We construct exact solutions of the Einstein-Dirac equation, which couples the gravitational field with an eigenspinor of the Dirac operator via the energy-momentum tensor. For this purpose we introduce a new field equation generalizing the notion of Killing spinors. The solutions of this spinorial field equation are called weak Killing spinors (WK-spinors). They are special solutions of the Einstein-Dirac equation and in dimension n=3 the two equations essentially coincide. It turns out that any Sasakian manifold with Ricci tensor related in some special way to the metric tensor as well as to the contact structure admits a WK-spinor. This result is a consequence of the investigation of special spinorial field equations on Sasakian manifolds (Sasakian quasi-Killing spinors). Altogether, in odd dimensions a contact geometry generates a solution of the Einstein-Dirac equation. Moreover, we prove the existence of solutions of the Einstein-Dirac equations that are not WK-spinors in all dimensions n > 8.
We study some similarities between almost product Riemannian structures and almost Hermitian structures. Inspired by the similarities, we prove lower eigenvalue estimates for the Dirac operator on compact Riemannian spin manifolds with locally product structures. We also provide some examples (limiting manifolds) for the limiting case of the estimates.
We prove a lower estimate for the first eigenvalue of the Dirac operator on a compact locally reducible Riemannian spin manifold with positive scalar curvature. We determine also the universal covers of the manifolds on which the smallest possible eigenvalue is attained.
In this paper, we get estimates on the higher eigenvalues of the Dirac operator on locally reducible Riemannian manifolds, in terms of the eigenvalues of the Laplace-Beltrami operator and the scalar curvature. These estimates are sharp, in the sense that, for the first eigenvalue, they reduce to the result of Alexandrov.
This paper contains a classification of all 3-dimensional manifolds with constant scalar curvature $S \not= 0$ that carry a non-trivial solution of the Einstein-Dirac equation.
We show the existence of nonsymmetric homogeneous spin Riemannian manifolds whose Dirac operator is like that on a Riemannian symmetric spin space. Such manifolds are exactly the homogeneous spin Riemannian manifolds $(M,g)$ which are traceless cyclic with respect to some quotient expression $M=G/K$ and reductive decomposition $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{m}$. Using transversally symmetric fibrations of noncompact type, we give a list of them.
Necessary and sufficient conditions for a Riemannian product to be conformally equivalent to an Einstein manifold are given. Such spaces which are complete are characterized.
We give a new upper bound for the smallest eigenvalues of the Dirac operator on a Riemannian flow carrying transversal Killing spinors. We derive an estimate on Sasakian and on 3-dimensional manifolds and partially classify those satisfying the limiting case. Finally, we compare our estimate with a lower bound in terms of a natural tensor depending on the eigenspinor.
In this paper, we describe the group SpinT (n) and give some properties of this group. We construct SpinT spinor bundle S by means of the spinor representation of the group SpinT (n) and define covariant derivative operator and Dirac operator on S. Finally, Schrodinger-Lichnerowicz-type formula is derived by using these operators.
We extract a new class of paracontact paracomplex Riemannian manifolds arising from certain cone construction, call it para-Sasaki-like Riemannian manifold and give explicit examples. We define a hyperbolic extension of a paraholomorphic paracomplex Riemannian manifold, which is a local product of two Riemannian spaces with equal dimensions, showing that it is a para-Sasaki-like Riemannian manifold. If the starting paraholomorphic paracomplex Riemannian manifold is complete Einstein with negative scalar curvature then its hyperbolic extension is a complete Einstein para-Sasaki-like Riemannian …
We consider local geometry of sub-pseudo-Riemannian structures on contact manifolds. We construct fundamental invariants of the structures and show that the structures give rise to Einstein-Weyl geometries in dimension 3, provided that certain additional conditions are satisfied.
In this paper, we first prove that homogeneous spaces $E_6/A_4$ and $E_6/A_1$ admit Einstein metrics which are $Ad(T\times A_1\times A_4)$-invariant, and then show that they admit Non-Riemannian Einstein-Randers metrics.
We ask a general question: what are locally homogeneous compact pseudo-Riemannian Einstein manifolds? We show that any standard compact Clifford-Klein form of a simple non-compact Lie group admits at least one Einstein metric. We conjecture that these are basically the only possible locally homogeneous Einstein pseudo-Riemannian compact manifolds using T. Kobayashi's conjecture as a guiding principle.
