Projection pencils of quadrics and Ivory��s theorem

Using selfadjoint regular endomorphisms, the authors of Stachel and Wallner (Sib Math J 45(4):785?C794, 2004) defined, for an indefinite inner product, a variant of the notion of confocality for the Euclidean space. Our aim is to give a definition that is a common generalization of the usual confocality, and the variant in Stachel and Wallner (Sib Math J 45(4):785?C794, 2004). We use this definition to prove a more general form of Ivory??s theorem.     

Affine biharmonic submanifolds in 3-dimensional pseudo-Hermitian geometry

The notion of biharmonic map between Riemannian manifolds is generalized to maps from Riemannian manifolds into affine manifolds. Hopf cylinders in 3-dimensional Sasakian space forms which are biharmonic with respect to Tanaka-Webster connection are classified. Dedicated to professor John C. Wood on his 60th birthday.     

The Oshima-Sekiguchi and Liouville theorems on Heintze groups

Let L be an elliptic operator on a Riemannian manifold M. A function F annihilated by L is said to be L-harmonic. F is said to have moderate growth if and only if F grows at most exponentially in the Riemannian distance. If M is a rank-one symmetric space and L is the Laplace-Beltrami operator for M, the Oshima-Sekiguchi theorem [T. Oshima, J. Sekiguchi, Eigenspaces of invariant differential operators on an affine symmetric space, Invent. Math. 57 (1980) 1-81] states that a L-harmonic function F has moderate growth if and only if F is the Poisson integral of a distribution on the Furstenberg boundary. In this work we prove that this result generalizes to a very large class of homogeneous Riemannian manifolds of negative curvature. We also (i) prove a Liouville type theorem that characterizes the “polynomial-like” harmonic functions which vanish on the boundary in terms of their growth properties, (ii) describe all “polynomial-like” harmonic functions, and (iii) give asymptotic expansions for the Poisson kernel. One consequence of this work is that every Schwartz distribution on the boundary is the boundary value for a L-harmonic function F which is uniquely determined modulo “polynomial-like” harmonic functions.     

Multiple closed geodesics on 3-spheres

This paper is devoted to a study on closed geodesics on Finsler and Riemannian spheres. We call a prime closed geodesic on a Finsler manifold rational, if the basic normal form decomposition (cf. [Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math. 187 (1999) 113-149]) of its linearized Poincaré map contains no 2×2 rotation matrix with rotation angle which is an irrational multiple of π, or irrational otherwise. We prove that if there exists only one prime closed geodesic on a d-dimensional irreversible Finsler sphere with d?2, it cannot be rational. Then we further prove that there exist always at least two distinct prime closed geodesics on every irreversible Finsler 3-dimensional sphere. Our method yields also at least two geometrically distinct closed geodesics on every reversible Finsler as well as Riemannian 3-dimensional sphere. We prove also such results hold for all compact simply connected 3-dimensional manifolds with irreversible or reversible Finsler as well as Riemannian metrics.     

The stress-energy tensor for biharmonic maps

Using Hilbert’s criterion, we consider the stress-energy tensor associated to the bienergy functional. We show that it derives from a variational problem on metrics and exhibit the peculiarity of dimension four. First, we use this tensor to construct new examples of biharmonic maps, then classify maps with vanishing or parallel stress-energy tensor and Riemannian immersions whose stress-energy tensor is proportional to the metric, thus obtaining a weaker but high-dimensional version of the Hopf Theorem on compact constant mean curvature immersions. We also relate the stress-energy tensor of the inclusion of a submanifold in Euclidean space with the harmonic stress-energy tensor of its Gauss map. S. Montaldo was supported by PRIN-2005 (Italy): Riemannian Metrics and Differentiable Manifolds. C. Oniciuc was supported by a CNR-NATO (Italy) fellowship and by the Grant CEEX, ET, 5871/2006 (Romania).     

具有正Ricci曲率和体积Pinching流形的一个球定理

M~n是一个紧致无边单连通的n(≥3)维Riemannian流形,S~n为R~(n+1)中的单位球面.本文所关注的流形满足截面曲率K_M≤1,而Ricci曲率Ric(M)≥(n+2)/4以及体积V(M)≤3/2(1+η)V(S~(2n)),这里η是一个仅和维数n有关的常数.最终将给出一个具有正的Ricci曲率的球定理新证明.     

