Almost complex structures on cotangent bundles and generalized geometry

We study a class of complex structures on the generalized tangent bundle of a smooth manifold M$M$ endowed with a torsion free linear connection, ∇$\nabla$. We introduce the concept of ∇$\nabla$-integrability and we study integrability conditions. In the case of the generalized complex structures introduced by Hitchin (2003) in [2], we compare the two concepts of integrability. Moreover, as an application, we describe almost complex structures on the cotangent bundle of M$M$ induced by complex structures on the generalized tangent bundle of M$M$.     

One-harmonic invariant vector fields on three-dimensional Lie groups

We determine all left-invariant vector fields on three-dimensional Lie groups which define harmonic sections of the corresponding tangent bundles, equipped with the complete lift metric.     

Symmetries of parabolic contact structures

We generalize the concept of locally symmetric spaces to parabolic contact structures. We show that symmetric normal parabolic contact structures are torsion-free and some types of them have to be locally flat. We prove that each symmetry given at a point with non-zero harmonic curvature is involutive. Finally we give restrictions on the number of different symmetries which can exist at such a point.     

Hopf fibration: Geodesics and distances

Here we study geodesics connecting two given points on odd-dimensional spheres respecting the Hopf fibration. This geodesic boundary value problem is completely solved in the case of three-dimensional sphere and some partial results are obtained in the general case. The Carnot–Carathéodory distance is calculated. We also present some motivations related to quantum mechanics.     

On the conformal geometry of transverse Riemann–Lorentz manifolds

Physical reasons suggested in [J.B. Hartle, S.W. Hawking, Wave function of the universe, Phys. Rev. D41 (1990) 1815–1834] for the Quantum Gravity Problem lead us to study type-changing metrics on a manifold. The most interesting cases are Transverse Riemann–Lorentz Manifolds. Here we study the conformal geometry of such manifolds.     

Geometric realizations,curvature decompositions,and Weyl manifolds

We show that any Weyl curvature model can be geometrically realized by a Weyl manifold.     

A left-symmetric algebraic approach to left invariant flat (pseudo-)metrics on Lie groups

Left invariant flat metrics on Lie groups are revisited in terms of left-symmetric algebras which correspond to affine structures. There is a left-symmetric algebraic approach with an explicit formula to the classification theorem given by Milnor. When the positive definiteness of the metric is replaced by nondegeneracy, there are many more examples of left invariant flat pseudo-metrics, which play important roles in several fields in geometry and mathematical physics. We give certain explicit constructions of these structures. Their classification in low dimensions and some interesting examples in higher dimensions are also given.     

On complete spacelike hypersurfaces with two distinct principal curvatures in Lorentz–Minkowski space

We investigate complete spacelike hypersurfaces in Lorentz–Minkowski space with two distinct principal curvatures and constant m$m$th mean curvature. By using Otsuki’s idea, we obtain the global classification result. As their applications, we obtain some characterizations for hyperbolic cylinders. We prove that the only complete spacelike hypersurfaces in Lorentz–Minkowski (n+1)$(n+1)$-spaces (n≥3$n\ge 3$) of nonzero constant m$m$th mean curvature (m≤n−1$m\le n-1$) with two distinct principal curvatures λ$\lambda$ and μ$\mu$ satisfying inf(λ−μ)²>0$inf{(\lambda -\mu )}^2>0$ are the hyperbolic cylinders. We also obtain a global characterization for hyperbolic cylinder Hⁿ⁻¹(c)×R${\mathbb{H}}^{n-1}\left(c\right)\times \mathbb{R}$ in terms of square length of the second fundamental form.     

Bounds on eigenvalues of the product and the Jordan product of two positive definite operators on a finite-dimensional Hilbert space

We show that R ²ⁿ with its standard symplectic structure is universal in that, subject to a mild topological restriction, essentially all symplectic manifolds can be obtained from it by reduction.Ford Foundation Fellow. Partially supported by NSF grant # DMS-8805699.Supported by the Netherlands Organization for Scientific Research (NWO).     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-Isolation Phenomenon for Complete Surfaces Arising from Yang–Mills Theory

Let M be a complete surface with parallel mean curvature in a complete simply connected space form F ^2+p (c) of constant curvature c. Denote by H and S the mean curvature and the squared length of the second fundamental form of M respectively. Motivated by L ²-isolation phenomenon in Yang–Mills theory, we prove that if , where c + H ² > 0, D(H,c) is an explicit positive constant depending on H and c, then , i.e., M is a totally umbilical sphere . Research supported by the Chinese NSF, Grant No. 10231010; Trans-Century Training Programme Foundation for Talents by the Ministry of Education of China.     

Minimal hypersurfaces in a nearly Kaehler 6-sphere

In this paper we show that for a compact minimal hypersurface M$M$ of constant scalar curvature in the unit sphere S⁶$S^6$ with the shape operator A$A$ satisfying ‖A‖²>5${\Vert A\Vert }^2>5$, there exists an eigenvalue λ>10$\lambda >10$ of the Laplace operator of the hypersurface M$M$ such that ‖A‖²=λ−5${\Vert A\Vert }^2=\lambda -5$. This gives the next discrete value of ‖A‖²${\Vert A\Vert }^2$ greater than 0 and 5.     

Gravitational waves without gravitons

The radiation connection induced on an isotropic hypersurface of a Lorentz manifold is described. Consequences for the energy tensor in Einstein field equation are analyzed. A cosmological interpretation is proposed.     

The first eigenvalue of the Dirac operator on locally reducible Riemannian manifolds

We prove a lower estimate for the first eigenvalue of the Dirac operator on a compact locally reducible Riemannian spin manifold with positive scalar curvature. We determine also the universal covers of the manifolds on which the smallest possible eigenvalue is attained.     

The Gauss map of minimal surfaces in the Anti-de Sitter space

It is proved that a pair of spinors satisfying a Dirac-type equation represent surfaces immersed in Anti-de Sitter space with prescribed mean curvature. Here, we consider Anti-de Sitter space as the Lie group SU_1,1${\text{<small-caps>SU</small-caps>}}_{1,1}$ endowed with a one-parameter family of left-invariant metrics where only one of them is bi-invariant and corresponds to the isometric embedding of Anti-de Sitter space as a quadric in R^2,2${\mathbb{R}}^{2,2}$. We prove that the Gauss map of a minimal surface immersed in SU_1,1${\text{<small-caps>SU</small-caps>}}_{1,1}$ is harmonic. Conversely, we exhibit a representation of minimal surfaces in Anti-de Sitter space in terms of a given harmonic map.     

Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in locally symmetric Lorentz spaces

In this paper, the complete spacelike hypersurface with constant normal scalar curvature in a locally symmetric Lorentz space is discussed. Several classified theorems are obtained using the operator _L1$L_1$ introduced by S.Y. Cheng and S.T. Yau (1977) [4].