Assume that the compact Riemannian spin manifold $(M^n,g)$ admits a $G$-structure with characteristic connection $\nabla$ and parallel characteristic torsion ($\nabla T=0$), and consider the Dirac operator $D^{1/3}$ corresponding to the torsion $T/3$. This operator plays an eminent role in the investigation of such manifolds and includes as special cases Kostant's ``cubic Dirac operator'' and the Dolbeault operator. In this article, we describe a general method of computation for lower bounds of the eigenvalues of $D^{1/3}$ by a clever deformation of the spinorial connection. In order to get explicit bounds, each geometric structure needs to be investigated separately; we do this in full generality in dimension 4 and for Sasaki manifolds in dimension 5.
The submanifold Dirac operator has been studied for this decade, which is closely related to Frenet-Serret and generalized Weierstrass relations. In this article, we will give a submanifold Dirac operator defined over a surface immersed in $\EE^4$ with U(1)-gauge field as torsion in the sense of the Frenet-Serret relation, which also has data of immersion of the surface in $\EE^4$.s
We study Dirac-harmonic maps from surfaces to manifolds with torsion, which is motivated from the superstring action considered in theoretical physics. We discuss analytic and geometric properties of such maps and outline an existence result for uncoupled solutions.
We study the Dirac spectrum on compact Riemannian spin manifolds $M$ equipped with a metric connection $\nabla$ with skew torsion $T\inΛ^3M$ by means of twistor theory. An optimal lower bound for the first eigenvalue of the Dirac operator with torsion is found that generalizes Friedrich's classical Riemannian estimate. We also determine a novel twistor and Killing equation with torsion and use it to discuss the case in which the minimum is attained in the bound.
We consider the Dirac equation in flat Minkowski 3-space and rewrite it as the Maxwell equation in Minkowski 4-space with torsion. The torsion tensor is defined as the dual of the electromagnetic vector potential. Our model clearly distinguishes the electron and the positron without resorting to "negative frequencies": we produce a real scalar invariant (charge) which indicates whether we are looking at an electron or a positron. Another interesting feature of our model is that the free electron and positron are identified with gradient type solutions of the standard (torsion free) Maxwell equ…
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.
Under standard local boundary conditions or certain global APS boundary conditions, we get lower bounds for the eigenvalues of the Dirac operator on compact spin manifolds with boundary. Limiting cases are characterized by the existence of real Killing spinors and the minimality of the boundary.
We extend the definition of the Nijenhuis torsion of an endomorphism of a Lie algebroid to that of a relation, and we prove that the torsion of the relation defined by a bi-Hamiltonian structure vanishes. Following Gelfand and Dorfman, we then define Dirac pairs, and we analyze the relationship of this general notion with the various kinds of compatible structures on manifolds, more generally on Lie algebroids.
In this paper, we consider the symmetries of the Dirac operator derived from a connection with skew-symmetric torsion. We find that the generalized conformal Killing-Yano tensors give rise to symmetry operators of the massless Dirac equation, provided an explicitly given anomaly vanishes. We show that this gives rise to symmetries of the Dirac operator in the case of strong Kahler with torsion (KT) and strong hyper-Kahler with torsion (HKT) manifolds.
Given a compact Riemannian spin manifold with positive scalar curvature, we find a family of connections $\nabla^{A_t}$ for $t\in[0,1]$ on a trivial vector bundle of sufficiently high rank, such that the first eigenvalue of the twisted Dirac operator $D_{A_t}$ is nonzero and becomes arbitrarily small as $t\to1$. However, if one restricts the class of twisting connections considered, then nonzero lower bounds do exist. We illustrate this fact by establishing a nonzero lower bound for the Dirac operator twisted by Hermitian-Einstein connections over Riemann surfaces.
We get optimal lower bounds for the eigenvalues of the submanifold Dirac operator on locally reducible Riemannian manifolds in terms of intrinsic and extrinsic expressions. The limiting-cases are also studied. As a corollary, one gets several known results in this direction.
We describe min-max formulas for the principal eigenvalue of a $V$-drift Laplacian defined by a vector field $V$ on a geodesic ball of a Riemannian manifold $N$. Then we derive comparison results for the principal eigenvalue with the one of a spherically symmetric model space endowed with a radial vector field, under pointwise comparison of the corresponding radial sectional and Ricci curvatures, and of the radial component of the vector fields. These results generalize the known case $V=0$.
We prove a CR Obata type result that if the first positive eigenvalue of the sub-Laplacian on a compact strictly pseudoconvex pseudohermitian manifold with a divergence free pseudohermitian torsion takes the smallest possible value then, up to a homothety of the pseudohermitian structure, the manifold is the standart Sasakian unit sphere. We also give a version of this theorem using the existence of a function with traceless horizontal Hessian on a complete, with respect to Webster's metric, pseudohermitian manifold.
