International Journal of Geometric Methods in Modern Physics

In this overview article, we study the first positive eigenvalue of the Dirac operator in a unit volume conformal class. In particular, we discuss the question whether the infimum is attained. In the first part, we explain the corresponding variational problem. In the following parts we discuss the relation to the spinorial mass endomorphism and an application to surfaces of constant mean curvature. The article also mentions some open problems and work in progress.

We describe the group of continous automorphisms of all simple infinite-dimensional linearly compact Lie superalgebras and use it in order to classify 𝔽-forms of these superalgebras over any field 𝔽 of characteristic zero.

The Dutch graphic artist Escher was very much interested in plane symmetry. He was fascinated by the problem of infinity and the representation of infinity on a sheet of paper. An exchange of ideas with Coxeter was the origin of the series of works called "Circle Limit". This article also describes the animation technique, which by adding time and movement to the images of Escher try to contribute to Escher's dream of reaching infinity.

In this expository article we discuss some ideas and results which might lead to a theory of infinite dimensional symmetric spaces where is an affine Kac–Moody group and the fixed point group of an involution (of the second kind). We point out several striking similarities of these spaces with their finite dimensional counterparts and discuss their geometry. Furthermore we sketch a classification and show that they are essentially in 1 : 1 correspondence with hyperpolar actions on compact simple Lie groups.

We study the geometry of hypersurfaces in manifolds with Ricci-flat holonomy group, on which we introduce a G-structure whose intrinsic torsion can be identified with the second fundamental form. The general problem of extending a manifold with such a G-structure so as to invert this construction is open, but results exist in particular cases, which we review. We list the five-dimensional nilmanifolds carrying invariant SU(2)-structures of this type, and present an example of an associated metric with holonomy SU(3).

We review the map between hypercomplex manifolds that admit a closed homothetic Killing vector (i.e. "conformal hypercomplex" manifolds) and quaternionic manifolds of one dimension less. This map is related to a method for constructing supergravity theories using superconformal techniques. An explicit relation between the structure of these manifolds is presented, including curvatures and symmetries. An important role is played by "ξ transformations," relating connections on quaternionic manifolds, and a new type " transformations" relating complex structures on conformal hypercomplex manifolds. In this map, the subclass of conformal hyper-Kähler manifolds is mapped to quaternionic-Kähler manifolds.

We survey results on infinitesimal deformations ("Jacobi fields") of harmonic maps, concentrating on (i) when they are integrable, i.e., arise from genuine deformations, and what this tells us, (ii) their relation with harmonic morphisms — maps which preserve Laplace's equation.

A flag manifold is a homogeneous space M = G/K, where G is a compact semisimple Lie group, and K the centralizer of a torus in G. Equivalently, M can be identified with the adjoint orbit Ad(G)w of an element w in the Lie algebra of G. We present several aspects of flag manifolds, such as their classification in terms of painted Dynkin diagrams, T-roots and G-invariant metrics, and Kähler metrics. We give a Lie-theoretic expression of the Ricci tensor in M, hence reducing the Einstein equation on flag manifolds into an algebraic system of equations, which can be solved in several cases. A flag manifold is also a complex manifold, and this dual representation as a real and a complex manifold is related to a similar property of an infinite-dimensional manifold, the loop space, which in fact can be viewed as a "universal" flag manifold.

A complete decomposition of the space of the curvature tensors over tensor product of vector spaces into simple modules under the action of the group G = GL(p, ℝ) ⊗ GL(q, ℝ) is given. We use these results to study geometry of manifolds with Grassmann structure and Grassmann manifolds endowed with a connection whose torsion is not zero. We show that Osc^r M a manifold is an example of a manifold with Grassmann structure. Owing to this fact, we consider results of Miron, Atanasiu, Anastasiei, Čomić and others from representation theory point of view and connect them with some results of Alekseevsky, Cortes, and Devchand, as well as of Machida and Sato, and others. New examples of connections with torsion defined on four-dimensional Grassmann manifold are given. Symmetries of curvatures for half-flat connections are also investigated. We use algebraic results to reveal obstructions to the existence of corresponding connections.

We prove that any special para-Kähler manifold is intrinsically an improper affine hypersphere. As a corollary, any para-holomorphic function F of n para-complex variables satisfying a non-degeneracy condition defines an improper affine hypersphere, which is the graph of a real function f of 2n variables. We give an explicit formula for the function f in terms of the para-holomorphic function F. Necessary and sufficient conditions for an affine hypersphere to admit the structure of a special para-Kähler manifold are given. Finally, it is shown that conical special para-Kähler manifolds are foliated by proper affine hyperspheres of constant mean curvature.

A specialp-form is a p-form which, in some orthonormal basis {e_μ}, has components φ_μ1⋯μp = φ(e_μ1,…,e_μp) taking values in {-1, 0, 1}. We discuss graphs which characterize such forms.

The object of this article is to study a torus action on a so-called Berger sphere. We also make some comments on polar actions on naturally reductive homogeneous spaces. Finally, we prove a rigidity-type theorem for Riemannian manifolds carrying a polar action with a fix point.

All candidates to the weakly-irreducible not irreducible holonomy algebras of Lorentzian manifolds are known. In the present paper metrics that realize all these candidates as holonomy algebras are given. This completes the classification of the Lorentzian holonomy algebras. Also new examples of metrics with the holonomy algebras g₂ ⋉ ℝ⁷ ⊂ 𝔰𝔬(1, 8) and 𝔰𝔭𝔦𝔫(7) ⋉ ℝ⁸ ⊂ 𝔰𝔬(1, 9) are constructed.

