Monochromatic metrics are generalized Berwald
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1 Monochromatic metrics are generalized Berwald Nina Bartelmeß Friedrich-Schiller-Universität Jena Closing Workshop GRK 1523 "Quantum and Gravitational Fields" 12/03/2018 ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
2 Contents 1 Introduction and Statement 2 History 3 Statement and Idea of Proof 4 Global Results 5 Landsberg Unicorn Problem ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
3 Overview 1 Introduction and Statement 2 History 3 Statement and Idea of Proof 4 Global Results 5 Landsberg Unicorn Problem ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
4 Introduction: Finsler metric Definition ([MT12]) A Finsler metric on a smooth manifold M is a continous function F : TM [0, ) such that for every point x M the restriction F x = F Tx M on the tangent space at x is a Minkowski norm, that is F x is positively homogenous, convex and it vanishes only at ξ = 0: 1 F x (λξ) = λf x0 (ξ) for all λ 0, ξ T x M 2 F x (ξ + η) F x0 (ξ) + F x0 (η) for all ξ, η T x M 3 F x (ξ) = 0 if and only if ξ = 0 Notation: F x (ξ) = F (x, ξ). ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
5 Introduction: Finsler metric (M, F ) is called Finsler manifold. ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
6 Introduction: Finsler metric (M, F ) is called Finsler manifold. F x is a norm if and only if it is reversible, i.e. F x (ξ) = F x ( ξ) ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
7 Introduction: Finsler metric (M, F ) is called Finsler manifold. F x is a norm if and only if it is reversible, i.e. F x (ξ) = F x ( ξ) indicatrices Ω x = {y T x M F x (y) = 1} enclose a strictly convex body Figure: 2-dimensional Finsler metric: unit balls are convex ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
8 Introduction: Examples Most famous example: Riemannian manifolds M n-dimensional smooth manifold, g Riemannian metric on M. This g defines a Finsler metric by F x (y) = g x (y, y) ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
9 Introduction: Examples Most famous example: Riemannian manifolds M n-dimensional smooth manifold, g Riemannian metric on M. This g defines a Finsler metric by F x (y) = g x (y, y) indicatrix is an ellipse which is symmetric with respect to the origin Figure: 2-dimensional Riemannian metric: all unit balls are ellipses ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
10 Introduction: Examples Motivational example: Windy surfaces and Randers metrics Take a surface with Riemannian metric g and a wind vector field W. ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
11 Introduction: Examples Motivational example: Windy surfaces and Randers metrics Take a surface with Riemannian metric g and a wind vector field W. Without wind, S x = {ξ T x M g x (ξ, ξ) = 1} measures how far one can move in a time unit With the wind W, in a time unit one can move to S x + W x ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
12 Introduction: Examples Motivational example: Windy surfaces and Randers metrics Take a surface with Riemannian metric g and a wind vector field W. Without wind, S x = {ξ T x M g x (ξ, ξ) = 1} measures how far one can move in a time unit With the wind W, in a time unit one can move to S x + W x Fact: There is a Finsler metric with indicatrix Ω x = S x + W x, precisely a so-called Randers metric: F (x, y) = g ij (x)y i y j + β i (x)y i }{{}}{{} Riemannian metric 1-form β=β i (x)dx i ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
13 Indicatrices Ω x in each T x M are shifted ellipses Figure: 2-dimensional Randers metric ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
14 Indicatrices Ω x in each T x M are shifted ellipses Figure: 2-dimensional Randers metric If β is small (g ij β i β j < 1), Ω x encloses the origin and F is a Finsler metric. ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
15 Introduction: generalized Berwald metric : connection on M, in local coordinates given by: Γ i jk parallel vector field: X Y = 0 parallel along a curve γ: γ Y = 0 parallel transport: P γ : T γ(t0)m T γ(t1)m, with γ : [t 0, t 1 ] M smooth curve Nina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
16 Introduction: generalized Berwald metric Definition ([MT12]) A Finsler metric F is called generalized Berwald metric if there exists an affine connection such that its associated parallel transport preserves the Finsler function. ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
17 Introduction: generalized Berwald metric Definition ([MT12]) A Finsler metric F is called generalized Berwald metric if there exists an affine connection such that its associated parallel transport preserves the Finsler function. let γ : [0, 1] M be a smooth curve from x to y and P γ : T x M T y M its associated parallel transport, then: F y (P γ (ξ)) = F x (ξ) for all ξ T x M ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
