Pseudo-Riemannian manifolds with recurrent spinor fields

The existence of a recurrent spinor field on a pseudo-Riemannian spin manifold (M,g) is closely related to the existence of a parallel 1-dimensional complex subbundle of the spinor bundle of (M,g). We characterize the following simply connected pseudo-Riemannian manifolds that admit these subbundles in terms of their holonomy algebras: Riemannian manifolds, Lorentzian manifolds, pseudo-Riemannian manifolds with irreducible holonomy algebras, and pseudo-Riemannian manifolds of neutral signature admitting two complementary parallel isotropic distributions.     

Embedding into manifolds with torsion

We introduce a class of special geometries associated to the choice of a differential graded algebra contained in ${\Lambda^*\mathbb{R}^n}$ . We generalize some known embedding results, that effectively characterize the real analytic Riemannian manifolds that can be realized as submanifolds of a Riemannian manifold with special holonomy, to this more general context. In particular, we consider the case of hypersurfaces inside nearly-K?hler and ??-Einstein?CSasaki manifolds, proving that the corresponding evolution equations always admit a solution in the real analytic case.     

Compact Lorentzian holonomy

We consider (compact or noncompact) Lorentzian manifolds whose holonomy group has compact closure. This property is equivalent to admitting a parallel timelike vector field. We give some applications and derive some properties of the space of all such metrics on a given manifold.     

Conformal holonomy of C-spaces, Ricci-flat, and Lorentzian manifolds

The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebra of a C-space is a Berger algebra. For Ricci-flat spaces we show how the conformal holonomy can be obtained by the holonomy of the ambient metric and get results for Riemannian manifolds and plane waves.     

Generalized Killing spinors on spheres

We study generalized Killing spinors on round spheres \(\mathbb {S}^n\) . We show that on the standard sphere \(\mathbb {S}^8\) any generalized Killing spinor has to be an ordinary Killing spinor. Moreover, we classify generalized Killing spinors on \(\mathbb {S}^n\) whose associated symmetric endomorphism has at most two eigenvalues and recover in particular Agricola–Friedrich’s canonical spinor on 3-Sasakian manifolds of dimension 7. Finally, we show that it is not possible to deform Killing spinors on standard spheres into genuine generalized Killing spinors.     

On non-simply connected manifolds with non-trivial parallel spinors

We examine the possibilities of the full holonomy groups of locally irreducible but not necessarily complete Riemannian spin manifolds admitting a non-trivial parallel spinor and discuss some applications of this classification.partially supported by NSERC Grant No. OPG0009421     

Conformal Operators on Weighted Forms; Their Decomposition and Null Space on Einstein Manifolds

There is a class of Laplacian like conformally invariant differential operators on differential forms ${L^\ell_k}$ which may be considered as the generalisation to differential forms of the conformally invariant powers of the Laplacian known as the Paneitz and GJMS operators. On conformally Einstein manifolds we give explicit formulae for these as factored polynomials in second-order differential operators. In the case that the manifold is not Ricci flat we use this to provide a direct sum decomposition of the null space of the ${L^\ell_k}$ in terms of the null spaces of mutually commuting second-order factors.     

Lorentzian Legendrian mean curvature flow

In this paper we generalize the Legendrian mean curvature flow to Lorentzian geometry. More precisely, we study the case, where the ambient manifold is a Lorentzian Sasaki $\eta $ -Einstein manifold. For Legendrian curves we establish convergence results in Theorems 1.1 and 1.2 and we derive estimates for the Legendrian angle for arbitrary dimensions in Theorem 1.3.     

Kernel based quadrature on spheres and other homogeneous spaces

Quadrature formulas for spheres, the rotation group, and other compact, homogeneous manifolds are important in a number of applications and have been the subject of recent research. The main purpose of this paper is to study coordinate independent quadrature (or cubature) formulas associated with certain classes of positive definite and conditionally positive definite kernels that are invariant under the group action of the homogeneous manifold. In particular, we show that these formulas are accurate—optimally so in many cases—and stable under an increasing number of nodes and in the presence of noise, provided the set $X$ of quadrature nodes is quasi-uniform. The stability results are new in all cases. In addition, we may use these quadrature formulas to obtain similar formulas for manifolds diffeomorphic to $\mathbb S ^n$ , oblate spheroids for instance. The weights are obtained by solving a single linear system. For $\mathbb S ^2$ , and the restricted thin plate spline kernel $r^2\log r$ , these weights can be computed for two-thirds of a million nodes, using a preconditioned iterative technique introduced by us.     