The main purpose of the paper is to prove that if a compact Riemannian manifold admits a gradient $ρ$-Einstein soliton such that the gradient Einstein potential is a non-trivial conformal vector field, then the manifold is isometric to the Euclidean sphere. We have showed that a Riemannian manifold satisfying gradient $ρ$-Einstein soliton with convex Einstein potential possesses non-negative scalar curvature. We have also deduced a sufficient condition for a Riemannian manifold to be compact which satisfies almost $η$-Ricci soliton (see Theorem 2).
The aim of this note is the study of Einstein condition for para-holomorphic Riemannian metrics in the para-complex geometry framework. Firstly, we make some general considerations about para-complex Riemannian manifolds (not necessarily para-holomorphic). Next, using an one-to-one correspondence between para-holomorphic Riemannian metrics and para-Kähler Norden metrics, we study the Einstein condition for a para-holomorphic Riemannian metric and the associated real para-Kähler Norden metric on a para-complex manifold. Finally, it is shown that every semi-simple para-complex Lie group inherits…
In this paper we prove a formula for the analytic index of a basic Dirac-type operator on a Riemannian foliation, solving a problem that has been open for many years. We also consider more general indices given by twisting the basic Dirac operator by a representation of the orthogonal group. The formula is a sum of integrals over blowups of the strata of the foliation and also involves eta invariants of associated elliptic operators. As a special case, a Gauss-Bonnet formula for the basic Euler characteristic is obtained using two independent proofs.
The holonomy algebras of Einstein not Ricci-flat pseudo-Riemannian manifolds of arbitrary signature are classified. As illustrating examples, the cases of Lorentzian manifolds, pseudo-Riemannian manifolds of signature $(2,n)$ and the para-quaternionic-Kählerian manifolds with non-zero scalar curvature are considered. Einstein not Ricci-flat metrics of signature $(2,n)$ with all possible holonomy algebras are given.
It is well-known that the spectrum of a $\text{spin}^{\mathbb{C}}$ Dirac operator on a closed Riemannian $\text{spin}^{\mathbb{C}}$ manifold $M^{2k}$ of dimension $2k$ for $k \in \mathbb{N}$ is symmetric. In this article, we prove that over an odd-dimensional Riemannian product $M_{1}^{2p} \times M_{2}^{2q+1}$ with a product $\text{spin}^{\mathbb{C}}$ structure for $p \geq 1, q \geq 0$, the spectrum of a $\text{spin}^{\mathbb{C}}$ Dirac operator given by a product connection is symmetric if and only if either the $\text{spin}^{\mathbb{C}}$ Dirac spectrum of $M_{2}^{2q+1}$ is symmetric or $(e^{…
In this paper we consider a class of Einstein warped product semi-Riemannian manifolds $\widehat{M} = M^{n}\times_{f}N^{m}$ with $n\geq 3$ and $m\geq 2$. For $\widehat{M}$ with compact base and Ricci-flat fiber, we prove that $\widehat{M}$ is simply a Riemannian product space. Then, when the base $M$ is conformal to a pseudo-Euclidean space which is invariant under the action of a $(n-1)$-dimensional translation group, we classify all such spaces. Furthermore, we get new examples of complete Einstein warped products Riemannian manifolds.
In this paper, we study gradient Einstein-type structure immersed into a Riemannian warped product manifold. We obtain some triviality results for the potential function and smooth map $u$. We investigate conditions for a gradient Einstein-type structure to be minimal, totally umbilical, or totally geodesic immersed into a warped product $I\times_f M^n$. Furthermore, we characterize rotational hypersurface into $\mathbb{R}\times_f\mathbb{R}^n$ has a gradient Einstein-type structure.
For closed manifolds endowed with a Riemannian foliation of codimension $4$, one can define a transversal Seiberg-Witten map. We show that there is a finite dimensional approximation for such a map. By such a method and under the condition that $H^1_b(M)\cap H^1(M,\mathbb Z)$ is a lattice of $H^1_b(M)$, we can define a foliated version of Bauer-Furuta invariant. Moreover, if the basic cohomological group is of zero dimension, we can give an estimate for the index of transversal Dirac operator of a foliated spin structure. Furthermore, under the condition that $H^\pm_b(M)=1$, we show the vanish…
In this paper, we use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on $S^3$ with ${\rm Ric} = 2 F^2$, ${\rm Ric}=0$ and ${\rm Ric}=- 2 F^2$, respectively. This family of metrics provide an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant, but the flag curvature is not.