18 Introduction: generalized Berwald metric Definition ([MT12]) A Finsler metric F is called generalized Berwald metric if there exists an affine connection such that its associated parallel transport preserves the Finsler function. let γ : [0, 1] M be a smooth curve from x to y and P γ : T x M T y M its associated parallel transport, then: F y (P γ (ξ)) = F x (ξ) for all ξ T x M if is torsion-free: Berwald metric ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
19 Introduction: monochromatic metric Definition ( [Bao07]) A Finsler metric F is called monochromatic if for every two points there exists a linear isomorphism between the tangent spaces that preserves the Finsler function. A(x): T x0 M T x M, ξ T x0 M : F (x, A(x)ξ) = F (x 0, ξ) There are no assumptions on the smoothness of A(x). ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
20 Theorem (V.S.Matveev, N.B.) Monochromatic metrics are generalized Berwald. ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
21 Theorem (V.S.Matveev, N.B.) Monochromatic metrics are generalized Berwald. the converse of this statement is trivial: ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
22 Theorem (V.S.Matveev, N.B.) Monochromatic metrics are generalized Berwald. the converse of this statement is trivial: Proof: there exists a connection such that the associated parallel transport preserves the Finsler function. ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
23 Theorem (V.S.Matveev, N.B.) Monochromatic metrics are generalized Berwald. the converse of this statement is trivial: Proof: there exists a connection such that the associated parallel transport preserves the Finsler function. This parallel transport is a linear isomorphism between two tangent spaces that preserves the Finsler function Statement. ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
24 Theorem (V.S.Matveev, N.B.) Monochromatic metrics are generalized Berwald. the converse of this statement is trivial: Proof: there exists a connection such that the associated parallel transport preserves the Finsler function. This parallel transport is a linear isomorphism between two tangent spaces that preserves the Finsler function Statement. we will see: if A(x) depends smoothly on x the statement that a monochromatic metric is generalized Berwald is trivial too ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
25 Overview 1 Introduction and Statement 2 History 3 Statement and Idea of Proof 4 Global Results 5 Landsberg Unicorn Problem ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
26 Riemann ( ) and Weyl ( ) already Weyl dealt with an equivalent concept in his commentary on Riemann s habilitation address (1919) Nina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
27 Riemann ( ) and Weyl ( ) already Weyl dealt with an equivalent concept in his commentary on Riemann s habilitation address (1919) Weyl s notion of a metrically homogeneous metric: [Lau65] Nina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
28 Riemann ( ) and Weyl ( ) already Weyl dealt with an equivalent concept in his commentary on Riemann s habilitation address (1919) Weyl s notion of a metrically homogeneous metric: [Lau65] A metric is called metrically homogeneous if there exist linear mappings onto each of the tangent spaces at different points such that F (x, ξ) = f (B i k(x)ξ k ) where f is a Minkowski norm in a fixed one of the tangent spaces. It is not clear if Weyl assumed that B(x) to depend smoothly on x. Nina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
29 Weyl s view on the universe looking for infinitesimal structures to describe space ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
30 Weyl s view on the universe looking for infinitesimal structures to describe space why only considering Riemannian metrics? Nina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
31 Weyl s view on the universe looking for infinitesimal structures to describe space why only considering Riemannian metrics? could also be a Finsler metric Nina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
32 Weyl s view on the universe looking for infinitesimal structures to describe space why only considering Riemannian metrics? could also be a Finsler metric isotropic point of view: universe is equal in all points Problem: a typical metric is not equal in all points Nina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
33 Weyl s view on the universe looking for infinitesimal structures to describe space why only considering Riemannian metrics? could also be a Finsler metric isotropic point of view: universe is equal in all points Problem: a typical metric is not equal in all points the Finsler metric we are looking for should be monochromatic/generalized Berwald, i.e. there exist isomorphisms between all tangent spaces that preserve the Finsler metric Nina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
