Riemann-Finsler geometry

By S. Chern, Zhongmin Shen. 2005

615 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/f8880186985de0102640da80d4091e5e3bddcd49

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In Astronomy and Astrophysics

Citations in the corpus

Projective spherically symmetric Finsler metrics with constant flag curvature in Rn (2010, background): “Let r = |x|, u = |y| and v = 〈x, y〉, it is similar with (α, β) metric to compute the fundamental tensor gij of Finsler metric F = φ(r, u, v)[8][2]:”
Weak Finsler structures and the Funk weak metric (2008, neutral): “Modern references on Finsler geometry include [1, 3, 4, 10].”
On a Class of Projectively Flat Metrics with Constant Flag Curvature (2008, neutral): “It is known that F is a Finsler metric if and only if ‖βx‖α < bo for any x ∈ M [7].”
On a class of Douglas metrics (2009, neutral): “The function φ = φ(s) is a C∞ positive function on an open interval (−bo, bo) satisfying φ(s)− sφ′(s) + (ρ(2) − s2)φ′′(s) > 0, |s| ≤ ρ < bo that guarantees that F = αφ(β/α) is a regular positive definite Finsler metric [7].”
Finsler spaces whose geodesics are orbits (2014, background): “Now according to a result of [10], F is a Minkowski norm on n if and only if f(s, t) satisfies the following conditions: fs > 0, ft > 0, fs + 2sfss > 0, ft + 2tftt > 0, fsft − 2ffst > 0.”
On a new class of Finsler metrics (2011, methodology): “Note that for a general (α, β)-metric, the variable b 2 is independent of y, so one can get the above formulas immediately from the corresponding ones of (α, β)-metrics given in [7].”
On Projectively Flat (α, β)-metrics (2009, background): “These two examples inspire us to study projectively flat Finsler metrics F = αφ(β/α) defined by a Riemannian metric α and a 1-form β. Metrics in this form are called (α, β)-metrics.”
On the definition and examples of Finsler metrics (2011, methodology): “The next result is related to one by Chern and Shen in [12], which is discussed as an Appendix (Subsection 4.3).”
Spherically symmetric Finsler metrics in $R^n$ (2010, background): “In fact, many classical Finsler metrics are spherically symmetric [2] [3].”
On two curvature-driven problems in Riemann–Finsler geometry (2007, background): “Many more interesting examples in this category are given in [Chern–Shen 2005], together with an informative exposition.”
The S-curvature of homogeneous Randers spaces (2009, neutral): “According to the formula of the Scurvature in local coordinate systems [2], we need to compute the following quantities at the origin: I.”