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 RF__^{n}:
Finsler spaces F^{n} = (*M*, *α* + *β*)
equipped with the Cartan non-linear connection, introduced by Roman S Ingarden.

* __Ingarden spaces IF__^{n}:
Finsler spaces F^{n} = (*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].

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send feedback and suggestions to bombelli at olemiss.edu – modified 26 dec 2020