关于减小体积的射影映射

Given a projective map F:M→N of a complete Riemannian manifold to a Riemannian manifold with the sectional curvature bounded above by a negative constant,we prove that f decreases volume up to a constant depending only on the curvatures of M and N.This generalizes the result due to Har‘el.     

RANDERS SPACES WITH SCALAR FLAG CURVATURE

Let(M, F) be an n-dimensional Randers space with scalar flag curvature. In this paper, we will introduce the definition of a weak Einstein manifold. We can prove that if(M, F) is a weak Einstein manifold, then the flag curvature is constant.     

On the Ricci curvature of a Randers metric of isotropic <Emphasis Type="Italic">S</Emphasis>-curvature

We derive the integral inequality of a Randers metric with isotropic S-curvature in terms of its navigation representation. Using the obtained inequality we give some rigidity results under the condition of Ricci curvature. In particular, we show the following result： Assume that an n-dimensional compact Randers manifold （M, F） has constant S-curvature c. Then （M, F） must be Riemannian if its Ricci curvature satisfies that Ric 〈 -（n - 1）c^2.     

Rigidity of closed submanifolds in a locally symmetric Riemannian manifold

Let M~n(n ≥ 4) be an oriented closed submanifold with parallel mean curvature in an(n + p)-dimensional locally symmetric Riemannian manifold N~(n+p). We prove that if the sectional curvature of N is positively pinched in [δ, 1], and the Ricci curvature of M satisfies a pinching condition, then M is either a totally umbilical submanifold, or δ = 1, and N is of constant curvature. This result generalizes the geometric rigidity theorem due to Xu and Gu[15].     

Evolution and monotonicity of eigenvalues under the Ricci flow

Let(M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. We derive the evolution equation for the eigenvalues of geometric operator-△φ+ c R under the Ricci flow and the normalized Ricci flow, where △φis the Witten-Laplacian operator, φ∈ C∞(M), and R is the scalar curvature with respect to the metric g(t). As an application, we prove that the eigenvalues of the geometric operator are nondecreasing along the Ricci flow coupled to a heat equation for manifold M with some Ricci curvature condition when c 14.     

MINIMAL SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD OF QUASI CONSTANT CURVATURE

A Riemannian manifold V~m which admits isometric imbedding into two spaces V~(m+p)ofdifferent constant curvatures is called a manifold of quasi constant curvature.TheRiemannian curvature of V~m is expressible in the formand conversely.In this paper it is proved that if M~n is any compact minimal submanifoldwithout boundary in a Riemannian manifold V~(n+p)of quasi constant curvature,then∫_(M~u)(2-1/p)σ~2-[na+1/2(b-丨b丨)(n+1)]σ+n(n-1)b~2*丨≥0,where σ is the square of the norm of the second fundamental form of M~n When V~(n+p)is amanifold of constant curvature,b=0,the above inequality reduces to that of Simons.     

Classification of Lagrangian Surfaces of Curvature e in Non-flat Lorentzian Complex Space Form M12（4ε）

It is well known that a totally geodesic Lagrangian surface in a Lorentzian complex space form M12（4ε） of constant holomorphic sectional curvature 4s is of constant curvature 6. A natural question is ＂Besides totally geodesic ones how many Lagrangian surfaces of constant curvature εin M12（46） are there？＂ In an earlier paper an answer to this question was obtained for the case e = 0 by Chen and Fastenakels. In this paper we provide the answer to this question for the case ε≠0. Our main result states that there exist thirty-five families of Lagrangian surfaces of curvature ε in M12（4ε） with ε ≠ 0. Conversely, every Lagrangian surface of curvature ε≠0 in M12（4ε） is locally congruent to one of the Lagrangian surfaces given by the thirty-five families.     

New geometric flows on Riemannian manifolds and applications to Schrödinger-Airy flows

In this paper,a class of new geometric flows on a complete Riemannian manifold is defined. The new flow is related to the generalized(third order) Landau-Lifshitz equation. On the other hand it could be thought of as a special case of the Schr¨odinger-Airy flow when the target manifold is a K¨ahler manifold with constant holomorphic sectional curvature. We show the local existence of the new flow on a complete Riemannian manifold with some assumptions on Ricci tensor. Moreover,if the target manifolds are Einstein or some certain type of locally symmetric spaces,the global results are obtained.     

Canonical Foliations of Certain Classes of Almost Contact Metric Structures

The purpose of this paper is to study the canonical foliations of an almost cosymplectic or almost Kenmotsu manifold M in a unified way. We prove that the canonical foliation F defined by the contact distribution is Riemannian and tangentially almost Kahler of codimension 1 and that F is tangentially Kahler if the manifold M is normal. Furthermore, we show that a semi-invariant submanifold N of such a manifold M admits a canonical foliation FN which is defined by the antiinvariant distribution and a canonical cohomology class c（N） generated by a transversal volume form for FN. In addition, we investigate the conditions when the even-dimensional cohomology classes of N are non-trivial. Finally, we compute the Godbillon Vey class for FN.     

