Manifolds with Many Rarita–Schwinger Fields

The Rarita–Schwinger operator is the twisted Dirac operator restricted to $$\nicefrac 32$$ 3 2 -spinors. Rarita–Schwinger fields are solutions of this operator which are in addition divergence-free. This is an overdetermined problem and solutions are rare; it is even more unexpected for there to be large dimensional spaces of solutions. In this paper we prove the existence of a sequence of compact manifolds in any given dimension greater than or equal to 4 for which the dimension of the space of Rarita–Schwinger fields tends to infinity. These manifolds are either simply connected Kähler–Einstein spin with negative Einstein constant, or products of such spaces with flat tori. Moreover, we construct Calabi–Yau manifolds of even complex dimension with more linearly independent Rarita–Schwinger fields than flat tori of the same dimension.

Conformally Killing Fields on 2-Symmetric Five-Dimensional Lorentzian Manifolds

The papers of many mathematicians are devoted to the study of conformally Killing vector fields. Being a natural generalization of the concept of Killing vector fields, these fields generate a Lie algebra corresponding to the Lie group of conformal transformations of the manifold. Moreover, they generate the class of locally conformally homogeneous (pseudo) Riemannian manifolds studied by V.V. Slavsky and E.D. Rodionov. Ricci solitons, which R. Hamilton first considered, are another important area of research. Ricci solitons are a generalization of Einstein's metrics on (pseudo) Riemannian manifolds. The Ricci soliton equation has been studied on various classes of manifolds by many mathematicians. In particular, a general solution of the Ricci soliton equation was found on 2-symmetric Lorentzian manifolds of low dimension, and the solvability of this equation in the class of 3-symmetric Lorentzian manifolds was proved. The Killing vector fields make it possible to find the general solution of the Ricci soliton equation in the case of the constancy of the Einstein constant in the Ricci soliton equation. However, the role of the Killing fields is played by conformally Killing vector fields for different values of the Einstein constant. In this paper, we investigate conformal Killing vector fields on 5-dimensional 2-symmetric Lorentzian manifolds. The general solution of the conformal analog of the Killing equation on five-dimensional locally indecomposable 2-symmetric Lorentzian manifolds is described in local coordinates, discovered by A.S. Galaev and D.V. Alekseevsky.

On the Betti and Tachibana Numbers of Compact Einstein Manifolds

Throughout the history of the study of Einstein manifolds, researchers have sought relationships between the curvature and topology of such manifolds. In this paper, first, we prove that a compact Einstein manifold ( M , g ) with an Einstein constant α > 0 is a homological sphere when the minimum of its sectional curvatures > α / ( n + 2 ) ; in particular, ( M , g ) is a spherical space form when the minimum of its sectional curvatures > α / n . Second, we prove two propositions (similar to the above ones) for Tachibana numbers of a compact Einstein manifold with α < 0 .

Einstein metrics, complex surfaces, and symplectic 4-manifolds

Which smooth compact 4-manifolds admit an Einstein metric with non-negative Einstein constant? A complete answer is provided in the special case of 4-manifolds that also happen to admit either a complex structure or a symplectic structure.

NEW SELF-DUAL EINSTEIN METRICS

A new family of euclidean Einstein metrics with self-dual Weyl tensor have been obtained using ideas from extended supersymmetries1,2. The basic supersymmetric formalism3, known as harmonic superspace, was adapted to the computation of self-dual Einstein metrics in 4. The resulting metric depends on 4 parameters besides the Einstein constant and has for isometry group U(1) × U(1), with hypersurface generating Killing vectors. In the limit of vanishing Einstein constant we recover a family of hyperkähler metrics within the Multicentre family 5 (in fact the most general one with two centres). Our results include the metrics of Plebanski and Demianski6 when these ones are restricted to be self-dual Weyl. From Flaherty's equivalence 7 these metrics can also be interpreted as a solution of the coupled Einstein-Maxwell field equations, for which we have given the Maxwell field strength forms2.