A Schur type lemma for the mean Berwald curvature in Finsler geometry

In this short paper, we study a symmetric covariant tensor in Finsler geometry, which is called the mean Berwald curvature. We first investigate the geometry of the fibres as the submanifolds of the tangent sphere bundle on a Finsler manifold. Then we prove that if the mean Berwald curvature is isotropic along fibres, then the Berwald scalar curvature is constant along fibres.     

S-curvature of isotropic Berwald metrics

Isotropic Berwald metrics are as a generalization of Berwald metrics. Shen proved that every Berwald metric is of vanishing S-curvature. In this paper, we generalize this fact and prove that every isotropic Berwald metric is of isotropic S-curvature. Let F = α + β be a Randers metric of isotropic Berwald curvature. Then it corresponds to a conformal vector field through navigation representation.     

All regular Landsberg metrics are Berwald

This article negatively answers the long standing problem concerning the existence of non-Berwald Landsberg metrics.     

Hypersurfaces with constant scalar curvature and constant mean curvature

Non-spherical hypersurfaces inE ⁴ with non-zero constant mean curvature and constant scalar curvature are the only hypersurfaces possessing the following property: Its position vector can be written as a sum of two non-constant maps, which are eigenmaps of the Laplacian operator with corresponding eigenvalues the zero and a non-zero constant.     

On Kropina metrics of scalar flag curvature

We classify Kropina metrics of weakly isotropic flag curvature in dimension greater than two. Moreover, we prove that every Einstein Kropina metric in dimension greater than two is a Ricci constant metric with vanishing S-curvature in different way from Zhang and Shen (2013) [14] and prove the three-dimensional rigidity theorem for an Einstein Kropina metric.     

Geography of 4-manifolds with positive scalar curvature

We discuss the geography problem of closed oriented 4-manifolds that admit a Riemannian metric of positive scalar curvature, and use it to survey mathematical work employed to address Gromov’s observation that manifolds with positive scalar curvature tend to be inessential by focusing on the four-dimensional case. We also point out an strengthening of a result of Carr and its extension to the non-orientable realm.     

The scalar curvature equation on

We give existence results for solutions of the prescribed scalar curvature equation on S³, when the curvature function is a positive Morse function and satisfies an index-count condition.     

A classification of unitary invariant weakly complex Berwald metrics of constant holomorphic curvature

In this paper, we give a characterization of strongly pseudoconvex complex Finsler metric F which is unitary invariant. A necessary and sufficient condition for F to be a weakly complex Berwald metric and a necessary and sufficient condition for F to be a weakly Kähler Finsler metric are given, respectively. We also give a classification of unitary invariant weakly complex Berwald metrics which are of constant holomorphic curvatures.     

Finsler metrics with constant （or scalar） flag curvature

A Finsler metric on a manifold M with its flag curvature K is said to be almost isotropic flag curvature if K =3c + σ where σ and c are scalar functions on M.In this paper,we establish the intrinsic re...     

A note on scalar curvature type problems in dimension 4

Let (M,g) be a 4-dimensional compact Riemannian manifold and let a,f be positive smooth functions on M. In this note, we prove that the problem Δgu+a(x)u=f(x)u³ always admits a positive solution, up to a conformal deformation of g. This leads to a geometric obstruction result for the prescribed scalar curvature problem.     

The main invariants of a complex finsler space

In this paper we extend the results obtained in [3], where are investigated the general settings of the two-dimensional complex Finsler manifolds, with respect to a local complex Berwald frame. The geometry of such manifolds is controlled by three real invariants which live on T'M: two horizontal curvature invariants K and W and one vertical curvature invariant I. By means of these invariants are defined both the horizontal and the vertical holomorphic sectional curvatures. The complex Landsberg and Berwald spaces are of particular interest. Complex Berwald spaces coincide with Kähler spaces, in the two – dimensional case. We establish the necessary and sufficient condition under which K is a constant and we obtain a characterization for the Kähler purely Hermitian spaces by the fact K = W = constant and I = 0. For the class of complex Berwald spaces we have K = W = 0. Finally, a classification of two-dimensional complex Finsler spaces for which the horizontal curvature satisfies a special property is obtained.     

The first Dirac eigenvalues on manifolds with positive scalar curvature

We show that on every compact spin manifold admitting a Riemannian metric of positive scalar curvature Friedrich's eigenvalue estimate for the Dirac operator can be made sharp up to an arbitrarily small given error by choosing the metric suitably.

     

Estimates and nonexistence of solutions of the scalar curvature equation on noncompact manifolds

This paper is to study the conformal scalar curvature equation on complete noncompact Riemannian manifold of nonpositive curvature. We derive some estimates and properties of supersolutions of the scalar curvature equation, and obtain some nonexistence results for complete solutions of scalar curvature equation.     

The rigidity theorem for harmonic maps from Berwald manifolds

In this paper, we can prove that any non‐degenerate strongly harmonic map ? from a compact Berwald manifold with nonnegative general Ricci curvature to a Landsberg manifold with non‐positive flag curvature must be totally geodesic, which generalizes the result of Eells and Sampson ([2]).     

de Sitter空间中具有常数量曲率的类空超曲面

研究了de Sitter空间中具有常数量曲率的类空超曲面,得到了曲面Mn关于截面曲率的一个刚性定理,并且额外获得关于de Sitter空间子流形的一个结论.     

THE MAIN INVARIANTS OF A COMPLEX FINSLER SPACE

In this paper we extend the results obtained in [3], where are investigated the general settings of the two-dimensional complex Finsler manifolds, with respect to a local complex Berwald frame. The geometry of such manifolds is controlled by three real invariants which live on T ′M : two horizontal curvature invariants K and W and one vertical curvature invariant I. By means of these invariants are defined both the horizontal and the vertical holomorphic sectional curvatures. The complex Landsberg and Berwald spaces are of particular interest. Complex Berwald spaces coincide with K¨ahler spaces, in the two-dimensional case. We establish the necessary and sufficient condition under which K is a constant and we obtain a characterization for the K¨ahler purely Hermitian spaces by the fact K = W = constant and I = 0. For the class of complex Berwald spaces we have K = W = 0.Finally, a classification of two-dimensional complex Finsler spaces for which the horizontal curvature satisfies a special property is obtained.     

Singular limit laminations, Morse index, and positive scalar curvature

For any 3-manifold M³ and any nonnegative integer g, we give here examples of metrics on M each of which has a sequence of embedded minimal surfaces of genus g and without Morse index bounds. On any spherical space form we construct such a metric with positive scalar curvature. More generally, we construct such a metric with Scal>0 (and such surfaces) on any 3-manifold which carries a metric with Scal>0.