On Asymptotic Behavior for Singularities of the Powers of Mean Curvature Flow

Let M^n be a smooth, compact manifold without boundary, and F0 ： M^n→ R^n＋1 a smooth immersion which is convex. The one-parameter families F（·, t） ： M^n× [0, T） → R^n＋1 of hypersurfaces Mt^n= F（·,t）（M^n） satisfy an initial value problem dF/dt （·,t） = -H^k（· ,t）v（· ,t）, F（· ,0） = F0（· ）, where H is the mean curvature and u（·,t） is the outer unit normal at F（·, t）, such that -Hu = H is the mean curvature vector, and k 〉 0 is a constant. This problem is called H^k-fiow. Such flow will develop singularities after finite time. According to the blow-up rate of the square norm of the second fundamental forms, the authors analyze the structure of the rescaled limit by classifying the singularities as two types, i.e., Type Ⅰ and Type Ⅱ. It is proved that for Type Ⅰ singularity, the limiting hypersurface satisfies an elliptic equation; for Type Ⅱ singularity, the limiting hypersurface must be a translating soliton.     

CR-structures of codimension 2 on tangent bundles in Riemann–Finsler geometry

We determine a 2-codimensional CR-structure on the slit tangent bundle \(T_0M\) of a Finsler manifold (M, F) by imposing a condition on the almost complex structure \(\Psi \) associated to F when restricted to the structural distribution of a framed f-structure. This condition is satisfied when (M, F) is of scalar flag curvature (particularly flat). In the Riemannian case (M, g) this last condition means that g is of constant curvature. This CR-structure is finally generalized by using one positive parameter but under more difficult conditions.     

The Weyl functional near the Yamabe invariant

For a compact manifold M ofdim M=n≥4, we study two conformal invariants of a conformal class C on M. These are the Yamabe constant Y_C(M) and the L^n/2-norm W_C(M) of the Weyl curvature. We prove that for any manifold M there exists a conformal class C such that the Yamabe constant Y_C(M) is arbitrarily close to the Yamabe invariant Y(M), and, at the same time, the constant W_C(M) is arbitrarily large. We study the image of the mapYW:C→(Y_C(M), W_C(M))∈R ² near the line {(Y(M), w)|w∈R}. We also apply our results to certain classes of 4-manifolds, in particular, minimal compact Kähler surfaces of Kodaira dimension 0, 1 or 2.     

A new class of almost complex structures

The purpose of this paper is to introduce a new class of almost complex structures J on a Riemannian manifold M by using a certain identity for the relationship between the tensor F _i ^j of J and the Riemann curvature tensor R _hijk of M. This class contains the Kählerian structures, and its relationship with some known classes of almost Hermitian structures defined by similar identities is discussed. For convenience we call each structure of this new class an almost C-structure, and a manifold with an almost C-structure an almost C-manifold. We obtain an analogue of F. Schur's theorem concerning the holomorphic sectional curvature of an almost Hermitian C-manifold, and some sufficient conditions for an almost Hermitian C-manifold to be Kählerian. We show that these results are also true for a manifold with a complex structure.     

A class of metrics and foliations on tangent bundle of Finsler manifolds

Let (M,F) be a Finsler manifold, and let TM ₀ be the slit tangent bundle of M with a generalized Riemannian metric G, which is induced by F. In this paper, we extract many natural foliations of (TM ₀,G) and study their geometric properties. Next, we use this approach to obtain new characterizations of Finsler manifolds with positive constant flag curvature. We also investigate the relations between Levi-Civita connection, Cartan connection, Vaisman connection, vertical foliation, and Reinhart spaces.     

Volume preserving curvature flows in Lorentzian manifolds

Let N be a (n + 1)-dimensional globally hyperbolic Lorentzian manifold with a compact Cauchy hypersurface ${\mathcal{S}_{0}}$ and F a curvature function, either the mean curvature H, the root of the second symmetric polynomial ${{\sigma}_{2}=\sqrt{H_{2}}}$ or a curvature function of class (K*), a class of curvature functions which includes the nth root of the Gaussian curvature ${{\sigma}_{n}= K^{\frac{1}{n}}}$ . We consider curvature flows with curvature function F and a volume preserving term and prove long time existence of the flow and exponential convergence of the corresponding graphs in the C ^∞-topology to a hypersurface of constant F-curvature, provided there are barriers. Furthermore we examine stability properties and foliations of constant F-curvature hypersurfaces.     

On the characteristic connection of gwistor space

We give a brief presentation of gwistor spaces, which is a new concept from G ₂ geometry. Then we compute the characteristic torsion T ^c of the gwistor space of an oriented Riemannian 4-manifold with constant sectional curvature k and deduce the condition under which T ^c is ?^c-parallel; this allows for the classification of the G ₂ structure with torsion and the characteristic holonomy according to known references. The case of an Einstein base manifold is envisaged.     

On the curvatures of Einstein spaces

For a pseudo-Riemannian manifold (M, g) of dimensionn3, we introduce a scalar curvature functionS(V) for non-degenerate subspacesV ofT _pM which is a generalization of the scalar curvature, and give some characterizations of Einstein spaces in terms of this scalar curvature function. We also give a characterization for spaces of constant curvature. As an application of our results, we show that the Ricci curvature or the sectional curvature of a Lorentz manifold is constant if the scalar curvature function for non-degenerate subspaces is bounded.Partially supported by the grants from TGRC.     

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).     

Spectral geometry for Riemannian foliations

Let F be a Riemannian foliation on a Riemannian manifold (M, g), with bundle-like metric g. Aside from the Laplacian △_g associated to the metric g, there is another differential operator, the Jacobi operator J▽, which is a second order elliptic operator acting on sections of the normal bundle. Its spectrum is discrete as a consequence of the compactness of M. Hence one has two spectra, spec (M, g) = spectrum of △_g (acting on functions), and spec (F, J▽) = spectrum of J▽. We discuss the following problem: Which geometric properties of a Riemannian foliation F on a Riemannian manifold (M, g) are determined by the two types of spectral invariants?