Compact Spin Manifold Research Articles

Inequalities of eigenvalues for the Dirac operator on compact complex spin submanifolds in complex projective spaces

For a compact complex spin manifold M with a holomorphic isometric embedding into the complex projective space, the authors obtain the extrinsic estimates from above and below for eigenvalues of the Dirac operator, which depend on the data of an isometric embedding of M. Further, from the inequalities of eigenvalues, the gaps of the eigenvalues and the ratio of the eigenvalues are obtained.

A spinorial analogue of Aubin’s inequality

Let (M, g, σ) be a compact Riemannian spin manifold of dimension ≥ 2. For any metric \(\tilde g\) conformal to g, we denote by \(\tilde\lambda\) the first positive eigenvalue of the Dirac operator on \((M, \tilde g, \sigma)\) . We show that$${\rm inf}_{\tilde{g} \in [g]} \tilde\lambda {\rm Vol}(M,\tilde g)^{1/n} \leq (n/2) {\rm Vol}(S^n)^{1/n}.$$This inequality is a spinorial analogue of Aubin’s inequality, an important inequality in the solution of the Yamabe problem. The inequality is already known in the case n ≥ 3 and in the case n = 2, ker D = {0}. Our proof also works in the remaining case n = 2, ker D ≠ {0}. With the same method we also prove that any conformal class on a Riemann surface contains a metric with \(2\tilde\lambda^2 \leq \tilde\mu\) , where \(\tilde\mu\) denotes the first positive eigenvalue of the Laplace operator.

Conformal structures in noncommutative geometry

It is well known that a compact Riemannian spin manifold (M, g) can be reconstructed from its canonical spectral triple (C^∞(M), L^2(M,ΣM), D) where ΣM denotes the spinor bundle and D the Dirac operator. We show that g can be reconstructed up to conformal equivalence from (C^∞(M), L^2(M,ΣM), \mathrm{sign}(D)) .

Energy–momentum tensor on foliations

In this paper, we give a new lower bound for the eigenvalues of the Dirac operator on a compact spin manifold. This estimate is motivated by the fact that in its limiting case a skew-symmetric tensor (see Eq. (1.6)) appears that can be identified geometrically with the O’Neill tensor of a Riemannian flow, carrying a transversal parallel spinor. The Heisenberg group which is a fibration over the torus is an example of this case. Sasakian manifolds are also considered to be particular examples of Riemannian flows. Finally, we characterize the 3-dimensional case by a solution of the Dirac equation.

L'opérateur de Dirac hypoelliptique

We announce the construction of a deformation of the Dirac operator on a compact spin manifold into a hypoelliptic Dirac operator on the total space of the tangent space. This construction gives an analogue for the Dirac operator of a related deformation we already gave for the de Rham complex. For simplicity, we only explain the construction in the case of complex manifolds. We define hypoelliptic Quillen metrics, which we compare to the classical Quillen metrics. To cite this article: J.-M. Bismut, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

Eigenvalue estimates for the Dirac operator with generalized APS boundary condition

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.

Solvmanifolds with integrable and non-integrable [formula omitted] structures

We show that a 7-dimensional non-compact Ricci-flat Riemannian manifold with Riemannian holonomy G 2 can admit non-integrable G 2 structures of type R ⊕ S 0 2 ( R 7 ) ⊕ R 7 in the sense of Fernández and Gray. This relies on the construction of some G 2 solvmanifolds, whose Levi-Civita connection is known to give a parallel spinor, admitting a 2-parameter family of metric connections with non-zero skew-symmetric torsion that has parallel spinors as well. The family turns out to be a deformation of the Levi-Civita connection. This is in contrast with the case of compact scalar-flat Riemannian spin manifolds, where any metric connection with closed torsion admitting parallel spinors has to be torsion-free.

The Hijazi inequality on manifolds with boundary

In this paper, we prove the Hijazi inequality on compact Riemannian spin manifolds under two boundary conditions: the condition associated with a chirality operator and the Riemannian version of the MIT bag condition. We then show that the limiting case is characterized as being a half-sphere for the first condition whereas the equality cannot be achieved for the second.

Prescribing eigenvalues of the Dirac operator

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.

?-Deformations as Compact Quantum Metric Spaces

Let M be a compact spin manifold with a smooth action of the n-torus. Connes and Landi constructed θ-deformations Mθ of M, parameterized by n×n real skew-symmetric matrices θ. The Mθ’s together with the canonical Dirac operator (D,Open image in new window) on M are an isospectral deformation of M. The Dirac operator D defines a Lipschitz seminorm on C(Mθ), which defines a metric on the state space of C(Mθ). We show that when M is connected, this metric induces the weak-* topology. This means that Mθ is a compact quantum metric space in the sense of Rieffel.