34 Laugwitz [Lau65] and Ichijyo [Ich76] Laugwitz (1965) and later independently Ichijyo (1976) studied metrically homogeneous metrics. They explicitly assume B(x) to depend smoothly on x. ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
35 Laugwitz [Lau65] and Ichijyo [Ich76] Laugwitz (1965) and later independently Ichijyo (1976) studied metrically homogeneous metrics. They explicitly assume B(x) to depend smoothly on x. Theorem(Laugwitz, Ichijyo) Every metrically homogeneous space is generalized Berwald. ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
36 Laugwitz [Lau65] and Ichijyo [Ich76] Laugwitz (1965) and later independently Ichijyo (1976) studied metrically homogeneous metrics. They explicitly assume B(x) to depend smoothly on x. Theorem(Laugwitz, Ichijyo) Every metrically homogeneous space is generalized Berwald. Proof: It holds F (x, ξ) = f (B i k(x)ξ k ) (metrically homogeneous) ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
37 Laugwitz [Lau65] and Ichijyo [Ich76] Laugwitz (1965) and later independently Ichijyo (1976) studied metrically homogeneous metrics. They explicitly assume B(x) to depend smoothly on x. Theorem(Laugwitz, Ichijyo) Every metrically homogeneous space is generalized Berwald. Proof: It holds F (x, ξ) = f (B i k(x)ξ k ) (metrically homogeneous) consider inverses A i k(x) where A i r B r k = δ i k ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
38 Laugwitz [Lau65] and Ichijyo [Ich76] Laugwitz (1965) and later independently Ichijyo (1976) studied metrically homogeneous metrics. They explicitly assume B(x) to depend smoothly on x. Theorem(Laugwitz, Ichijyo) Every metrically homogeneous space is generalized Berwald. Proof: It holds F (x, ξ) = f (B i k(x)ξ k ) (metrically homogeneous) consider inverses A i k(x) where A i r B r k = δ i k A i k(x) are isometric linear mappings from the tangent space in x 0 to the other points x M construct a length preserving parallel transport/connection out of that: Γ i jr = A i k,j(x)b k r ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
39 Overview 1 Introduction and Statement 2 History 3 Statement and Idea of Proof 4 Global Results 5 Landsberg Unicorn Problem ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
40 Assumptions and statement consider Finsler manifolds such that two tangent spaces are isomorphic as normed spaces (monochromatic) F (x, A(x)ξ) = F (x 0, ξ), A(x): T x0 M T x M, ξ T x0 M ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
41 Assumptions and statement consider Finsler manifolds such that two tangent spaces are isomorphic as normed spaces (monochromatic) F (x, A(x)ξ) = F (x 0, ξ), A(x): T x0 M T x M, ξ T x0 M we do not assume the field A(x) to be smooth (in contrary to the statements pointed out so far) ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
42 Assumptions and statement consider Finsler manifolds such that two tangent spaces are isomorphic as normed spaces (monochromatic) F (x, A(x)ξ) = F (x 0, ξ), A(x): T x0 M T x M, ξ T x0 M we do not assume the field A(x) to be smooth (in contrary to the statements pointed out so far) we simply assume the Finsler function F to be a smooth enough function Theorem (V.S.Matveev, N.B. [MB17]) Monochromatic metrics are generalized Berwald. ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
43 Idea of Proof Step 1: locally one can choose A x such that it depends smoothly on x use of the Implicit Function Theorem Step 2: construction of the associated connection more or less repetition of the proof of Laugwitz and Ichijyo use of the partition of unity to obtain a global connection Nina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
44 Overview 1 Introduction and Statement 2 History 3 Statement and Idea of Proof 4 Global Results 5 Landsberg Unicorn Problem ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
45 Existence of generalized Berwald metrics on 2- and 3-dimensional closed manifolds Theorem: The only two-dimensional closed manifolds admitting non-riemannian generalized Berwald metrics are the Torus and the Klein bottle. ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
46 Existence of generalized Berwald metrics on 2- and 3-dimensional closed manifolds Theorem: The only two-dimensional closed manifolds admitting non-riemannian generalized Berwald metrics are the Torus and the Klein bottle. Theorem: Every three-dimensional compact manifold with Euler characteristic zero admits a non-riemannian generalized Berwald metric. ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
47 Overview 1 Introduction and Statement 2 History 3 Statement and Idea of Proof 4 Global Results 5 Landsberg Unicorn Problem ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