An intrinsic characterization of developable surfaces

We consider the classical theorem saying that if f: M → R³ is a Riemannian surface in R³ without planar points and with vanishing Gaussian curvature, then there is an open dense subset M′ of M such that around each point of M′ the surface f is a cylinder or a cone or a tangential developable. As we shall show below, the theorem, in fact, belongs to affine geometry. We give an affine proof of this theorem. The proof works in Riemannian geometry as well. We use the proof for solving the realization problem for a certain class of affine connections on 2-dimensional manifolds. In contrast with Riemannian geometry, in affine geometry, cylinders, cones as well as tangential developables can be characterized intrinsically, i.e. by means of properties of any nowhere flat induced connection. According to the characterization we distinguish three classes of affine connections on 2-dimensional manifolds, i.e. cylindric, conic and TD-connections.     

On geometric vector fields of Minkowski spaces and their applications

As it is well-known, a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. By the help of its second order partial derivatives we can introduce a Riemannian metric on the vector space and the indicatrix hypersurface S:=F⁻¹(1) can be investigated as a Riemannian submanifold in the usual sense.Our aim is to study affine vector fields on the vector space which are, at the same time, affine with respect to the Funk metric associated with the indicatrix hypersurface. We give an upper bound for the dimension of their (real) Lie algebra and it is proved that equality holds if and only if the Minkowski space is Euclidean. Criteria of the existence is also given in lower dimensional cases. Note that in case of a Euclidean vector space the Funk metric reduces to the standard Cayley-Klein metric perturbed with a nonzero 1-form.As an application of our results we present the general solution of Matsumoto's problem on conformal equivalent Berwald and locally Minkowski manifolds. The reasoning is based on the theory of harmonic vector fields on the tangent spaces as Riemannian manifolds or, in an equivalent way, as Minkowski spaces. Our main result states that the conformal equivalence between two Berwald manifolds must be trivial unless the manifolds are Riemannian.     

Double Bubbles for Immiscible Fluids in ℝ n

We use a new approach that we call unification to prove that standard weighted double bubbles in n-dimensional Euclidean space minimize immiscible fluid surface energy, that is, surface area weighted by constants. The result is new for weighted area, and also gives the simplest known proof to date of the (unit weight) double bubble theorem (Hass et al., Electron. Res. Announc. Am. Math. Soc., 1(3):98–102, 1995; Hutchings et al., Ann. Math., 155(2):459–489, 2002; Reichardt, J. Geom. Anal., 18(1):172–191, 2008). As part of the proof, we introduce a striking new symmetry argument for showing that a minimizer must be a surface of revolution.     

向量在几何中的应用

近期将欧氏平面E2上的正弦定理和余弦定理推广到三维欧氏空间E3中,建立了E3中四面体空间角正弦定理、二面角正弦定理和四面体余弦定理,利用向量给出了三维余弦定理和三维正弦定理的简单证明.     

A remark on locally homogeneous Riemannian spaces

A locally homogeneous Riemannian space is called non-regular if it is not locally isometric to any globally homogeneous Riemannian space. We show that no non-regular space has non positive Ricci tensor and that a theorem by Alkseevski-Kimelfeld may be extended to the class of locally homogeneous spaces: i.e. any locally homogeneous Riemannian space with zero Ricci tensor is locally euclidean.     

A Liouville-type theorem for biharmonic maps between complete Riemannian manifolds with small energies

We prove a Liouville-type theorem for biharmonic maps from a complete Riemannian manifold of dimension \(n\) that has a lower bound on its Ricci curvature and positive injectivity radius into a Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the \(L^p\)-norm of the tension field is bounded and the n-energy of the map is sufficiently small, then every biharmonic map must be harmonic, where \(2<p<n\).     

