A note on the fundamental group of Finsler manifolds with nonnegative flag curvature

In this note we prove that the fundamental group of any forward complete Finsler manifold with nonnegative flag curvature is finitely generated provided the line integral of T-curvature is small. In particular, the fundamental group of any forward complete Berwald manifold with nonnegative flag curvature is finitely generated.     

The existence of harmonic maps from Finsler manifolds of Riemannian manifolds

In this paper,we consider the existence of harmonic maps from a Finsler manifold and study the characterisation of harmonic maps,in the spirit of Ishihara.Using heat quation method we show that any map from a compact Finsler manifold M to a compact Riemannian manifold with non-positive sectional curvature can be deformed into a harmonic map which has minimum energy in its homotopy class.     

The existence of harmonic maps from Finsler manifolds to Riemannian manifolds

In this paper,we consider the existence of harmonic maps from a Finsler man-ifold and study the characterisation of harmonic maps,in the spirit of lshihara.Using heatequation method we show that any map from a compact Finsler manifold M to a com-pact Riemannian manifold with non-positive sectional curvature can be deformed into aharmonic map which has minimum energy in its homotopy class.     

Foliations on the tangent bundle of Finsler manifolds

Let M be a smooth manifold with Finsler metric F,and let T M be the slit tangent bundle of M with a generalized Riemannian metric G,which is induced by F.In this paper,we prove that (i) (M,F) is a Landsberg manifold if and only if the vertical foliation F V is totally geodesic in (T M,G);(ii) letting a:= a(τ) be a positive function of τ=F 2 and k,c be two positive numbers such that c=2 k(1+a),then (M,F) is of constant curvature k if and only if the restriction of G on the c-indicatrix bundle IM (c) is bundle-like for the horizontal Liouville foliation on IM (c),if and only if the horizontal Liouville vector field is a Killing vector field on (IM (c),G),if and only if the curvature-angular form Λ of (M,F) satisfies Λ=1-a 2/R on IM (c).     

Second variation of harmonic maps between Finsler manifolds

The first and second variation formulas of the energy functional for a nonde-generate map between Finsler manifolds is derived. As an application, some nonexistence theorems of nonconstant stable harmonic maps from a Finsler manifold to a Riemannian manifold are given.     

Hodge decomposition theorem on strongly Kahler Finsler manifolds Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Faran posed an open problem about analysis on complex Finsler spaces: Is there an analogue of the (?)-Laplacian? Is there an analogue of Hodge theory? Under the assumption that (M,F) is a compact strongly Kahler Finsler manifold, we define a (?)-Laplacian on the base manifold. Our result shows that the well-known Hodge decomposition theorem in Kahler manifolds is still true in the more general compact strongly Kahler Finsler manifolds.     

Hodge theorem for the natural projection of complex horizontal Laplacian on complex Finsler manifolds

Let M be a compact complex manifold with a complex Finsler metric F. We define a natural projection of complex horizontal Laplacian on M: it is independent of the fiber coordinate. By using Sobolev space theory and spectral resolution theory in Hilbert space, we prove the Hodge theorem for the natural projection of complex horizontal Laplacian on M.     

On the maximum principle on complete Finsler manifolds

In this paper, we generalize Omori–Yau maximum principle to Finsler geometry. As an application, we obtain some Liouville-type theorems of subharmonic functions on forward complete Finsler manifolds.     

On the fundamental equations of homogeneous Finsler spaces

By introducing the notion of single colored Finsler manifold, we deduce the curvature formulas of a homogeneous Finsler space. It results in a set of fundamental equations that are more elegant than the Riemannian case. Several applications of the equations are also supplied.     

On fundamental groups of symplectically aspherical manifolds II: Abelian groups

We describe all abelian groups which can appear as the fundamental groups of closed symplectically aspherical manifolds. The proofs use the theory of symplectic Lefschetz fibrations.      

