Projective connections

We compare the approaches of E. Cartan and of T.Y. Thomas and J.H.C. Whitehead to the study of ‘projective connections’. Although the quoted phrase has quite different meanings in the two contexts considered, we show that a class of projectively equivalent symmetric affine connections–or, more generally, sprays–on a manifold (the latter meaning) gives rise, in a global way, to a unique Cartan connection on a principal bundle over the manifold (the former meaning). The principal bundle on which the Cartan connection is defined is itself a geometric object, and exists independently of any particular connection. In the course of the discussion we derive a Cartan normal projective connection for a system of second-order ordinary differential equations (extending the results of Cartan from a single equation to many) and we generalize the concept of a normal Thomas–Whitehead connection from affine to general sprays.     

The existence of homogeneous geodesics in homogeneous pseudo-Riemannian and affine manifolds

It is well known that, in any homogeneous Riemannian manifold, there is at least one homogeneous geodesic through each point. For the pseudo-Riemannian case, even if we assume reductivity, this existence problem is still open. The standard way to deal with homogeneous geodesics in the pseudo-Riemannian case is to use the so-called “Geodesic Lemma”, which is a formula involving the inner product. We shall use a different approach: namely, we imbed the class of all homogeneous pseudo-Riemannian manifolds into the broader class of all homogeneous affine manifolds (possibly with torsion) and we apply a new, purely affine method to the existence problem. In dimension 2, it was solved positively in a previous article by three authors. Our main result says that any homogeneous affine manifold admits at least one homogeneous geodesic through each point. As an immediate corollary, we prove the same result for the subclass of all homogeneous pseudo-Riemannian manifolds.     

Topological and affine classification of complete noncompact flat 4-manifolds

In this paper we give topological and affine classification of complete noncompact flat 4-manifolds. In particular, we show that the number of diffeomorphism classes of them is equal to 44. The affine classification uses the results of [M. Sadowski, Affinely equivalent complete flat manifolds, Cent. Eur. J. Math. 2 (2) (2004) 332–338]. The affine and the topological equivalence classes are the same for flat manifolds not homotopy equivalent to S¹,T²$S^1,T^2$ or the Klein bottle.     

Definite affine spheres and loop groups

We obtain the Weierstrass-type representation and the dressing transformation for definite affine spheres in this paper. As an application of the Weierstrass-type representation, we construct the entire family of finite-Symes type affine spheres.     

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.     

Affine Yang-Mills theory and affine self-dual Chern-Simons Solitons

A four-dimensional affine Yang-Mills theory, i.e. Yang—Mills gauge theory with values in an affine Kac-Moody algebra, is constructed which can give rise to a spontaneous breakdown of the affine symmetry. The affine self-dual Yang-Mills equation (which is a special kind of affine YM theory) in four dimensions is dimensionally reduced to the affine self-dual Chem-Simons equation in two dimensions. The latter is shown to have soliton solutions which satisfy the conformal affine Toda equations.K. C. Wong Research Award Winner; address after 28 October 1992: Department of Mathematics, University of Queensland, Brisbane, Queensland 4072, Australia.     

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.     

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.     

Kirillov’s character formula,the holomorphic Peter–Weyl theorem,and the Blattner–Kostant–Sternberg pairing

By means of the orbit method we show that, for a compact Lie group, the Blattner–Kostant–Sternberg pairing map, with the constants being appropriately fixed, is unitary. Along the way we establish a holomorphic Peter–Weyl theorem for the complexification of a compact Lie group. Among our crucial tools is Kirillov’s character formula. The basic observation is that the Weyl vector is lurking behind the Kirillov character formula, as well as behind the requisite half-form correction on which the Blatter–Kostant–Sternberg-pairing for the compact Lie group relies, and thus produces the appropriate shift which, in turn, controls the unitarity of the BKS-pairing map. Our methods are independent of heat kernel harmonic analysis, which is used by B. C. Hall to obtain a number of these results [B.C. Hall, The Segal–Bargmann Coherent State Transform for compact Lie groups, J. Funct. Anal. 122 (1994) 103–151; B.C. Hall, Geometric quantization and the generalized Segal–Bargmann transform for Lie groups of compact type, Comm. Math. Phys. 226 (2002) 233–268, quant.ph/0012015].     