We associate to a homogeneous manifold M = G/H, with a simple spectrum of the isotropy representation, a compact convex polytope P_M which is the Newton polytope of the rational function s(t) and that to each invariant metric t of M associates its scalar curvature. We estimate the number of isolate invariant holomorphic Einstein metrics (up to homothety) on M^ℂ = G^ℂ/H^ℂ. Using the results of A. G. Kouchnirenko and D. N. Bernstein, we prove that , where ν(M) is the integer volume of P_M, and give conditions when the defect . In case when G is a compact semisimple Lie group, the positiveness of d(M) is related with the existence of Ricci-flat holomorphic metric on a complexification of a noncompact homogeneous space M_γ = G_γ/H_P which is a contraction of M and is associated with a proper face γ of P_M.

A discussion is given of the sectional curvature function on a four-dimensional Lorentz manifold and, in particular, on the space–time of Einstein's general relativity theory. Its tight relationship to the metric tensor is demonstrated and some of its geometrical and algebraic properties evaluated. The concept of a sectional curvature preserving symmetry, in the form of a certain smooth vector field, is introduced and discussed.

This is a survey article for the relationship between spherical CR-manifolds and Bochner flat Kähler manifolds. We shall give a uniformization of Bochner flat Kähler manifolds. A Bochner flat locally conformal Kähler manifold is a locally conformal Kähler manifold with vanishing Bochner curvature tensor. We shall apply our result to Bochner flat locally conformal Kähler manifolds.

Let M be a G-manifold and ω a G-invariant exact m-form on M. We indicate when these data allow us to construct a cocycle on a group G with values in the trivial G-module ℝ, and when this cocycle is nontrivial.

We study the equivalence problem of submanifolds with respect to a transitive pseudogroup action. The corresponding differential invariants are determined via formal theory and lead to the notions of l-variants and l-covariants, even in the case of non-integrable pseudogroup. Their calculation is based on the cohomological machinery: we introduce a complex for covariants, define their cohomology and prove the finiteness theorem. This implies the well-known Lie–Tresse theorem about differential invariants. We also generalize this theorem to the case of pseudogroup action on differential equations.

It is well known that starting with real structure, the Cayley–Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions p = 2, 4 and 8, respectively, but the procedure fails for p = 16 in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are n = 2⁷. Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower type related to complex and Pauli structures and, in relation to the iteraton process p → p + 2 → p + 4 → ⋯, they have constructed 2⁴-dimensional "bipetals" for p = 9 and 2⁷-dimensional "bisepals" for p = 13. The objects constructed appear to have an interesting property of periodicity related to the gradating function on the fractal diagonal interpreted as the "pistil" and a family of pairs of segments parallel to the diagonal and equidistant from it, interpreted as the "stamens." The present paper aims at an effective, explicit determination of the periods and expressing them in terms of complex and quaternionic structures, thus showing the quaternionic background of that periodicity. The proof of the Periodicity Theorem is given in the case where the index of the generator of the algebra in question exceeds the order of the initial algebra. In contrast to earlier results, the fractal bundle flower structure, in particular sepals, bisepals, perianths, and calyces are not introduced ab initio; they are quoted a posteriori, when they are fully motivated. The same concerns canonical two-layer pairing of prianth sepals and the relationship of fractal bundles of algebraic structure with the Hurwitz problem.

In this paper we show some results on homogeneous CR manifolds, proved by introducing their associated CR algebras. In particular, we give different notions of nondegeneracy (generalizing the usual notion for the Levi form) which correspond to geometrical properties for the corresponding manifolds. We also give distinguished equivariant CR fibrations for homogeneous CR manifolds. In the second part of the paper we apply these results to minimal orbits for the action of a real form of a semisimple Lie group Ĝ on a flag manifold Ĝ/Q.

We collect some properties of the moment map relative to the isometric and holomorphic action of a compact Lie group G on a (compact) Kähler (Einstein) manifold; in particular, we study some invariants which only depend on the cohomology class of the invariant Kähler form and then specialize to the complex projective space when the group G is simple and acts linearly.

We compute explicit transgression forms for the Euler and Pontrjagin classes of a Riemannian manifold M of dimension 4 under a conformal change of the metric, or a change to a Riemannian connection with torsion. These formulas describe the singular set of some connections with singularities on compact manifolds as a residue formula in terms of a polynomial of invariants. We give some applications for minimal submanifolds of Kähler manifolds. We also express the difference of the first Chern class of two almost complex structures, and in particular an obstruction to the existence of a homotopy between them, by a residue formula along the set of anti-complex points. Finally we take the first steps in the study of obstructions for two almost quaternionic-Hermitian structures on a manifold of dimension 8 to have homotopic fundamental forms or isomorphic twistor spaces.

A generalization to the almost complex setting of a well-known result by Webster is given. Namely, we prove that if Γ is a strongly pseudoconvex hypersurface in an almost complex manifold (M, J), then the conormal bundle of Γ is a totally real submanifold of (T* M, 𝕁), where 𝕁 is the lifted almost complex structure on T* M defined by Ishihara and Yano.

Some event horizons in space–times that are invariant under an isometric action, considered first by Carter, are called isometry horizons, especially Killing horizons. In this paper, isometry horizons in spherically symmetric space–times are considered. It is shown that these isometry horizons are all Killing horizons.

We consider generalized Calabi–Yau manifolds and we give a formula for the Maslov class of a Lagrangian submanifold of a generalized Calabi–Yau manifold. In particular, we characterize the Lagrangian submanifolds with vanishing Maslov class. In the 6-dimensional case, we refine our definition. Finally, we construct some examples.