Finsler几何中体积形式与子流形

本文对一般的Finsler体积元讨论了Finsler子流形几何,证明了关于任意Finsler体积元, Minkowski空间中不存在闭定向的极小子流形,关键在于对任意的Finsler体积元,沈忠民的方法仍然有效．对于特殊Randers空间中的子流形,给出了其体积增长估计,从而得到了Randers空间可以极小浸入到特殊Randers空间的一个必要条件．     

Growth of fundamental groups and isoembolic volume and diameter

Some properties of fundamental groups of Riemannian manifolds will be studied without a lower bound assumption on Ricci curvature. The main method is to relate the local packing to global packing instead of using the Bishop-Gromov relative volume comparison. This method allows us to control the volume growth of the universal cover and yields bounds on the number of generators of in terms of some isoembolic geometric invariants of .

     

Commensurability of Hyperbolic Manifolds with Geodesic Boundary

Suppose , let M ₁, M ₂ be n-dimensional connected complete finite-volume hyperbolic manifolds with nonempty geodesic boundary, and suppose that π₁ (M ₁) is quasi-isometric to π₁ (M ₂) (with respect to the word metric). Also suppose that if n=3, then ∂M ₁ and ∂M ₂ are compact. We show that M ₁ is commensurable with M ₂. Moreover, we show that there exist homotopically equivalent hyperbolic 3-manifolds with non-compact geodesic boundary which are not commensurable with each other. We also prove that if M is as M ₁ above and G is a finitely generated group which is quasi-isometric to π₁ (M), then there exists a hyperbolic manifold with geodesic boundary M′ with the following properties: M′ is commensurable with M, and G is a finite extension of a group which contains π₁ (M′) as a finite-index subgroupMathematics Subject Classification (2000). Primary: 20F65; secondary: 30C65, 57N16     

Integral formulas for differential forms of type (p,q) on complex Finsler manifolds

Using the invariant integral kernel introduced by Demailly and Laurent-Thiebaut, complex Finsler metric and nonlinear connection associating with Chern-Finsler connection, we research the integral representation theory on complex Finsler manifolds. The Koppelman and Koppelman-Leray formulas are obtained, and the -equations are solved.     

Weighted diffeomorphism groups of Riemannian manifolds

In this paper, we define locally convex vector spaces of weighted vector fields and use them as model spaces for Lie groups of weighted diffeomorphisms on Riemannian manifolds. We prove an easy condition on the weights that ensures that these groups contain the compactly supported diffeomorphisms. We finally show that for the special case where the manifold is the euclidean space, these Lie groups coincide with the ones constructed in the author’s earlier work (Walter, 2012).     

Surgery of closed manifolds with dihedral fundamental group

In the paper the obstruction groups to obtaining simple homotopy equivalence by surgery from normal degree 1 maps of closed manifolds with dihedral fundamental group are computed. The cases of trivial orientation for the dihedral group and nontrivial orientation for the order 2 cyclic subgroup are considered. New results concerning the Browder-Livesey groups and natural maps ofL-groups arising in index 2 inclusions of the cyclic group into the dihedral group are obtained.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 238–250, August, 1998.The work of the first-named author was partially supported by of the Grant of the President of the Russian Federation, grant No. 96-15-96841. The work of the second-named author was partially suported the Ministry of Science and Technology of the Republic of Sloveniya, grant No. J1-7039-0101-95.     

关于Finsler子流形的平均曲率

通过使用由射影球丛诱导的体积元来研究Finsler子流形几何,推导了体积泛函的第一变分公式。给出了Finsler子流形的平均曲率形式和第二基本形式的定义,该定义在Riemannian情形下与通常的概念一致．此外,通过推导射影球丛纤维上的散度公式。给出了平均曲率形式的一种非常简洁的等价表示,并得到一些关于Minkowski空间中Finsler子流形的有趣的结果．     

Classification of negatively pinched manifolds with amenable fundamental groups

We give a diffeomorphism classification of pinched negatively curved manifolds with amenable fundamental groups, namely, they are precisely the Möbius band, and the products of R with the total spaces of flat vector bundles over closed infranilmanifolds.     

On fundamental groups related to the Hirzebruch surface F_1

Given a projective surface and a generic projection to the plane,the braid monodromy factorization(and thus,the braid monodromy type)of the complement of its branch curve is one of the most important topological invariants,stable on deformations.From this factorization,one can compute the fundamental group of the complement of the branch curve,either in C~2 or in CP~2.In this article,we show that these groups,for the Hirzebruch surface F_1,(a,b),are almost-solvable.That is, they are an extension of a solvable group,which strengthen the conjecture on degeneratable surfaces.