We investigate the second Dirac eigenvalue on Riemannian manifolds admitting a Killing spinor. In small dimensions the whole Dirac spectrum depends on special eigenvalues on functions and 1-forms. We compute and discuss the formulas in dimension $n = 7$.
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 note, we prove lower and upper bounds for Dirac operators of submanifolds in certain ambient manifolds in terms of conformal and extrinsic quantities.
For a compact spin manifold $M$ isometrically embedded into Euclidean space, we derive the extrinsic estimates from above and below for eigenvalues of the Dirac operators, which depend on the second fundamental form of the embedding. We also show the bounds of the ratio of the eigenvalues.
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 prove a lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold depending on the scalar curvature as well as a chosen Codazzi tensor. The inequality generalizes the classical estimate from [2].
In this paper we will prove new extrinsic upper bounds for the eigenvalues of the Dirac operator on an isometrically immersed surface $M^2 \hookrightarrow {\Bbb R}^3$ as well as intrinsic bounds for 2-dimensional compact manifolds of genus zero and genus one. Moreover, we compare the different estimates of the eigenvalue of the Dirac operator for special families of metrics.
In two previous papers, we started a study of the first eigenvalue of the Dirac operator on compact spin symmetric spaces, providing, for symmetric spaces of "inner" type, a formula giving this first eigenvalue in terms of the algebraic data of the groups involved. We conclude here that study by giving the explicit expression of the first eigenvalue for "outer" compact spin symmetric spaces.
We review some recent results concerning lower eigenvalues estimates for the Dirac operator [6, 7]. We show that Friedrich's inequality can be improved via certain well-chosen symmetric tensors and provide an application to Sasakian spin manifolds.
In this paper, we extend the Hijazi inequality, involving the Energy-Momentum tensor, for the eigenvalues of the Dirac operator on $Spin^c$ manifolds without boundary. The limiting case is then studied and an example is given.
In this note we show that every compact spin manifold of dimension $\geq 3$ can be given a Riemannian metric for which a finite part of the spectrum of the Dirac operator consists of arbitrarily prescribed eigenvalues with multiplicity 1.
Under two boundary conditions, the generalized Atiyah-Patodi-Singer boundary condition and the modified generalized -Atiyah-Patodi-Singer boundary condition, we get the lower bounds for the eigenvalues of the fundamental Dirac operator on compact spin manifolds with nonempty boundary.
We give a lower bound for the eigenvalues of the Dirac operator on a compact domain of a Riemannian spin manifold under the $\MIT$ bag boundary condition. The limiting case is characterized by the existence of an imaginary Killing spinor.
Using Weitzenböck techniques on any compact Riemannian spin manifold we derive inequalities that involve a real parameter and join the eigenvalues of the Dirac operator with curvature terms. The discussion of these inequalities yields vanishing theorems for the kernel of the Dirac operator $D$ and lower bounds for the spectrum of $D^2$ if the curvature satisfies certain conditions.
We consider the Dirac operator on compact quaternionic Kaehler manifolds and prove a lower bound for the spectrum. This estimate is sharp since it is the first eigenvalue of the Dirac operator on the quaternionic projective space.
In this paper, we give an optimal lower bound for the eigenvalues of the basic Dirac operator on a quaternion-Kahler foliation. The limiting case is characterized by the existence of quaternion-Kahler Killing spinors. We end this paper by giving some examples.
Eigenvalue estimate for the Dirac-Witten operator is given on bounded domains (with smooth boundary) of spacelike hypersurfaces satisfying the dominant energy condition, under four natural boundary conditions (MIT, APS, modified APS, and chiral conditions). This result is a generalisation of Friedrich's inequality for the usual Dirac operator. The limiting cases are also investigated.
We prove a new lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold by refined Weitzenböck techniques. It applies to manifolds with harmonic curvature tensor and depends on the Ricci tensor. Examples show how it behaves compared to other known bounds.
We give lower bounds for the eigenvalues of the submanifold Dirac operator in terms of intrinsic and extrinsic curvature expressions. We also show that the limiting cases give rise to a class generalizing that of Killing spinors. We conclude by translating these results in terms of intrinsic twisted Dirac operators.
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 new lower bounds for the first eigenvalue of the Dirac operator on compact manifolds whose Weyl tensor or curvature tensor, respectively, is divergence free. In the special case of Einstein manifolds, we obtain estimates depending on the Weyl tensor.
We give a survey of results relating the restricted holonomy of a Riemannian spin manifold with lower bounds on the spectrum of its Dirac operator, giving a new proof of a result originally due to Kirchberg.
