On the Flag Curvature of Finsler Manifolds

Abstract

In this paper, firstly we prove that all R-quadratic manifolds with nonzero scalar flag curvature must be Riemannian spaces with constant sectional curvature. We introduce the definition of the scalar curvature condition. We can prove that if (M, F) is a Randers space with constant flag curvature K satisfying the scaler curvature condition, then (M, F) is either a locally Minkowskian space with $K=0$ or a Riemannian space with constant sectional curvature $K\ne 0$.

Keywords

R-quadratic manifolds / Randers spaces / Scalar curvature condition / Flag curvature / 53C60 / 53B40

References

[1] Akbar-ZadehH. Sur les espaces de Finsler $\grave{a}$ courbures sectionnelles constantes. Acad. Roy. Belg. Bull. Cl. Sci., 1988, 74: 281-322

[2] Bao, D., Robles, C.: On Ricci curvature and flag curvature in Finsler geometry. In” A Sampler of Finsler geometry MSRI series Camb. Univ. Press 50, 197–259 (2004)

[3] BaoD, ChernSS, ShenZAn introduction to Riemann-Finsler geometry, Graduate Text in Mathematics 200, 2000, New York. Springer-Verlag.

[4] BerwaldL. ${\ddot{U}}$ber die n-dimensionalen Geometrien konstanter Kr$\ddot{u}$mmung, in denen die Geraden die k$\ddot{u}$rzesten sind. Math. Z., 1929, 30: 449-469

[5] BryantRFinsler structures on the 2-sphere satisfying $K=1$, Finsler Geometry, Contemporary Mathematics 196, 1996, Amer. Math. Soc.. Providence, RI. 2740

[6] KimC, YimJ. Finsler manifolds with positive constant flag curvature. Geom. Dedicata, 2003, 98: 47-56

[7] Li, J.: Compact hypersurfaces in Randers space. Sc. Norm. Super. Pisa Cl. Sci, Ann (2020). https://doi.org/10.2422/2036-2145.201711-005

[8] MatsumotoMFoundations of Finsler geometry and special Fisler spaces, 1986, Japan. Kaiseisha Press.

[9] ShenZ. Projectively flat Randers metrics of constant flag curvature. Math. Ann., 2003, 325: 19-30

Funding

National Natural Science Foundation of China(11871405)

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature