The current state of play in the Landsberg-Berwald problem of Finsler geometry

Authors

  • M. Crampin Orchard Rising, Herrings Lane, Burnham Market, Norfolk, UK

DOI:

https://doi.org/10.17398/

Keywords:

Finsler spaces, Landsberg spaces, Berwald spaces

Abstract

A progress report on the (still unresolved) Landsberg-Berwald problem of Finsler geometry: whether there can be non-Berwaldian regular Landsberg spaces.

Downloads

Download data is not yet available.

References

T. Aikou, Some remarks on Berwald manifolds and Landsberg manifolds, Acta Math. Acad. Paedagog. Nyházi. (N.S.) 26 (2010), 139 – 148.

T. Aikou, Averaged Riemannian metrics and connections with application to locally conformal Berwald manifolds, Publ. Math. Debrecen 81 -(1-2) (2012), 179 – 198.

H. Akbar-Zadeh, Sur les espaces de Finsler à courbures sectionnelles constantes, Acad. Roy. Belg. Bull. Cl. Sci. (5) 74 (1988), 271 – 322.

S. Bácsó, F. Ilosvay, B. Kis, Landsberg spaces with common geodesics, Publ. Math. Debrecen 42 (1-2) (1993), 139 – 144.

S. Bácsó, R. Yoshikawa, Weakly-Berwald spaces, Publ. Math. Debrecen 61 (1-2) (2002), 219 – 231.

D. Bao, On two curvature-driven problems in Riemann-Finsler geometry, in “ Finsler geometry, Sapporo 2005 – in memory of Makoto Matsumoto ”, Adv. Stud. Pure Math., 48, Mathematical Society of Japan, Tokyo, 2007, 19 – 71.

D. Bao, S.-S. Chern, Z. Shen, “ An introduction to Riemann-Finsler geometry ”, Grad. Texts in Math., 200, Springer-Verlag, New York, 2000.

P. Centore, Volume forms in Finsler spaces, Houston J. Math. 25 (1999), 625 – 640.

X. Cheng, The (α, β)-metrics of scalar flag curvature, Differential Geom. Appl. 35 (2014), 361 – 369.

M. Crampin, On Landsberg spaces and the Landsberg-Berwald problem, Houston J. Math. 37 (4) (2011), 1103 – 1124.

M. Crampin, On the construction of Riemannian metrics for Berwald spaces by averaging, Houston J. Math. 40 (3) (2014), 737 – 750.

M. Crampin, A condition for a Landsberg space to be Berwaldian, Publ. Math. Debrecen 93 (1-2) (2018), 143 – 155.

M. Crampin, Finsler spaces of (α, β) type and semi-C-reducibility, Publ. Math. Debrecen 98 (3-4) (2021), 419 – 454.

M. Crampin, Invariant volumes, weakly-Berwald Finsler spaces, and the Landsberg-Berwald problem, Publ. Math. Debrecen 100 (1-2) (2022), 101 – 118.

M. Crampin, Isometries and geodesic invariants of Finsler spaces of (α, β) type, 2022, DOI:10.13140/RG.2.2.10839.55203.

M. Crampin, On parallel transport in Finsler spaces, Publ. Math. Debrecen 92 (3-4) (2023), 343 – 370.

M. Crampin, S-curvature, E-curvature, and Berwald scalar curvature of Finsler spaces, Differential Geom. Appl. 92 (2024), paper no. 102080, 12 pp.

M. Crampin, D.J. Saunders, Holonomy of a class of bundles with fibre metrics, Publ. Math. Debrecen 81 (1-2) (2012), 199 – 234.

S. Deng, Z. Hou, The group of isometries of a Finsler space, Pacific J. Math. 207 (1) (2002), 149 – 155.

S.G. Elgendi, On the classification of Landsberg spherically symmetric Finsler metrics, Int. J. Geom. Methods Mod. Phys. 18 (14) (2021), paper no. 2150232, 26 pp.

H. Feng, Y. Han, M. Li, An equivalence theorem of a class of Minkowski norms and its applications, Sci. China Math. 64 (2021), 1429 – 1446.

H. Feng, M. Li, An equivalence theorem for a class of Minkowski spaces, 2018, arXiv:1812.11938v2.

Y. Ichijyō, Finsler manifolds modeled on a Minkowski space, J. Math. Kyoto Univ. 16 (1976), 639 – 652.

Y. Ichijyō, On holonomy mappings associated with a nonlinear connection, J. Math. Tokushima Univ. 17 (1983), 1 – 9.

L. Kozma, On holonomy groups of Landsberg manifolds, Tensor (N.S.) 62 (1) (2000), 87 – 90.

B. Li, Z. Shen, Ricci curvature tensor and non-Riemannian quantities, Canad. Math. Bull. 58 (3) (2015), 530 – 537.

M. Li, Equivalence theorems of Minkowski spaces and an application in Finsler geometry, Acta Math. Sinica (Chin. Ser.) 62 (2) (2019), 177 – 190 (in Chinese: for the English version see arXiv:1504.04475v2 (2018)).