48 Landsberg Unicorn Problem one of the main open problems in Finsler geometry ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
49 Landsberg Unicorn Problem one of the main open problems in Finsler geometry Landsberg metric: a Finsler metric is of Landbserg type, when the Landsberg curvature vanishes ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
50 Landsberg Unicorn Problem one of the main open problems in Finsler geometry Landsberg metric: a Finsler metric is of Landbserg type, when the Landsberg curvature vanishes Question: Does vanishing of the Landsberg curvature imply that the metric is Berwald? ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
51 Landsberg Unicorn Problem one of the main open problems in Finsler geometry Landsberg metric: a Finsler metric is of Landbserg type, when the Landsberg curvature vanishes Question: Does vanishing of the Landsberg curvature imply that the metric is Berwald? Landsberg curvature can be expressed via the Berwald curvature ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
52 Landsberg Unicorn Problem one of the main open problems in Finsler geometry Landsberg metric: a Finsler metric is of Landbserg type, when the Landsberg curvature vanishes Question: Does vanishing of the Landsberg curvature imply that the metric is Berwald? Landsberg curvature can be expressed via the Berwald curvature and Berwald metrics are also characterised by vanishing Berwald curvature ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
53 Landsberg Unicorn Problem one of the main open problems in Finsler geometry Landsberg metric: a Finsler metric is of Landbserg type, when the Landsberg curvature vanishes Question: Does vanishing of the Landsberg curvature imply that the metric is Berwald? Landsberg curvature can be expressed via the Berwald curvature and Berwald metrics are also characterised by vanishing Berwald curvature every Berwald metric must be of Landsberg type ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
54 Landsberg Unicorn Problem one of the main open problems in Finsler geometry Landsberg metric: a Finsler metric is of Landbserg type, when the Landsberg curvature vanishes Question: Does vanishing of the Landsberg curvature imply that the metric is Berwald? Landsberg curvature can be expressed via the Berwald curvature and Berwald metrics are also characterised by vanishing Berwald curvature every Berwald metric must be of Landsberg type main opinion at the end of 20 th century: Landsberg metrics that are not Berwald do not exist! we call them unicorns ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
55 Hunting unicorns.. [BCS00] hunting unicorns: searching for Landsberg metrics that are not of Berwald type ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
56 Hunting unicorns.. [BCS00] hunting unicorns: searching for Landsberg metrics that are not of Berwald type solutions were presented, but mistakes were found ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
57 Hunting unicorns.. [BCS00] hunting unicorns: searching for Landsberg metrics that are not of Berwald type solutions were presented, but mistakes were found breakthrough [Asa06]: Asanov produces y local examples of unicorns in dimensions 3 ina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
58 Hunting unicorns.. [BCS00] hunting unicorns: searching for Landsberg metrics that are not of Berwald type solutions were presented, but mistakes were found breakthrough [Asa06]: Asanov produces y local examples of unicorns in dimensions 3 still no such unicorns for dimension 2 nor global results known Nina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
59 Thank you... Thank you very much for your attention! Nina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
60 G.S. Asanov. Finsleroid-Finsler spaces of positive-definite and relativistic types. Rep. Math. Phys., (58), D. Bao. On two curvature-driven problems in Riemann-Finsler geometry. Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo, (48), David Bao, Shiin-Shen Chern, and Zhongmin Shen. An Introduction to Riemann-Finsler Geometry. Springer Science & Business Media, Berlin Heidelberg, first edition edition, Yoshihiro Ichijyo. Finsler manifolds modeled on a Minkowski space. J. Math. Kyoto Univ., (16), Detlef Laugwitz. Differential and Riemannian Geometry. Academic Press, New York, Nina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
61 V. Matveev and N. Bartelmeß. Monochromatic metrics are generalized Berwald. to appear in: Differential Geometry and its Applications, arxiv: v1. V. Matveev and M. Troyanov. The Binet-Legendre metric in Finsler geometry. Geometry and Topology, (16), arxiv: Nina Bartelmeß (Friedrich-Schiller-Universität Jena Closing Monochromatic Workshop GRK metrics 1523 are "Quantum generalized andberwald Gravitational Fields") 12/03/ / 27
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