Para-CR Structures of Codimension 2 on Tangent Bundles in Riemann–Finsler Geometry

We determine a 2-codimensional para-CR structure on the slit tangent bundle T0 M of a Finsler manifold（M,F） by imposing a condition regarding the almost paracomplex structure P associated to F when restricted to the structural distribution of a framed para-f-structure.This condition is satisfied when（M,F） is of scalar flag curvature（particularly constant） or if the Riemannian manifold（M,g） is of constant curvature.     

不动点集为$RP(2^m)\sqcup P(2^m,\,2n-1)$的对合

Let （M^2m＋4n＋k-2, T） be a smooth closed manifold with a smooth involution T whose fixed point set is RP（2^m） ∪ P（2^m, 2n - 1） （m 〉 3, n 〉 0）. For 2n ≥ 2^m, （M^2m＋4n＋k-2, T） is bordant to （P（2^m, RP（2n））, To）.     

紧致流形上Lichnerowicz方程的梯度估计(英文)

Let(M,g) be a smooth compact Riemannian manifold of dimension n.Denote△f=△-▽f.▽ the weighted Laplacian operator,where f is a smooth real valued function on M.When N is finite and the N-Bakry-Emery Ricci tensor is bounded from below by a constant,we establish local gradient estimates for positive solutions of the following simple Lichnerowicz equation△fu+cu~(-α)=0 on a compact Riemannian manifold,where α is a positive constant and c is a smooth function.     

Teichmller space of negatively curved metrics on complex hyperbolic manifolds is not contractible

We prove that for all n = 4k- 2 and k 2 there exists a closed smooth complex hyperbolic manifold M with real dimension n having non-trivial π1(T0(M)). T0(M) denotes the Teichm¨uller space of all negatively curved Riemannian metrics on M, which is the topological quotient of the space of all negatively curved metrics modulo the space of self-diffeomorphisms of M that are homotopic to the identity.     

RESEARCH ANNOUNCEMENTS ON "UPPER BOUND ESTIMATES OF EIGENVALUES FOR HORMANDER OPERATORS ON NON-EQUIREGULAR SUB-RIEMANNIAN MANIFOLDS"

1 Introduction and Main Results For n≥2,let M be a n-dimensional connected smooth manifold.We take a fixed smooth measure on M with strictly positive density,and write dx as the measure when we integrate.Also,we simply denote the measure of a set E?M by|E|.     

Boundability of Open Manifolds

An open manifold in question is assumed to bea noncompact manifold with compact boundary (probably empty). Let M be a connected open n-dimensional smooth manifold. Take the one-point-compactification of M, it will be denoted by M=M∪{*}, where * is the infinite point. Since M has countable basis, M is metrizable and {*} is a closed subset and a G_δset. By Urysohn lemma, there exists a continuous function f:→[O,1] with f~-1(O)=M and f~-1(1)=*.Choose a sui-     

UNIFORM ESTIMATES OF SOLUTIONS OF δ^--EQUATION ON STEIN MANIFOLDS

By means of the Hermitian metric and Chern connection, Qiu [4] obtained the Koppelman-Leray-Norguet formula for （p, q） differential forms on an open set with C^1 piecewise smooth boundary on a Stein manifold, and under suitable conditions gave the solutions of δ^--equation on a Stein manifold. In this article, using the method of Range and Siu [5], under suitable conditions, the authors complicatedly calculate to give the uniform estimates of solutions of δ^--equation for （p, q） differential forms on a Stein manifold.     

Boundary reducible handle additions on simple 3-manifolds

Let M be a simple manifold, and F be a component of δM of genus two. For a slope γ on F, we denote by M（γ） the manifold obtained by attaching a 2-handle to M along a regular neighborhood of γon F. In this paper, we shall prove that there is at most one separating slope γ on F such that M（γ） is δ-reducible.     