Conformal Riemannian Maps between Riemannian Manifolds,Their Harmonicity and Decomposition Theorems

Riemannian maps were introduced by Fischer (Contemp. Math. 132:331–366, 1992) as a generalization isometric immersions and Riemannian submersions. He showed that such maps could be used to solve the generalized eikonal equation and to build a quantum model. On the other hand, horizontally conformal maps were defined by Fuglede (Ann. Inst. Fourier (Grenoble) 28:107–144, 1978) and Ishihara (J. Math. Kyoto Univ. 19:215–229, 1979) and these maps are useful for characterization of harmonic morphisms. Horizontally conformal maps (conformal maps) have their applications in medical imaging (brain imaging)and computer graphics. In this paper, as a generalization of Riemannian maps and horizontally conformal submersions, we introduce conformal Riemannian maps, present examples and characterizations. We show that an application of conformal Riemannian maps can be made in weakening the horizontal conformal version of Hermann’s theorem obtained by Okrut (Math. Notes 66(1):94–104, 1999). We also give a geometric characterization of harmonic conformal Riemannian maps and obtain decomposition theorems by using the existence of conformal Riemannian maps.     

Non existence of quasi-harmonic spheres

Let M and N be compact Riemannian manifolds. To prove the global existence and convergence of the heat flow for harmonic maps between M and N, it suffices to show the nonexistence of harmonic spheres and nonexistence of quasi-harmonic spheres. In this paper, we prove that, if the universal covering of N admits a nonnegative strictly convex function with polynomial growth, then there are no quasi-harmonic spheres nor harmonic spheres. This generalizes the famous Eells–Sampson’s theorem (Am J Math 86:109–169, [7]).     

Stability of compact constant mean curvature hypersurfaces in a wide class of Riemannian manifolds

Barbosa, do Carmo and Eschenburg proved in Barbosa and do Carmo (Math Z 185(3):339?C353, 1984), Barbosa et?al. (Math Z 197(1): 123?C138, 1988) that the only stable compact hypersurfaces of constant mean curvature immersed in space forms are geodesic spheres. We give a more general result in a wide class of Riemannian manifold including space forms. The interest lies mostly on the simplicity of the proof.     

Special metrics and triality

We investigate a new 8-dimensional Riemannian geometry defined by a generic closed and coclosed 3-form with stabiliser PSU(3), and which arises as a critical point of Hitchin's variational principle. We give a Riemannian characterisation of this structure in terms of invariant spinor-valued 1-forms, which are harmonic with respect to the twisted Dirac operator ? on Δ⊗Λ¹. We establish various obstructions to the existence of topological reductions to PSU(3). For compact manifolds, we also give sufficient conditions for topological PSU(3)-structures that can be lifted to topological SU(3)-structures. We also construct the first known compact example of an integrable non-symmetric PSU(3)-structure. In the same vein, we give a new Riemannian characterisation for topological quaternionic Kähler structures which are defined by an Sp(1)⋅Sp(2)-invariant self-dual 4-form. Again, we show that this form is closed if and only if the corresponding spinor-valued 1-form is harmonic for ? and that these equivalent conditions produce constraints on the Ricci tensor.     

A curvature identity on a 6-dimensional Riemannian manifold and its applications

We derive a curvature identity that holds on any 6-dimensional Riemannian manifold, from the Chern-Gauss-Bonnet theorem for a 6-dimensional closed Riemannian manifold. Moreover, some applications of the curvature identity are given. We also define a generalization of harmonic manifolds to study the Lichnerowicz conjecture for a harmonic manifold “a harmonic manifold is locally symmetric” and provide another proof of the Lichnerowicz conjecture refined by Ledger for the 4-dimensional case under a slightly more general setting.     

Sufficient conditions for open manifolds to be diffeomorphic to Euclidean spaces

Let M be a complete non-compact connected Riemannian n-dimensional manifold. We first prove that, for any fixed point p∈M, the radial Ricci curvature of M at p is bounded from below by the radial curvature function of some non-compact n-dimensional model. Moreover, we then prove, without the pointed Gromov-Hausdorff convergence theory, that, if model volume growth is sufficiently close to 1, then M is diffeomorphic to Euclidean n-dimensional space. Hence, our main theorem has various advantages of the Cheeger-Colding diffeomorphism theorem via the Euclidean volume growth. Our main theorem also contains a result of do Carmo and Changyu as a special case.     

A THEOREM OF LIOUVlLLEfS TYPE ON HARMONIC MAPS WITH FINITE OR SLOWLY DIVERGENT ENERGY

Some theorems of Liouville''s type on harmonic maps from Euclidean space of conformal flat space with finite or slowly divergent energy have been obtained by the first-named author and H.C.J. Sealey, respectively. In this paper , a more general theorem is proved, which includes their results as special cases. The technique is to use a conservation law for harmonic maps.