A generalization of Weyl's theorem on projectively related affine connections

From a physical point of view, the geodesics in a four-dimensional Lorentzian spacetime are really significant only as point sets. In 1921 Weyl proved that two torsion-free covariant derivative operators D_M and on a manifold M have the same geodesics with possibly different parametrizations if and only if there is a 1-form α on M such that , where 1 is the identity (1,1) tensor on M. By a theorem of Ambrose, Palais and Singer [1], torsion-free covariant derivative operators are generated by affine sprays, and vice versa. More generally, any (not necessarily affine) spray induces a number of covariant derivatives in the tangent bundle τ of M, or in the pull-back bundle τ∗τ. We show that in the context of sprays, similarly to Weyl's relation, a correspondence between the Yano derivatives can be detected.     

Loop Group-Valued CS and WZNW models,integrable systems,and self-dual Yang-Mills theory

Affine CS and WZNW theories with values in infinite-dimensional (loop) groups are proposed. It appears that the affine CS theory naturally introduces a spectral parameter into a CS theory. The Sinh-Gordon, KdV, and nonlinear Schrödinger equations are obtained, via Hamiltonian reductions, from the affine WZNW. It is shown that the self-dual Yang-Mills (SDYM) equation is related to the equation of motion of the affine WZNW and, thus, symmetry algebra underlying the SDYM can be identified with the affine two-loop Kac-Moody algebra of the affine WZNW.K. C. Wong Research Award Winner, address after 28 October 1992: Department of Mathematics, University of Queensland, Brisbane, Queensland 4072, Australia.     

Affine structures and KV-cohomology

We study transversely affine foliations with affine leaves from the point of view of Koszul–Vinberg modules. We have found a cohomological condition which assures that these structures determine an affine structure on the ambient manifold. The theoretical part is supplemented by suitable examples.     

Some rigidity theorems for locally symmetrical Finsler manifolds

In this paper we study some rigidity properties for locally symmetrical Finsler manifolds and obtain some results. We obtain the local equivalent characterization for a Finsler manifold to be locally symmetrical and prove that any locally symmetrical Finsler manifold with nonzero flag curvature must be Riemannian. We also generalize a rigidity result due to Akbar-Zadeh.     

The quantum Weyl group and the universal quantumR-matrix for affine lie algebraA1(1)

The universal quantumR-matrix is obtained in the case of the affine Kac-Moody Lie algebra sl(2).     

Minimal Lagrangian isotropic immersions in indefinite complex space forms

Let Mⁿ be an n-dimensional (n≥3) minimal Lagrangian isotropic submanifold in an indefinite complex space form. We show that the dimension of Mⁿ satisfies n=3r+2 with r, a positive integer. When n<14, we give a classification of such submanifolds.     

A family of Poisson non-compact symmetric spaces

We study Poisson symmetric spaces of group type with Cartan subalgebra “adapted” to the Lie cobracket.     

Normalization and prescribed extrinsic scalar curvature on lightlike hypersurfaces

In a recent paper [C. Atindogbé, Scalar curvature on lightlike hypersurfaces, Appl. Sci. 11 (2009) 9–18], the present author considered the concept of extrinsic (induced) scalar curvature on lightlike hypersurfaces. This scalar quantity has been studied on lightlike hypersurfaces equipped with a given normalization. But a very important problem was left open: How to characterize the set of all normalizations admitting a prescribed extrinsic scalar curvature? In this paper, we provide various responses to this question, supported by examples.     

Solutions of the Yang–Baxter equations on quadratic Lie groups: The case of oscillator groups

A Lie group is called quadratic if it carries a bi-invariant semi-Riemannian metric. Oscillator Lie groups constitute a subclass of the class of quadratic Lie groups. In this paper, we determine the Lie bialgebra structures and the solutions of the classical Yang–Baxter equation on a generic class of oscillator Lie algebras. Moreover, we show that any solution of the generalized classical Yang–Baxter equation (resp. classical Yang–Baxter equation) on a quadratic Lie group determines a left invariant locally symmetric (resp. flat) semi-Riemannian metric on the corresponding dual Lie groups.