Almost hermitian structures with parallel torsion

The characteristic connection of an almost hermitian structure is a hermitian connection with totally skew-symmetric torsion. The case of parallel torsion in dimension 6 is of particular interest. In this work, we give a full classification of the algebraic types of the torsion form, and, on the basis of this, undertake a systematic investigation into the possible geometries. Numerous naturally reductive spaces are constructed and classified, and examples on nilmanifolds given.     

Grading of spinor bundles and gravitating matter in noncommutative geometry

The gravitating matter is studied within the framework of noncommutative geometry. The noncommutative Einstein-Hilbert action on the product of a four-dimensional manifold with discrete space gives models of matter fields coupled to the standard Einstein gravity. The matter multiplet is encoded in the Dirac operator which yields a representation of the algebra of universal forms. The general form of the Dirac operator depends on a choice of the grading of the corresponding spinor bundle. A choice is given, which leads to the nonlinear vectorσ-model coupled to the Einstein gravity.     

The classification of 4-dimensional non-degenerate affine hypersurfaces with parallel cubic form

In this paper, we complete the classification of 4-dimensional non-degenerate affine hypersurfaces with parallel cubic form with respect to the Levi-Civita connection of the affine Berwald–Blaschke metric.     

Some calibrated surfaces in manifolds with density

Hyperplanes, hyperspheres and hypercylinders in Rⁿ${\mathbb{R}}^n$ with suitable densities are proved to be weighted area-minimizing by a calibration argument.     

Einstein four-manifolds with skew torsion

We develop a notion of Einstein manifolds with skew torsion on compact, orientable Riemannian manifolds of dimension four. We prove an analogue of the Hitchin–Thorpe inequality and study the case of equality. We use the link with self-duality to study the moduli space of 1-instantons on S⁴$S^4$ for a family of metrics defined by Bonneau.     

On a generalized 1-harmonic equation and the inverse mean curvature flow

We introduce and study generalized 1-harmonic equations (1.1). Using some ideas and techniques in studying 1-harmonic functions from Wei (2007)  [1], and in studying nonhomogeneous 1-harmonic functions on a cocompact set from Wei (2008)  [2, (9.1)], we find an analytic   quantity w$w$ in the generalized 1-harmonic equations  (1.1) on a domain in a Riemannian n$n$-manifold that affects the behavior of weak solutions of  (1.1), and establish its link with the geometry   of the domain. We obtain, as applications, some gradient bounds and nonexistence results for the inverse mean curvature flow, Liouville theorems for p$p$-subharmonic functions of constant p$p$-tension field, p≥n$p\ge n$, and nonexistence results for solutions of the initial value problem of inverse mean curvature flow.     

Killing spinor initial data sets

A 3+1 decomposition of the twistor and valence-2 Killing spinor equation is made using the space-spinor formalism. Conditions on initial data sets for the Einstein vacuum equations are given so that their developments contain solutions to the twistor and/or Killing equations. These lead to the notions of twistor and Killing spinor initial data. These notions are used to obtain a characterisation of initial data sets whose developments are of Petrov type N or D.     

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.     

Warped product Kähler manifolds and Bochner–Kähler metrics

Using as an underlying manifold an alpha-Sasakian manifold, we introduce warped product Kähler manifolds. We prove that if the underlying manifold is an alpha-Sasakian space form, then the corresponding Kähler manifold is of quasi-constant holomorphic sectional curvatures with a special distribution. Conversely, we prove that any Kähler manifold of quasi-constant holomorphic sectional curvatures with a special distribution locally has the structure of a warped product Kähler manifold whose base is an alpha-Sasakian space form. As an application, we describe explicitly all Bochner–Kähler metrics of quasi-constant holomorphic sectional curvatures. We find four families of complete metrics of this type. As a consequence, we obtain Bochner–Kähler metrics generated by a potential function of distance in complex Euclidean space and of time-like distance in the flat Kähler–Lorentz space.     

