Finsler Geometry

In General > s.a. Gauss-Bonnet Theorem; lie group.
* Idea: "Riemannian geometry without the quadratic restriction" (S S Chern); The concept was included in Riemann's 1854 memoir, but was studied in detail for the first time only in Finsler's 1919 thesis; It can be locally anisotropic, and has been used to model/explain anisotropy in cosmology.
\$ Def: A Finsler geometry (manifold) is a differentiable manifold M with a Finsler norm, a positive-definite, smooth function F: M → $$\mathbb R$$ which is homogeneous of degree 1 and subadditive, i.e., F(λv) = λ F(v) and F(v+w) ≤ F(v) + F(w).
@ General references: Busemann BAMS(50); Rund 59 [historical preface]; Asanov 85; Matsumoto 86; Beil IJTP(89) [class of metrics]; Bejancu 90; Chern NAMS(96); Antonelli 99; Bao et al 00; Shen 01; Antonelli ed-03 [handbook]; Chern & Shen 05; Tamássy DG&A(08) [relationship with metric spaces]; Szilasi 13 [connections and sprays]; Shen & Shen 16 [intro].
@ Related topics: Kozma et al RPMP(06) [twisted products]; Bejancu & Farran RPMP(06) [tangent bundles, positive constant curvature]; Mo DG&A(09) [non-Riemannian invariant H]; Kouretsis et al MMAS(14)-a1301-proc [parallel displacements].
> Online resources: M Dahl's page; MathWorld page; Wikipedia page.
Related topics: see finsler geometry and physics [including Finsler spacetime].

Additional Structure and Special Cases > s.a. spacetime boundaries.
* Randers spaces RFn: Finsler spaces Fn = (M, α + β) equipped with the Cartan non-linear connection, introduced by Roman S Ingarden.
* Ingarden spaces IFn: Finsler spaces Fn = (M, α + β) equipped with the Lorentz non-linear connection, introduced by Radu Miron.
* Berwald spaces: A Finsler space is of Berwald type if its Chern-Rund connection defines an affine connection on the underlying manifold; For positive-definite metrics Szabo's metrizability theorem states that a Berwald space is affinely equivalent to a Riemann space, meaning that its affine connection is the Levi-Civita connection of some Riemannian metric; This result does not extend to indefinite-signature metrics, whose affine structure is instead that of a metric-affine geometry with vanishing torsion; > s.a. Encyclopedia of Math page.
@ Spinors, connections: Vacaru in(96)dg; Vargas & Torr GRG(96); Solov'yov & Vladimirov IJTP(01)mp [N-spinors]; Ikeda RPMP(05); Youssef et al JEMS-a0805 [torsion and curvature of a connection]; Minguzzi IJGMP(14)-a1405 [connections].
@ Homogeneous manifolds: Deng & Hou JPA(04), JPA(06); Latifi & Razavi RPMP(06).
@ Isometries: Li et al a1001 [Killing equation]; Habibi & Razavi JGP(10) [weakly symmetric]; Hohmann JMP(16)-a1505 [symmetry-generating vector fields]; Gallego Torromé & Piccione HJM-a2007 [Lie group structure].
@ Randers spaces: Cheng & Shen 12; Rafie-Rad IJGMP(13) [Riemann curvature]; Brody et al JGP(16)-a1507 [geodesics, Riemannian geometry approach]; Gibbons a1708 [and null geodesics in a stationary Lorentzian spacetime and other relationships].
@ Special cases: Miron RPMP(04), RPMP(06) [Ingarden spaces]; Mo & Yang DG&A(06) [isotropic S-curvature]; Chen et al JGP(13) [a class of Ricci-flat Finsler metrics]; Youssef & Soleiman a1405 [Finsler spaces of scalar curvature], a1610 [more special types].
@ Related topics: Józefowicz & Wolak DG&A(08) [Finslerian foliations of compact manifolds are Riemannian]; Kothawala GRG(14)-a1406 [intrinsic and extrinsic curvatures].

Generalizations > s.a. non-commutative geometry; Riemann-Cartan Structure; types of fiber bundles.
@ Pseudo-Finsler structures: Skákala & Visser IJMPD(10)-a0806, JPCS(09)-a0810 [and birefringent optics], JGP(11) [and bimetric spacetimes].
@ Finsler-Lagrange spaces: Vacaru a0707, JGP(10)-a0709, IJGMP(08)-a0801, Sigma(08)-a0806 [rev, general relativity and string theory]; Miron RPMP(06).
@ Other generalizations: Tanaka PhD-a1310 [Kawaguchi geometry]; Caponio et al a1407 [wind Finslerian structure].