Finsler Geometry| 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].
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 26 dec 2020