On the Flag Curvature of Finsler Metrics of Scalar Curvature

By Xinyue Chen, X. Mo, Zhongmin Shen. 2003

128 citations recorded by the source corpus.

The flag curvature of a Finsler metric is called a Riemannian quantity because it is an extension of sectional curvature in Riemannian geometry. In Finsler geometry, there are several non‐Riemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the S‐curvature, which all vanish for Riemannian metrics. It is important to understand the geometric meanings of these quantities. In the paper, Finsler metrics of scalar curvature (that is, the flag curvature is a scalar function on the slit tangent bundle) are studied and the flag curvature is partially determined when certain non‐Riemannian quantities are isotropic. Using the obtained formula for the flag curvature, locally projectively flat Randers metrics with isotropic S‐curvature are classified.

Subfields: Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/dd5baff9354ddd6a38852efea26067505e41d2da

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In Astronomy and Astrophysics

Citations in the corpus

On Some Non-Riemannian Quantities in Finsler Geometry (2013, background): “1 ([4, 12]) Let F be a Finsler metric of scalar flag curvature on an ndimensional manifold M.”
Finsler metrics of scalar flag curvature with special non-Riemannian curvature properties (2008, background): “The first row is the relationship between the S-curvature and the flag curvature which is established in [4].”
A class of Finsler metrics with isotropic S-curvature (2009, methodology): “where σ = σ(x) and c̃ = c̃(x) are scalar functions with c − c̃ = constant [4].”
On Randers metrics with isotropic S-curvature (2008, background): “It is a non-Riemannian quantity which interacts with the flag curvature in a delicate way [5] [10] [19].”
RANDERS METRICS OF SCALAR FLAG CURVATURE (2009, background): “Thus the flag curvature is the analogue of sectional curvature in Riemannian geometry (see [4, 6]).”