Let $G/K$ be a simply connected spin compact inner irreducible symmetric space, endowed with the metric induced by the Killing form of $G$ sign-changed. We give a formula for the square of the first eigenvalue of the Dirac operator in terms of a root system of $G$. As an example of application, we give the list of the first eigenvalues for the spin compact irreducible symmetric spaces endowed with a quaternion-Kähler structure.
We show that on every compact spin manifold admitting a Riemannian metric of positive scalar curvature Friedrich's eigenvalue estimate for the Dirac operator can be made sharp up to an arbitrarily small given error by choosing the metric suitably.
We compare the eigenvalues of the Dirac and Laplace operator on a two-dimensional torus with respect to the trivial spin structure. In particular, we compute their variation up to order 4 upon deformation of the flat metric, study the corresponding Hamiltonian and discuss several families of examples.
We derive upper eigenvalue bounds for the Dirac operator of a closed hypersurface in a manifold with Killing spinors such as Euclidean space, spheres or hyperbolic space. The bounds involve the Willmore functional. Relations with the Willmore inequality are briefly discussed. In higher codimension we obtain bounds on the eigenvalues of the Dirac operator of the submanifold twisted with the spinor bundle of the normal bundle.
In this paper, we prove Kirchberg inequalities for any kahler spin foliations. Their limiting cases are then characterized as being transversal minimal Einstein foliations. The key point is to introduce the transversal kahlerian twistor operators.
We re-visit the eigenvalue estimate of the Dirac operator on spin manifolds with boundary in terms of the first eigenvalues of conformal Laplace operator as well as the conformal mean curvature operator. These problems were studied earlier by Hijazi-Montiel-Zhang and Raulot and we re-prove them under weaker assumption that a boundary chirality operator exists. Moreover, on these spin manifolds with boundary, we show that any $C^{3,α}$ conformal compactification of some Poincare-Einstein metric must be the standard hemisphere when the first nonzero eigenvalue of the Dirac operator achieves its …
It has recently been conjectured that the eigenvalues $λ$ of the Dirac operator on a closed Riemannian spin manifold $M$ of dimension $n\ge 3$ can be estimated from below by the total scalar curvature: $$ λ^2 \ge \frac{n}{4(n-1)} \cdot \frac{\int_M S}{vol(M)}. $$ We show by example that such an estimate is impossible.
In this note, we prove an optimal upper bound for the first Dirac eigenvalue of some hypersurfaces in Euclidean space by combining a positive mass theorem and the construction of quasi-spherical metrics. As a direct consequence of this estimate, we obtain an asymptotic expansion for the first eigenvalue of the Dirac operator on large spheres in three dimensional asymptotically flat manifolds. We also study this expansion for small geodesic spheres in a three dimensional Riemannian manifold. We finally discuss how this method can be adapted to yield similar results in the hyperbolic space.
We prove that on a compact $n$-dimensional spin manifold admitting a non-trivial harmonic 1-form of constant length, every eigenvalue $λ$ of the Dirac operator satisfies the inequality $λ^2 \geq \frac{n-1}{4(n-2)}\inf_M Scal$. In the limiting case the universal cover of the manifold is isometric to $R\times N$ where $N$ is a manifold admitting Killing spinors.
We give new estimates for the eigenvalues of the hypersurface Dirac operator in terms of the intrinsic energy-momentum tensor, the mean curvature and the scalar curvature. We also discuss their limiting cases as well as the limiting cases of the estimates obtained by X. Zhang and O. Hijazi in [13] and [10]. We compare these limiting cases with those corresponding to the Friedrich and Hijazi inequalities. We conclude by comparing these results to intrinsic estimates for the Dirac-Schrödinger operator D_f = D - f/2.
This review article intends to introduce the reader to non-integrable geometric structures on Riemannian manifolds and invariant metric connections with torsion, and to discuss recent aspects of mathematical physics--in particular superstring theory--where these naturally appear. Connections with skew-symmetric torsion are exhibited as one of the main tools to understand non-integrable geometries. To this aim a a series of key examples is presented and successively dealt with using the notions of intrinsic torsion and characteristic connection of a $G$-structure as unifying principles. % The G…
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.
A universal lower bound for the first positive eigenvalue of the Dirac operator on a compact quaternionic Kaehler manifold M of positive scalar curvature is calculated. It is shown that it is equal to the first positive eigenvalue on the quaternionic projective space. For this, the horizontal tangent bundle on the canonical SO(3)-bundle over M is equipped with a hyperkaehlerian structure and the corresponding splitting of the horizontal spinor bundle is considered. The desired estimate is obtained by looking at hyperkaehlerian twistor operators on horizontal spinors.
