Georges Habib

Energy–momentum tensor on foliations

Abstract 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.

Eigenvalues of the transversal Dirac operator on Kahler foliations

Abstract In this paper, we prove Kirchberg-type inequalities for any Kahler spin foliation. Their limiting-cases are then characterized as being transversal minimal Einstein foliations. The key point is to introduce the transversal Kahlerian twistor operators.

A spectral estimate for the Dirac operator on Riemannian flows

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 both Sasakian and 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.

The Energy-Momentum tensor on low dimensional Spin c manifolds

On a compact surface endowed with any Spinc structure, we give a formula involving the Energy-Momentum tensor in terms of geometric quantities. A new proof of a Bar-type inequality for the eigenvalues of the Dirac operator is given. The round sphere 𝕊2 with its canonical Spinc structure satisfies the limiting case. Finally, we give a spinorial characterization of immersed surfaces in 𝕊2 × ℝ by solutions of the generalized Killing spinor equation associated with the induced Spinc structure on 𝕊2 × ℝ.

Riemannian flows and adiabatic limits

We show the convergence properties of the eigenvalues of the Dirac operator on a spin manifold with a Riemannian flow when the metric is collapsed along the flow.

A new upper bound for the Dirac operator on hypersurfaces

We prove a new upper bound for the first eigenvalue of the Dirac operator of a compact hypersurface in any Riemannian spin manifold carrying a non-trivial twistor spinor without zeros on the hypersurface. The upper bound is expressed as the first eigenvalue of a drifting Schrodinger operator on the hypersurface. Moreover, using a recent approach developed by O. Hijazi and S. Montiel, we completely characterize the equality case when the ambient manifold is the standard hyperbolic space.

Skew Killing spinors

We study the existence of a skew Killing spinor on 2- and 3-dimensional Riemannian spin manifolds. We establish the integrability conditions and prove that these spinor fields correspond to twistor spinors in the two dimensional case while, up to a conformal change of the metric, they correspond to parallel spinors in the three dimensional case.