本刊英文版Vol.30(2014),No.9论文摘要(英文)

<正>Least Number of Periodic Points of Self-maps of Lie Groups Jerzy JEZIERSKI Abstract There are two algebraic lower bounds of the number of n-periodic points of a self-map f:M→M of a compact smooth manifold of dimension at least 3:NF_n(f)=min{#Fix(g~n);g~f;g is continuous}and NJD_n(f)=min{#Fix(g~n);g~f;g is smooth}.In general,NJD_n(f)may be much greater than NF_n(f).If M is a torus,then the invariants are     

REGULARITY OF HARMONIC MAPS INTO POSITIVELY CURVED MANIFOLDS

Let M be a compact Riemannian manifold of dimension m, N a complete Amply connected δ-pinched Riemannian manifold of dimension n. There exists a constant d(n). It is proved that if m≤d(n), then every minimizing map from M into N is smooth in the interior of M. If m=d(n)+1, such a map has at most diserete singular set and in general the Hausdorff dimension of the singular set is at most m-d(n)-1.     

SUB-SIGNATURE OPERATORS,η-INVARIANTS AND A RIEMANN-ROCH THEOREM FOR FLAT VECTOR BUNDLES

§0. Introduction Let M be an odd dimensional closed oriented manifold. Let ρbe a unitary represen-tation of the fundamental group of M. By using their η-invariant, Atiyah-Patodi-Singer [2,3] introduced an R/Z-valued smooth invariant associated to …     

Diffeomorphisms with C 1-stably average shadowing

Let M be a closed smooth manifold M, and let f : M → M be a diffeomorphism. In this paper, we consider a nontrivial transitive set Λ of f . We show that if f has the C1-stably average shadowing property on Λ, then Λ admits a dominated splitting.     

The distribution of the periodic orbits with open deviation in C~(1+α)non-uniformly hyperbolic systems

Let f : M → M be a C~(1+α) diffeomorphism on a smooth compact Riemannian manifold M and Λ be a Pesin set associated with the ergodic hyperbolic measure μ. Then f : Λ→Λ forms a non-uniformly hyperbolic system. We concern with the distribution of the periodic orbits whose time averages are apart from the space average of μ. Finally, we derive a large deviation result for these periodic orbits with open deviation property.     

A NOTE ON EQUIVARIANT VECTOR FIELDS

§1. Introduction. Throughout this note, G is a finite group, M is a compact connected smooth on-dimensional manifold with or without boundary M, and G acts smoothly on M. We follow the standard notations ([B], [tD]). The isotropy subgroup of a point     

Variational properties of quadratic curvature functionals

In this paper, we investigate a class of quadratic Riemannian curvature functionals on closed smooth manifold M of dimension n ≥3 on the space of Riemannian metrics on M consisting of metrics with unit volume.We study the stability of these functionals at the metric with constant sectional curvature as its critical point.     

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).     

LOWER BOUNDS FOR TNE FIRST GAP OF EIGENVALUES IN THE SCHRDINGER OPERATOR ON SIMPLE RIEMANNIAN MANIFOLDS

We give a lower bound for the first gap λ_2—λ_1 of the twolowerst eigenvalues of the Schr(o|¨)dinger operator-△+W(p) with the Dirichletboundary condition and a strictly convex potential W(p)on M in which M is acompact simple Riemannian manifold with smooth strictly convex boundary (?)MHere a compact Riemannian manifold M is said to be simple if M～(?)M istopologically R~2.We prove thatλ_2-λ_1≥(π~2)/(d~2)+min{0,-(n-1)K}where d is the diameter of M and-(n-1)K,(K≥0)the lower bound of theRicci curvature of M.This work generalizes the results in the classical Eucli-dean situation due to Singer,Wong and Yau,Yu and Zhong to a kind of curvedRiemannian manifold.     

The Riemannian Manifolds with Boundary and Large Symmetry

Seventy years ago, Myers and Steenrod showed that the isometry group of a Riemannian manifold without boundary has a structure of Lie group. In 2007, Bagaev and Zhukova proved the same result for a Riemannian orbifold. In this paper, the authors first show that the isometry group of a Riemannian manifold M with boundary has dimension at most 1/2 dim M（dim M - 1）. Then such Riemannian manifolds with boundary that their isometry groups attain the preceding maximal dimension are completely classified.     

Weinstein Conjecture and Holomorphic Spheres

The Weinstein conjecture is one of the central problems in the symplectic geo-metry. Let (M,ω) be a symplectic manifold. Given a compact smooth bypersurfaceS(M,ω) we denote by (S) the collection of all periodic Hamiltonian trajectoriesou S. In [6] A. Weinstein proposed the following conjecture. Conjecture If S is a compact bypersurface of contact type with H~1(S;R) =