Globally hyperbolic Lorentzian spaces with special holonomy groups

We prove that each special Lorentzian holonomy group (with the exception of those including the isotropy groups of Kähler symmetric spaces) can be realized as the holonomy group of a globally hyperbolic Lorentzian manifold.     

Convergence of Vector Bundles with Metrics of Sasaki-Type

If a sequence of Riemannian manifolds, X _i , converges in the pointed Gromov–Hausdorff sense to a limit space, X _∞, and if E _i are vector bundles over X _i endowed with metrics of Sasaki-type with a uniform upper bound on rank, then a subsequence of the E _i converges in the pointed Gromov–Hausdorff sense to a metric space, E _∞. The projection maps π _i converge to a limit submetry π _∞ and the fibers converge to its fibers; the latter may no longer be vector spaces but are homeomorphic to \(\mathbb {R}^{k}/G\) , where G, henceforth called the wane group, is a closed subgroup of O(k) that depends on the basepoint and that is defined using the holonomy groups on the vector bundles. The norms μ _i =∥?∥ _i converge to a map μ _∞ compatible with the rescaling in \(\mathbb {R}^{k}/G\) and the \(\mathbb {R}\) -action on E _i converges to an \(\mathbb {R}\) -action on E _∞ compatible with the limiting norm. A natural notion of parallelism is given to the limiting spaces by considering curves whose length is unchanged under the projection. The class of such curves is invariant under the \(\mathbb {R}\) -action and each such curve preserves norms. The existence of parallel translation along rectifiable curves with arbitrary initial conditions is also exhibited. Also, necessary conditions for uniqueness of parallel translates are given in terms of the wane groups. In the special case when the sequence of vector bundles has a uniform lower bound on holonomy radius (as in a sequence of collapsing flat tori to a circle), the limit fibers are vector spaces. Under the opposite extreme, e.g., when a single compact n-dimensional manifold is rescaled to a point, the limit fiber is \(\mathbb {R}^{n}/H\) where H is the closure of the holonomy group of the compact manifold considered. Both these examples have uniqueness of parallel translates. However, examples for non-uniqueness are also produced by looking at isolated conical singularities.     

Cocalibrated structures on Lie algebras with a codimension one Abelian ideal

Cocalibrated G₂-structures and cocalibrated ${{\rm G}_2^*}$ -structures are the natural initial values for Hitchin’s evolution equations whose solutions define (pseudo)-Riemannian manifolds with holonomy group contained in Spin(7) or Spin₀(3, 4), respectively. In this article, we classify 7-D real Lie algebras with a codimension one Abelian ideal which admit such structures. Moreover, we classify the 7-D complex Lie algebras with a codimension one Abelian ideal which admit cocalibrated ${({\rm G}_2)_{\mathbb{C}}}$ -structures.     

Parallel pure spinors and holonomy

We characterize the spin pseudo-Riemannian manifolds which admit parallel pure spinors by their holonomy groups. In particular, we study the Lorentzian case. To cite this article: A. Ikemakhen, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On geometry of flat complete strictly causal Lorentzian manifolds

A flat complete causal Lorentzian manifold is called strictly causal if the past and future of its every point are closed near this point. We consider the strictly causal manifolds with unipotent holonomy groups and assign to a manifold of this type four nonnegative integers (a signature) and a parabola in the cone of positive definite matrices. Two manifolds are equivalent if and only if their signatures coincide and the corresponding parabolas are equal (up to a suitable automorphism of the cone and an affine change of variable). Also, we give necessary and sufficient conditions distinguishing the parabolas of this type among all parabolas in the cone.     