Towards physically motivated proofs of the Poincaré and geometrization conjectures

Although the Poincaré and the geometrization conjectures were recently proved by Perelman, the proof relies heavily on properties of the Ricci flow previously investigated in great detail by Hamilton. Physical realization of such a flow can be found, for instance, in the work by Friedan [D. Friedan, Nonlinear models in 2+ε$2+\epsilon$ dimensions, Ann. Phys. 163 (1985) 318–419]. In his work the renormalization group flow for a nonlinear sigma model in 2+ε$2+\epsilon$ dimensions was obtained and studied. For ε=0$\epsilon =0$, by approximating the β$\beta$-function for such a flow by the lowest order terms in the sigma model coupling constant, the equations for Ricci flow are obtained. In view of such an approximation, the existence of this type of flow in Nature is questionable. In this work, we find totally independent justification for the existence of Ricci flows in Nature. This is achieved by developing a new formalism extending the results of two-dimensional conformal field theories (CFT’s) to three and higher dimensions. Equations describing critical dynamics of these CFT’s are examples of the Yamabe and Ricci flows realizable in Nature. Although in the original works by Perelman some physically motivated arguments can be found, their role in his proof remain rather obscure. In this paper, steps are made toward making these arguments more explicit, thus creating an opportunity for developing alternative, more physically motivated, proofs of the Poincaré and geometrization conjectures.     

The Ricci flow approach to homogeneous Einstein metrics on flag manifolds

We give a global picture of the normalized Ricci flow on generalized flag manifolds with two or three isotropy summands. The normalized Ricci flow for these spaces reduces to a parameter-dependent system of two or three ordinary differential equations, respectively. Here, we present a qualitative study of these systems’ global phase portrait, which uses techniques of dynamical systems theory. This study allows us to draw conclusions about the existence and the analytical form of invariant Einstein metrics on such manifolds and seems to offer a better insight to the classification problem of invariant Einstein metrics on compact homogeneous spaces.     

Deformations of 2k-Einstein structures

It is shown that the space of infinitesimal deformations of 2k-Einstein structures is finite dimensional on compact non-flat space forms. Moreover, spherical space forms are shown to be rigid in the sense that they are isolated in the corresponding moduli space.     

Hirota equation as an example of an integrable symplectic map

The Hamiltonian formalism is developed for the sine-Gordon model on the spacetime light-like lattice, first introduced by Hirota. The evolution operator is explicitly constructed in the quantum variant of the model and the integrability of the corresponding classical finite-dimensional system is established.     

Complete explicit classification of parallel Lorentz surfaces in arbitrary pseudo-Euclidean spaces

A Lorentz surface of an indefinite space form is called parallel if its second fundamental form is parallel. Such surfaces are locally invariant under the reflection with respect to the normal space at each point. Parallel surfaces are important in geometry as well as in physics since extrinsic invariants of such surfaces do not change from point to point. Recently, parallel spacelike surfaces in an arbitrary indefinite space form are classified in Chen (2010) [20]. Moreover, parallel Lorentz surfaces in 4D indefinite space forms are completely classified in a series of recent articles Chen (submitted for publication) [16], Chen (submitted for publication) [17], Chen (in press) [18], Chen (2010) [19], Chen and Van der Veken (2009) [15] (see also Graves (1979) [12], Graves (1979) [13] and Magid (1984) [14] for some partial results). In this paper, we achieve the complete classification of parallel Lorentz surfaces in a pseudo-Euclidean space with arbitrary codimension and arbitrary index.     

Real hypersurfaces in complex two-plane Grassmannians with commuting Ricci tensor

In this paper, first we introduce the full expression for the Ricci tensor of a real hypersurface M$M$ in complex two-plane Grassmannians _G2(C^m+2)$G_2\left({\mathbb{C}}^{m+2}\right)$ from the equation of Gauss. Next we prove that a Hopf hypersurface in complex two-plane Grassmannians _G2(C^m+2)$G_2\left({\mathbb{C}}^{m+2}\right)$ with commuting Ricci tensor is locally congruent to a tube of radius r$r$ over a totally geodesic _G2(C^m+1)$G_2\left({\mathbb{C}}^{m+1}\right)$. Finally it can be verified that there do not exist any Hopf Einstein hypersurfaces in _G2(C^m+2)$G_2\left({\mathbb{C}}^{m+2}\right)$.     

Connections on Lie algebroids and on derivation-based noncommutative geometry

In this paper, we show how connections and their generalizations on transitive Lie algebroids are related to the notion of connections in the framework of the derivation-based noncommutative geometry. In order to compare the two constructions, we emphasize the algebraic approach of connections on Lie algebroids, using a suitable differential calculus. Two examples allow this comparison: on the one hand, the Atiyah Lie algebroid of a principal fiber bundle and, on the other hand, the space of derivations of the algebra of endomorphisms of an SL(n,C)-vector bundle. Gauge transformations are also considered in this comparison.