We give results about the L^2 kernel and the spectrum of the Dirac operator on a complete Riemannian manifold which is conformally equivalent to the interior of a Riemannian manifold with nonempty boundary.
We study the spectrum of the Dirac operator $D$ on pseudo-Riemannian spin manifolds of signature $(p,q)$, considered as an unbounded operator in the Hilbert space $L^2_ξ(S)$. The definition of $L^2_ξ(S)$ involves the choice of a $p$-dimensional time-like subbundle $ξ\subset TM$. We establish a sufficient criterion for the spectra of $D$ induced by two maximal time-like subbundles $ξ_1,ξ_2\subset TM$ to be equal. If the base manifold $M$ is compact, the spectrum does not depend on $ξ$ at all. We then proceed by explicitely computing the full spectrum of $D$ for $\mathbb R^{p,q}$, the flat torus…
We study warped products semi-Riemannian Einstein manifolds. We consider the case in that the base is conformal to an n-dimensional pseudo Euclidean space and invariant under the action of an translation group. We provide all such solutions in the case Ricci flat when the base is conformal to an n-dimensional pseudo-Euclidean space, invariant under the action of an translation group and the fiber F is Ricci flat. In particular, we obtain explicit solutions, in the case vacuum, for the Einstein field equation.
We answer in the affirmative the question posed by Conti and Rossi on the existence of nilpotent Lie algebras of dimension 7 with an Einstein pseudo-metric of nonzero scalar curvature. Indeed, we construct a left-invariant pseudo-Riemannian metric $g$ of signature $(3, 4)$ on a nilpotent Lie group of dimension 7, such that $g$ is Einstein and not Ricci-flat. We show that the pseudo-metric $g$ cannot be induced by any left-invariant closed $G_2^*$-structure on the Lie group. Moreover, some results on closed and harmonic $G_2^*$-structures on an arbitrary 7-manifold $M$ are given. In particular,…
We show that a connection with skew-symmetric torsion satisfying the Einstein metricity condition exists on an almost contact metric manifold exactly when it is D-homothetic to a cosymplectic manifold. In dimension five, we get that the existence of a connection with skew torsion satisfying the Einstein metricity condition is equivalent to the existence of a Sasaki-Einstein 5-manifold and vice versa, any Sasaki-Einstein 5-manifold generates a two parametric family of connections with skew torsion satisfying the Einstein metricity condition. Formulas for the curvature and the Ricci tensors of t…
In this paper we prove that the compact Lie group $G_2$ admits a left-invariant Einstein metric that is not geodesic orbit. In order to prove the required assertion, we develop some special tools for geodesic orbit Riemannian manifolds. It should be noted that a suitable metric is discovered in a recent paper by I. Chrysikos and Y. Sakane, where the authors proved also that this metric is not naturally reductive.
We characterize the Dirac structures that are parallel with respect to Gualtieri's canonical connection of a generalized Riemannian metric. On the other hand, we discuss Dirac structures that are images of generalized tangent structures. These structures turn out to be Dirac structures that, if seen as Lie algebroids, have a symplectic structure. Particularly, if compatibility with a generalized Riemannian metric is required, the symplectic structure is of the Kaehler type.
