In this paper, we study Zermelo navigation on Riemannian manifolds and use that to solve a long standing problem in Finsler geometry, namely the complete classification of strongly convex Randers metrics of constant flag curvature.

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Zermelo navigation on Riemannian manifolds

Introduction 198 1. Flag and Ricci Curvatures 199 1.1. Finsler metrics 199 1.2. Flag curvature 207 1.3. Ricci curvature 214 2. Randers Metrics in Their Defining Form 218 2.1. Basics 218 2.2. Characterising Einstein–Randers metrics 220 2.3. Characterising constant flag curvature Randers metrics 228 3. Randers Metrics Through Zermelo Navigation 230 3.1. Zermelo navigation 230 3.2. Navigation description of curvature conditions 236 3.3. Issues resolved by the navigation description 241 4. Einstein Metrics of Nonconstant Flag Curvature 244 4.1. Examples with Riemannian–Einstein navigation data 244 4.2. Examples with Kähler–Einstein navigation data 247 4.3. Rigidity and a Ricci-flat example 251 5. Open Problems 254 5.1. Randers and beyond 254 5.2. Chern’s question 255 5.3. Geometric flows 256 Acknowledgments 256 References 256

Ricci and Flag Curvatures in Finsler Geometry

The monograph deals with the theory of conformal, geodesic,
holomorphically projective, F-planar and others mappings and
transformations of manifolds with affine connection,
Riemannian, Kahler and Riemann-Finsler manifolds. Concretely,
the monograph treats the following: basic concepts of
topological spaces, the theory of manifolds with affine
connection (particularly, the problem of semigeodesic
coordinates), Riemannian and Kahler manifolds (reconstruction
of a metric, equidistant spaces, variational problems in
Riemannian spaces, SO(3)-structure as a model of statistical
manifolds, decomposition of tensors), the theory of
differentiable mappings and transformations of manifolds (the
problem of metrization of affine connection, harmonic
diffeomorphisms), conformal mappings and transformations
(especially conformal mappings onto Einstein spaces, conformal
transformations of Riemannian manifolds), geodesic mappings
(GM; especially geodesic equivalence of a manifold with affine
connection to an equiaffine manifold), GM onto Riemannian
manifolds, GM between Riemannian manifolds (GM of equidistant
spaces, GM of Vn(B) spaces, its field of symmetric linear
endomorphisms), GM of special spaces, particularly Einstein,
Kahler, pseudosymmetric manifolds and their generalizations,
global geodesic mappings and deformations, GM between
Riemannian manifolds of different dimensions, global GM,
geodesic deformations of hypersurfaces in Riemannian spaces,
some applications of GM to general relativity, namely three
invariant classes of the Einstein equations and geodesic
mappings, F-planar mappings of spaces with affine connection,
holomorphically projective mappings (HPM) of Kahler manifolds
(fundamental equations of HPM, HPM of special Kahler manifolds,
HPM of parabolic Kahler manifolds, almost geodesic mappings,
which generalize geodesic mappings, Riemann-Finsler spaces and
their geodesic mappings, geodesic mappings of Berwald spaces
onto Riemannian spaces.

Differential Geometry of Special Mappings

In this paper, we study a class of Finsler metrics defined by a Riemannian metric and a 1-form We characterize these metrics with isotropic S-curvature

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A class of Finsler metrics with isotropic S-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 this paper, we study Finsler metrics of scalar curvature (i.e., the flag curvature is a scalar function on the slit tangent bundle) and partially determine the flag curvature when certain non-Riemannian quantities are isotropic. Using the obtained formula for the flag curvature, we classify locally projectively flat Randers metrics with isotropic S-curvature.

On the Flag Curvature of Finsler Metrics of Scalar Curvature

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Randers metrics with special curvature properties

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.

/pdf/on-the-flag-curvature-of-finsler-metrics-of-scalar-curvature-26d709yb72.pdf

We show that if two Riemannian metrics \({\tilde g}\)and g are pointwiseprojectively equivalent and the Ricci curvatures satisfy \(\widetilde{Ric}\)≤ Ric, then the projective equivalence is trivialprovided that g is complete. In this case, \({\tilde g}\)is parallel with respect to g and the Riemann curvatures of g and \({\tilde g}\) are equal.The Ricci curvature condition can be weakened when the manifold iscompact. This rigidity theorem actually holds for more general geometricstructures, such as Finsler metrics and sprays. In this paper, we willalso discuss several examples and show that the completeness of g cannot be dropped.

A Comparison Theorem on the Ricci Curvature in Projective Geometry

In this paper, we discuss the relationship between the flag curvature and some non-Riemannian quantities of Finsler metrics of scalar curvature. In particular, we characterize projectively flat Finsler metrics with isotropic S-curvature.

Xinyue Chen

Papers

On the Flag Curvature of Finsler Metrics of Scalar Curvature

Randers metrics with special curvature properties

On the Flag Curvature of Finsler Metrics of Scalar Curvature

A Comparison Theorem on the Ricci Curvature in Projective Geometry

Finsler metrics with special curvature properties