M. Li, L. Zhang, Properties of Berwald scalar curvature, Front. Math. China 15 (6) (2020), 1143 – 1153.

M. Matsumoto, On C-reducible Finsler spaces, Tensor (N.S.) 24 (1972), 29 – 37.

M. Matsumoto, Theory of Finsler spaces with (α, β)-metric, Rep. Math. Phys. 31 (1992), 43 – 83.

M. Matsumoto, An improved proof of Numata and Shibata’s theorems on Finsler spaces of scalar curvature, Publ. Math. Debrecen 64 (3-4) (2004), 489 – 500.

M. Matsumoto, C. Shibata, On semi-C-reducibility, T -tensor= 0, and S4-likeness of Finsler spaces, J. Math. Kyoto Univ. 19 (1979), 301–314.

V.S. Matveev, M. Troyanov, The Binet-Legendre metric in Finsler geometry, Geom. Topol. 16 (2012), 2135 – 2170.

X. Mo, L. Zhou, The curvatures of spherically symmetric Finsler metrics in R n , 2014, arXiv:1202.4543.

Z. Muzsnay, The Euler-Lagrange PDE and Finsler metrizability, Houston J. Math. 32 (1) (2006), 79 – 98.

Z. Muzsnay, P.T. Nagy, Finsler manifolds with non-Riemannian holonomy, Houston J. Math. 38 (1) (2012), 77 – 92.

B. Najafi, Z. Shen, A. Tayebi, Finsler metrics of scalar flag curvature with special non-Riemannian curvature properties, Geom. Dedicata 131 (2008), 87 – 97.

S. Numata, On Landsberg spaces of scalar curvature, J. Korean Math. Soc. 12 (2) (1975), 97 – 100.

Z. Shen, “ Differential geometry of spray and Finsler spaces ”, Kluwer Academic Publishers, Dordrecht, 2001.

Z. Shen, On R-quadratic Finsler spaces, Publ. Math. Debrecen 58 (1-2) (2001), 263 – 274.

Z. Shen, Landsberg curvature, S-curvature and Riemann curvature, in “ A sampler of Riemann-Finsler geometry ” (edited by David Bao, Robert L. Bryant, Shiing-Shen Chern and Zhongmin Shen), Math. Sci. Res. Inst. Publ., 50, Cambridge University Press, Cambridge, 2004, 303 – 355.

Z. Shen, On a class of Landsberg metrics in Finsler geometry, Canad. J. Math. 61 (2009), 1357 – 1374.

C. Shibata, On the curvature tensor R hijk of Finsler spaces of scalar curvature, Tensor (N.S.) 32 (1978), 311 – 317.

J. Szilasi, R. Lovas, D. Kertész, “ Connections, sprays and Finsler structures ”, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2014.

J. Szilasi, R.L. Lovas, D.Cs. Kertész, Several ways to a Berwald manifold — and some steps beyond, Extracta Math. 26 (2011), 89 – 130.

A. Tayebi, A survey on unicorns in Finsler geometry, J. Math. Com. 2 (2) (2021), 239 – 250.

R.G. Torrome, F. Etayo, On a rigidity condition for Berwald spaces, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. 104 (1) (2010), 69 – 80.

Sz. Vattamány, Projection onto the indicatrix bundle of a Finsler manifold, Publ. Math. Debrecen 58 (1-2) (2001), 193 – 221.

Cs. Vince, A new proof of Szabó’s theorem on the Riemann-metrizability of Berwald manifolds, Acta Math. Acad. Paedagog. Nyházi. (N.S.) 21 (2005), 199 – 204.

M. Xu, S. Deng, The Landsberg equation of a Finsler space, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22 (1) (2021), 31 – 51.

M. Xu, V.S. Matveev, Proof of Laugwitz conjecture and Landsberg unicorn conjecture for Minkowski norms with SO(k) × SO(n − k)-symmetry, Canad. J. Math. 74 (5) (2022), 1486 – 1516.

K. Yano, “ The theory of Lie derivatives and its applications ”, Bibliotheca Mathematica 3, North-Holland, Amsterdam, 1957.

C. Yu, H. Zhu, On a new class of Finsler metrics, Differential Geom. Appl. 29 (2) (2011), 244 – 254.

L. Zhou, Spherically symmetric Finsler metrics in R n , Publ. Math. Debrecen 80 (1-2) (2012), 67 – 77.

S. Zhou, J. Wang, B. Li, On a class of almost regular Landsberg metrics, Sci. China Math. 62 (2019), 935 – 960.

Downloads

Published

2024-02-20

Issue

Section

Articles in press

How to Cite

The current state of play in the Landsberg-Berwald problem of Finsler geometry. (2024). Extracta Mathematicae. https://doi.org/10.17398/