Anti-holomorphic involutions and almost complex structures on four-manifold

Let σ be an anti-holomorphic involution on almost complex manifoldX, the existence of almost complex structure onY=X/σ is discussed. Project supported by the National Natural Science Foundation of China (Grant No. 19771010).     

Reduction and submanifolds of generalized complex manifolds

We recall the presentation of the generalized, complex structures by classical tensor fields, while noticing that one has a similar presentation and the same integrability conditions for generalized, paracomplex and subtangent structures. This presentation shows that the generalized, complex, paracomplex and subtangent structures belong to the realm of Poisson geometry. Then, we prove geometric reduction theorems of Marsden-Ratiu and Marsden-Weinstein type for the mentioned generalized structures and give the characterization of the submanifolds that inherit an induced structure via the corresponding classical tensor fields.     

Poisson fiber bundles and coupling Dirac structures

Poisson fiber bundles are studied. We give sufficient conditions for the existence of a Dirac structure on the total space of a Poisson fiber bundle endowed with a compatible connection. We also provide some examples.      

Projective geometry and the theory of physical structures

In this paper, we establish connection between s-metric physical structures of rank (s + 3, 2) and projective geometry. In particular, we find explicit functional relations determining phenomenological symmetry. For s = 1, this relation is expressed in terms of the anharmonic ratio of four points. We prove that these functional relations lead to the group of projective transformations.     

Reduction of the n-dimensional static vacuum Einstein equation and generalized Schwarzschild solutions

We consider the static vacuum Einstein spacetime when the spatial factor is conformal to a n-dimensional pseudo-Euclidean space. The most general ansatz that reduces the resulting system of partial differential equations to a system of ordinary differential equations is completely described. We obtain the entire set of solutions of the reduced system, where the classical Schwarzschild solution arises as a particular solution. In addition, we show that the Riemannian spatial factors associated to these solutions are foliated by parallel hypersurfaces of constant mean curvature.     

Generalized Calabi-Yau manifolds and the chiral de Rham complex

We show that the chiral de Rham complex of a generalized Calabi-Yau manifold carries N=2 supersymmetry. We discuss the corresponding topological twist for this N=2 algebra. We interpret this as an algebroid version of the super-Sugawara or Kac-Todorov construction.     

Reduction,Linearization, and Stability of Relative Equilibria for Mechanical Systems on Riemannian Manifolds

Consider a Riemannian manifold equipped with an infinitesimal isometry. For this setup, a unified treatment is provided, solely in the language of Riemannian geometry, of techniques in reduction, linearization, and stability of relative equilibria. In particular, for mechanical control systems, an explicit characterization is given for the manner in which reduction by an infinitesimal isometry, and linearization along a controlled trajectory “commute.” As part of the development, relationships are derived between the Jacobi equation of geodesic variation and concepts from reduction theory, such as the curvature of the mechanical connection and the effective potential. As an application of our techniques, fiber and base stability of relative equilibria are studied. The paper also serves as a tutorial of Riemannian geometric methods applicable in the intersection of mechanics and control theory. F. Bullo’s research supported in part by grant CMS 0442041 from the USA National Science Foundation. A.D. Lewis’ research supported in part by grant from the Natural Sciences and Engineering Research Council of Canada.     

Matched pairs of Courant algebroids

We introduce the notion of matched pairs of Courant algebroids and give several examples arising naturally from complex manifolds, holomorphic Courant algebroids, and certain regular Courant algebroids. We also consider the matched sum of two Dirac subbundles, one in each of two Courant algebroids forming a matched pair.     

Leibniz 2-Algebras and Twisted Courant Algebroids

Let Y be an integral projective curve whose singularities are of type A_k, i.e. with only tacnodes and planar (perhaps non-ordinary) cusps. Set g:= p_a(Y). Here we study the Brill - Noether theory of spanned line bundles on Y. If the singularities are bad enough, we show the existence of spanned degree d line bundles, L, with h⁰(Y, L) ≥ r + 1 even if the Brill - Noether number ρ(g, d, r) < 0. We apply this result to prove that genus g curves with certain singularities cannot be hyperplane section of a simple K3 surface S ? P ^g.     

On higher analogues of Courant algebroids

In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle TM ⊕∧nT*M for an m-dimensional manifold. As an application, we revisit Nambu-Poisson structures and multisymplectic structures. We prove that the graph of an (n + 1)-vector field π is closed under the higher-order Dorfman bracket iff π is a Nambu-Poisson structure. Consequently, there is an induced Leibniz algebroid structure on ∧nT*M. The graph of an (n+1)-form ω is closed under...     

Generalized complex and Dirac structures on homogeneous spaces

The aim of this paper is to study generalized complex geometry (Hitchin, 2002) [6] and Dirac geometry (Courant, 1990) [3], (Courant and Weinstein, 1988) [4] on homogeneous spaces. We offer a characterization of equivariant Dirac structures on homogeneous spaces, which is then used to construct new examples of generalized complex structures. We consider Riemannian symmetric spaces, quotients of compact groups by closed connected subgroups of maximal rank, and nilpotent orbits in sln(R). For each of these cases, we completely classify equivariant Dirac structures. Additionally, we consider equivariant Dirac structures on semisimple orbits in a semisimple Lie algebra. Here equivariant Dirac structures can be described in terms of root systems or by certain data involving parabolic subalgebras.     

Lie algebroid structures on double vector bundles and representation theory of Lie algebroids

A VB-algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB-algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in special cases reproduce characteristic classes constructed by Crainic and Fernandes. We give a complete classification of regular VB-algebroids, and in the process we obtain another characteristic class of Lie algebroids that does not appear in the ordinary representation theory of Lie algebroids.     

Leibniz代数胚上动力系统的轨道范例与图示

首先陈述Leibniz代数胚上的动力系统和在局部坐标系下该动力系统的方程,在此基础上,给出了动力系统轨道的范例和图示.     

Nonsmooth differential geometry and algebras of generalized functions

Algebras of generalized functions offer possibilities beyond the purely distributional approach in modelling singular quantities in nonsmooth differential geometry. This article presents an introductory survey of recent developments in this field and highlights some applications in mathematical physics.     

Long time behaviour for generalized complex Ginzburg–Landau equation

In this paper, the two-dimensional generalized complex Ginzburg–Landau equation (CGL)

u_t=ρu−Δφ(u)−(1+iγ)Δu−νΔ²u−(1+iμ)|u|^2σu+αλ₁(|u|²u)+β(λ₂)|u|²