Generalized quaternionic manifolds

We initiate the study of the generalized quaternionic manifolds by classifying the generalized quaternionic vector spaces, and by giving two classes of nonclassical examples of such manifolds. Thus, we show that any complex symplectic manifold is endowed with a natural (nonclassical) generalized quaternionic structure, and the same applies to the heaven space of any three-dimensional Einstein–Weyl space. In particular, on the product \(Z\) of any complex symplectic manifold \(M\) and the sphere, there exists a natural generalized complex structure, with respect to which \(Z\) is the twistor space of  \(M\) .     

Closed geodesics and holonomies for Kleinian manifolds

For a rank one Lie group G and a Zariski dense and geometrically finite subgroup \({\Gamma}\) of G, we establish the joint equidistribution of closed geodesics and their holonomy classes for the associated locally symmetric space. Our result is given in a quantitative form for geometrically finite real hyperbolic manifolds whose critical exponents are big enough. In the case when \({G={\rm PSL}_2 (\mathbb{C})}\) , our results imply the equidistribution of eigenvalues of elements of Γ in the complex plane. When \({\Gamma}\) is a lattice, the equidistribution of holonomies was proved by Sarnak and Wakayama in 1999 using the Selberg trace formula.     

Spherical symmetrization and the first eigenvalue of geodesic disks on manifolds

Given a manifold \(M\) , we build two spherically symmetric model manifolds based on the maximum and the minimum of its curvatures. We then show that the first Dirichlet eigenvalue of the Laplace–Beltrami operator on a geodesic disk of the original manifold can be bounded from above and below by the first eigenvalue on geodesic disks with the same radius on the model manifolds. These results may be seen as extensions of Cheng’s eigenvalue comparison theorems, where the model constant curvature manifolds have been replaced by more general spherically symmetric manifolds. To prove this, we extend Rauch’s and Bishop’s comparison theorems to this setting.     

Conical Ricci-flat nearly para-Kähler manifolds

In this paper, we classify conical Ricci-flat nearly para-Kähler manifolds having isotropic Nijenhuis tensor. More precisely, we give a bijective correspondence between this class of nearly para-Kähler manifolds and local cones $M_1 \times (a,b)$ over para-Sasaki-Einstein manifolds $(M_1,g_1,T)$ carrying a parallel 3-form with isotropic support. Moreover, we show that the cone over a para-Sasaki-Einstein five-manifold $(M_1,g_1,T)$ admits a family of parallel 3-forms with isotropic support. As an application our result yields first examples of Ricci-flat (non-flat) nearly para-Kähler structures.     

New HKT manifolds arising from quaternionic representations

We give a procedure for constructing an 8n-dimensional HKT Lie algebra starting from a 4n-dimensional one by using a quaternionic representation of the latter. The strong (respectively, weak, hyper-K?hler, balanced) condition is preserved by our construction. As an application of our results we obtain a new compact HKT manifold with holonomy in ${SL(n,\mathbb{H})}$ which is not a nilmanifold. We find in addition new compact strong HKT manifolds. We also show that every K?hler Lie algebra equipped with a flat, torsion-free complex connection gives rise to an HKT Lie algebra. We apply this method to two distinguished 4-dimensional K?hler Lie algebras, thereby obtaining two conformally balanced HKT metrics in dimension 8. Both techniques prove to be an effective tool for giving the explicit expression of the corresponding HKT metrics.     

Symplectic Dolbeault operators on Kähler manifolds

For a Kähler manifold $M$ , the “symplectic Dolbeault operators” are defined using the symplectic spinors and associated Dirac operators, in complete analogy to how the usual Dolbeault operators, $\bar{\partial }$ and $\bar{\partial }^*$ , arise from Dirac operators on the canonical complex spinors on $M$ . We give special attention to two special classes of Kähler manifolds: Riemann surfaces and flag manifolds ( $G/T$ for $G$ a simply-connected compact semisimple Lie group and $T$ a maximal torus). For Riemann surfaces, the symplectic Dolbeault operators are elliptic and we compute their indices. In the case of flag manifolds, we will see that the representation theory of $G$ plays a role and that these operators can be used to distinguish (as Kähler manifolds) between the flag manifolds corresponding to the Lie algebras $B_n$ and $C_n$ . We give a thorough analysis of these operators on $\mathbb{C } P^1$ (the intersection of these classes of spaces), where the symplectic Dolbeault operators have an especially interesting structure.