It is a well-known fact that on a bounded spectral interval the Dirac spectrum can be described locally by a non-decreasing sequence of continuous functions of the Riemannian metric. In the present article we extend this result to a global version. We think of the spectrum of a Dirac operator as a function from the integers to the reals and endow the space of all spectra with an arsinh-uniform metric. We prove that the spectrum of the Dirac operator depends continuously on the Riemannian metric. As a corollary, we obtain the existence of a non-decreasing family of functions on the space of all…
We consider a Riemannian spin manifold (M,g) with a fixed spin structure. The zero sets of solutions of generalized Dirac equations on M play an important role in some questions arising in conformal spin geometry and in mathematical physics. In this setting the mass endomorphism has been defined as the constant term in an expansion of Green's function for the Dirac operator. One is interested in obtaining metrics, for which it is not zero. In this thesis we study the dependence of the zero sets of eigenspinors of the Dirac operator on the Riemannian metric. We prove that on closed spin manifol…
Riemannian manifolds with non-zero Killing spinors are Einstein manifolds. Klaus Kröncke proved that all complete Riemannian manifolds with imaginary Killing spinors are (linearly) strictly stable in \cite{Kro15}. In this paper, we obtain a new proof for this stability result by using a Bochner type formula in \cite{DWW05} and \cite{Wan91}. Moreover, existence of real Killing spinors is closely related to the Sasaki-Einstein structure. A regular Sasaki-Einstein manifold is essentially the total space of a certain principal $S^{1}$-bundle over a Kähler-Einstein manifold. We prove that if the ba…
Semi-Riemannian $\text{spin}^c$ manifolds carrying generalized Killing spinors and the classification of Riemannian $\text{spin}^c$ manifolds admitting a type I imaginary generalized Killing spinor
We develop variation formulas on almost-product (e.g. foliated) pseudo-Riemannian manifolds, and we consider variations of metric preserving orthogonality of the distributions. These formulae are applied to Einstein-Hilbert type actions: the total mixed scalar curvature and the total extrinsic scalar curvature of a distribution. The obtained Euler-Lagrange equations admit a number of solutions, e.g., twisted products, conformal submersions and isoparametric foliations. The paper generalizes recent results about the actions on codimension-one foliations for the case of arbitrary (co)dimension.
We study the existence of projectable $G$-invariant Einstein metrics on the total space of $G$-equivariant fibrations $M=G/L\to G/K$, for a compact connected semisimple Lie group $G$. We obtain necessary conditions for the existence of such Einstein metrics in terms of appropriate Casimir operators, which is a generalization of the result by Wang and Ziller about Einstein normal metrics. We describe binormal Einstein metrics which are the orthogonal sum of the normal metrics on the fiber and on the base. The special case when the restriction to the fiber and the projection to the base are also…
We prove $R$-bisectoriality and boundedness of the $H^\infty$-functional calculus in $L^p$ for all $1<p<\infty$ for the Hodge-Dirac operator associated with Witten Laplacians on complete Riemannian manifolds with non-negative Bakry-Emery Ricci curvature on $k$-forms.
We show the smoothness of weakly Dirac-harmonic maps from a closed spin Riemann surface into stationary Lorentzian manifolds, and obtain a regularity theorem for a class of critical elliptic systems without anti-symmetry structures.
In this work we study the existence of homogeneous Einstein metrics on the total space of homogeneous fibrations such that the fibers are totally geodesic manifolds. We obtain the Ricci curvature of an invariant metric with totally geodesic fibers and some necessary conditions for such a metric to be Einstein in terms of Casimir operators. Some particular cases are studied, for instance, for normal base or fiber, symmetric fiber, Einstein base or fiber, for which the Einstein equations are manageable. We show the existence of new invariant Einstein metrics on homogeneous bisymmetric fibrations…
Given a Riemannian space $N$ of dimension $n$ and a field $D$ of symmetric endomorphisms on $N$, we define the extension $M$ of $N$ by $D$ to be the Riemannian manifold of dimension $n+1$ obtained from $N$ by a construction similar to extending a Lie group by a derivation of its Lie algebra. We find the conditions on $N$ and $D$ which imply that the extension $M$ is Einstein. In particular, we show that in this case, $D$ has constant eigenvalues; moreover, they are all integer (up to scaling) if $\det D \ne 0$. They must satisfy certain arithmetic relations which imply that there are only fini…
In this paper, we study an important class of Finsler metrics--square metrics. We give two expressions of such metrics in terms of a Riemannian metric and a 1-form. We show that Einstein square metrics can be classified up to the classification of Einstein Riemannian metrics.
On a given closed connected manifold of dimension two, or greater, we consider the squared $L^2$-norm of the scalar curvature functional over the space of constant volume Riemannian metrics. We prove that its critical points have constant scalar curvature, and use this to show that a metric is a solution of the critical point equation if, and only if, it is either Einstein, or scalar flat.
