling (2016, methodology): “We refer to [48, 49, 50, 47, 51, 4, 52, 53, 54, 8, 55, 56, 57, 5, 58, 59, 60] for details on Randers spaces and on the Zermelo navigation problem.”
On the Flag Curvature of Finsler Metrics of Scalar Curvature (2003, background): “tisfy a system of PDEs [4] (i.e., equation (35) below with c(x) = c). Recently, Bao-Robles-Shen have classiﬁed Randers metrics of constant curvature via the navigation problem in Riemannian manifolds [5]. A Finsler metric is said to be locally projectively ﬂat if at any point there is a local coordi”
Finsler manifolds with nonpositive flag curvature and constant S-curvature (2003, background): “. An n-dimensional Finsler metric is said to have constant S-curvature if S= (n+1)cF for some constant c. It is known that every Randers metric of constant ﬂag curvature has constant S-curvature [3], [4]. This is one of our motivations to consider Finsler metrics of constant S-curvature. In this pap”
Nonreversible Finsler Metrics of Positive Flag Curvature (2004, methodology): “For example, in contrast to the Riemannian case, there are Finsler metrics of constant positive flag curvature on spheres that are not isometric to the standard Riemannian metric; see [Bao et al. 2003; Bryant 2002; Shen 2002] for their characterization.”
On a new class of Finsler metrics (2011, methodology): “By Zermelo’s viewpoint [2], we can obtain new Minkowski norms by shifting the indicatrix of an (α, β)-norm.”
ON COMPLEX FINSLER SPACES WITH RANDERS METRIC (2009, methodology): “It was not our goal here to study complex Randers spaces with constant holomorphic curvature or the Zermelo navigation problem ([8]), to give some examples.”
Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry (2008, background): “It would appear that we have not recovered all of the constant flag curvature manifolds of [35], since they find in addition the possibility that h is flat and W is a homothety.”