Zhongmin Shen

Lectures on Finsler geometry

Finsler Spaces Finsler m Spaces Co-Area Formula Isoperimetric Inequalities Geodesics and Connection Riemann Curvature Non-Riemannian Curvatures Structure Equations Finsler Spaces of Constant Curvature Second Variation Formula Geodesics and Exponential Map Conjugate Radius and Injectivity Radius Basic Comparison Theorems Geometry of Hypersurfaces Geometry of Metric Spheres Volume Comparison Theorems Morse Theory of Loop Spaces Vanishing Theorems for Homotopy Groups Spaces of Finsler Spaces.

Differential geometry of spray and finsler spaces

Introduction. 1. Minkowski Spaces. 2. Finsler Spaces. 3. SODEs and Variational Problems. 4. Spray Spaces. 5. S-Curvature. 6. Non-Riemannian Quantities. 7. Connections. 8. Riemann Curvature. 9. Structure Equations of Sprays. 10. Structure Equations of Finsler Metrics. 11. Finsler Spaces of Scalar Curvature. 12. Projective Geometry. 13. Douglas Curvature and Weyl Curvature. 14. Exponential Maps. Bibliography. Index.

On the Flag Curvature of Finsler Metrics of Scalar Curvature

The flag curvature of a Finsler metric is called a Riemannian quantity because it is an extension of sectional curvature in Riemannian geometry. In Finsler geometry, there are several non-Riemannian quantities such as the (mean) Cartan torsion, the (mean) Landsberg curvature and the S-curvature, which all vanish for Riemannian metrics. It is important to understand the geometric meanings of these quantities. In the paper, Finsler metrics of scalar curvature (that is, the flag curvature is a scalar function on the slit tangent bundle) are studied and the flag curvature is partially determined when certain non-Riemannian quantities are isotropic. Using the obtained formula for the flag curvature, locally projectively flat Randers metrics with isotropic S-curvature are classified.

Projectively flat Finsler metrics of constant flag curvature

Finsler metrics on an open subset in R n with straight geodesics are said to be projective. It is known that the flag curvature of any projective Finsler metric is a scalar function of tangent vectors (the flag curvature must be a constant if it is Riemannian). In this paper, we discuss the classification problem on projective Finsler metrics of constant flag curvature. We express them by a Taylor expansion or an algebraic formula. Many examples constructed in this paper can be used as models in Finsler geometry.

Finsler Metrics of Constant Positive Curvature on the Lie Group S3

Guided by the Hopf fibration, a family (indexed by a positive constant

On a Class of Landsberg Metrics in Finsler Geometry

K

On projectively related Einstein metrics in Riemann-Finsler geometry

) of right invariant Riemannian metrics on the Lie group

On a Class of Projectively Flat Metrics with Constant Flag Curvature

S^3

ON A CLASS OF PROJECTIVELY FLAT FINSLER METRICS WITH CONSTANT FLAG CURVATURE

is singled out. Using the Yasuda–Shimada paper as an inspiration, a privileged right invariant Killing field of constant length is determined for each

On the Volume of Unit Tangent Spheres in a Pinsler Manifold

K > 1