Eigenvalues of Dirac operators for hyperbolic degenerations

We study the behaviour of the spectrum of the Dirac operator for sequences of compact hyperbolic spin manifolds whose limit is non-compact. If the spectrum of the limit manifold is discrete we show that this spectrum is approximated by the spectra of the compact manifolds.

The first Dirac eigenvalues on manifolds with positive scalar curvature

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.

Un problème de type Yamabe sur les variétés spinorielles compactes

Let ( M, g, σ) be a compact spin manifold of dimension n⩾2. Let λ 1 +( g ̃ ) be the smallest positive eigenvalue of the Dirac operator in the metric g ̃ ∈[g] conformal to g. We then define λ min +(M,[g],σ)= inf g ̃ ∈[g] λ 1 +( g ̃ ) Vol(M, g ̃ ) 1/n . We show that 0<λ min +(M,[g],σ)⩽λ min +( S n) . We find sufficient conditions for which we obtain strict inequality λ min +(M,[g],σ)<λ min +( S n) . This strict inequality has applications to conformal spin geometry. To cite this article: B. Ammann et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Eigenvalue estimates for the Dirac operator and harmonic 1-forms of constant length

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⩾ n−1 4(n−2) inf M Scal . In the limiting case the universal cover of the manifold is isometric to R×N where N is a manifold admitting Killing spinors. To cite this article: A. Moroianu, L. Ornea, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Curvature dependent lower bounds for the first eigenvalue of the Dirac operator

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 D2 if the curvature satisfies certain conditions.

Eigenvalue Boundary Problems for the Dirac Operator

On a compact Riemannian spin manifold with mean-convex boundary, we analyse the ellipticity and the symmetry of four boundary conditions for the fundamental Dirac operator including the (global) APS condition and a Riemannian version of the (local) MIT bag condition. We show that Friedrich's inequality for the eigenvalues of the Dirac operator on closed spin manifolds holds for the corresponding four eigenvalue boundary problems. More precisely, we prove that, for both the APS and the MIT conditions, the equality cannot be achieved, and for the other two conditions, the equality characterizes respectively half-spheres and domains bounded by minimal hypersurfaces in manifolds carrying non-trivial real Killing spinors.

Eigenvalue estimates of the Dirac operator depending on the Ricci tensor

We prove a new lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold by refined Weitzenbock 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.

A spin-conformal lower bound of the first positive Dirac eigenvalue

Let D be the Dirac operator on a compact spin manifold M. Assume that 0 is in the spectrum of D. We prove the existence of a lower bound on the first positive eigenvalue of D depending only on the spin structure and the conformal type.

Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations

We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of $\Rb^n$. They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features. The new examples include the instanton algebra and the NC-4-spheres $S^4_{\theta}$. The noncommutative algebras $\Ac=C^{\ify} (S^{4}_{\theta})$ of functions on NC-spheres are solutions to the vanishing, $ {\rm ch}_j (e) = 0, j < 2 $, of the Chern character in the cyclic homology of $\Ac$ of an idempotent $e \in M_4 (\Ac), e^2 = e, e = e^*$. The universal noncommutative space defined by this equation is a noncommutative Grassmanian defined by very non trivial cubic relations. This space ${\rm Gr}$ contains the suspension of a NC-3-sphere intimately related to quantum group deformations ${\rm SU}_q (2)$ of ${\rm SU} (2)$ but for unusual values (complex values of modulus one) of the parameter $q$ of $q$-analogues, $q=\exp (2\pi i \t)$. We then construct the noncommutative geometry of $S_{\t}^4$ as given by a spectral triple $(\Ac, \Hc, D)$ and check all axioms of noncommutative manifolds. The Dirac operator $D$ on the noncommutative 4-spheres $S_{\t}^4$ gives a solution to the basic quartic equation defining the `volume form' $ < (e - {1/2}) [D,e]^4 > = \g_5$, where $<$ is the projection on the commutant of $4 \ts 4$ matrices. Finally, we show that any compact Riemannian spin manifold whose isometry group has rank $r \geq 2$ admits isospectral deformations to noncommutative geometries.

Eigenvalues of the Dirac Operator on Manifolds¶with Boundary

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.