We establish the factorization of Dirac operators on Riemannian submersions of compact spin$^c$ manifolds in unbounded KK-theory. More precisely, we show that the Dirac operator on the total space of such a submersion is unitarily equivalent to the tensor sum of a family of Dirac operators with the Dirac operator on the base space, up to an explicit bounded curvature term. Thus, the latter is an obstruction to having a factorization in unbounded KK-theory. We show that our tensor sum represents the bounded KK-product of the corresponding KK-cycles and connect to the early work of Connes and Sk…
All known examples of homogeneous Einstein metrics of negative Ricci curvature can be realized as left-invariant Riemannian metrics on solvable Lie groups. After defining a notion of maximal symmetry among left-invariant Riemannian metrics on a Lie group, we prove that any left-invariant Einstein metric of negative Ricci curvature on a solvable Lie group is maximally symmetric. This theorem is motivated both by the Alekseevskii Conjecture and by the question of stability of Einstein metrics under the Ricci flow. We also address questions of existence of maximally symmetric left-invariant Riema…
In this paper we prove that geodesic mappings of (pseudo-) Riemannian manifolds preserve the class of differentiability \hbox{$(C^r, r\geq1)$}. Also, if the Einstein space $V_n$ admits a non trivial geodesic mapping onto a \hbox{(pseudo-)} Riemannian manifold $\bar V_n\in C^1$, then $\bar V_n$ is an Einstein space. If a four-dimensional Einstein space with non constant curvature globally admits a geodesic mapping onto a (pseudo-) Riemannian manifold $\bar V_4\in C^1$, then the mapping is affine and, moreover, if the scalar curvature is non vanishing, then the mapping is homothetic, i.e. $\bar …
We study the Dirac spectrum on compact Riemannian spin manifolds $M$ equipped with a metric connection $\nabla$ with skew torsion $T\inΛ^3 M$ in the situation where the tangent bundle splits under the holonomy of $\nabla$ and the torsion of $\nabla$ is of `split' type. We prove an optimal lower bound for the first eigenvalue of the Dirac operator with torsion that generalizes Friedrich's classical Riemannian estimate.
In Kähler-Einstein case of positive scalar curvature and even complex dimension, an improved lower bound for the first eigenvalue of the Dirac operator is given. It is shown by a general construction that there are manifolds for which this new lower bound itself is the first eigenvalue.
We derive the general formulas for a special configuration of the sequential warped product semi-Riemannian manifold to be Einstein, where the base-manifold is the product of two manifolds both equipped with a conformal metrics. Subsequently we study the case in which these two manifolds are conformal to a $n_1$-dimensional and $n_2$-dimensional pseudo-Euclidean space, respectively. For the latter case, we prove the existence of a family of solutions that are invariant under the action of a $(n_1-1)$-dimensional group of transformations to the case of positive constant Ricci curvature ($λ>0$).
Using the method of Witten deformation, we express the basic index of a transversal Dirac operator over a Riemannian foliation as the sum of integers associated to the critical leaf closures of a given foliated bundle map.
We study the Einstein-Dirac equation as well as the weak Killing equation on Riemannian spin manifolds with codimension one foliation. We prove that, for any manifold $M^n$ admitting real Killing spinors (resp. parallel spinors), there exist warped product metrics $\barη$ on $M^n \times {\mathbb R}$ such that $(M^n \times {\mathbb R}, \barη)$ admit Einstein spinors (resp. weak Killing spinors). To prove the result we split the Einstein-Dirac equation into evolution equations and constraints, by means of Cartan's frame formalism, and apply the local preservation property of constraints.
We construct 4-dimensional Riemannian Lie groups carrying left-invariant conformal foliations with minimal leaves of codimension 2. We show that these foliations are holomorphic with respect to an (integrable) Hermitian structure which is not K\" ahler. We then prove that the Riemannian Lie groups constructed are {\it not} Einstein manifolds. This answers an important open question in the theory of complex-valued harmonic morphisms from Riemannian 4-manifolds.
A method, due to Élie Cartan, is used to give an algebraic classification of the non-reductive homogeneous pseudo-Riemannian manifolds of dimension four. Only one case with Lorentz signature can be Einstein without having constant curvature, and two cases with (2,2) signature are Einstein of which one is Ricci-flat. If a four-dimensional non-reductive homogeneous pseudo-Riemannian manifold is simply connected, then it is shown to be diffeomorphic to $\real^4$. All metrics for the simply connected non-reductive Einstein spaces are given explicitly. There are no non-reductive pseudo-Riemannian h…
