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The Riemannian product $${\mathbb{M}}_1(c_1) \times {\mathbb{M}}_2(c_2)$$ , where $${\mathbb{M}}_i(c_i)$$ denotes the 2-dimensional space form of constant sectional curvature $$c_i \in {\mathbb{R}}$$ , has two different $${\mathrm{Spin}^{\mathrm{c}}}$$ structures carrying each a parallel spinor. The restriction of these two parallel spinor fields to a 3-dimensional hypersurface M characterizes the isometric immersion of M into $${\mathbb{M}}_1(c_1) \times {\mathbb{M}}_2(c_2)$$ . As an application, we prove that totally umbilical hypersurfaces of $${\mathbb{M}}_1(c_1) \times {\mathbb{M}}_1(c_1)$$ and totally umbilical hypersurfaces of $${\mathbb{M}}_1(c_1) \times {\mathbb{M}}_2(c_2)$$ ( $$c_1 \ne c_2$$ ) having a local structure product are of constant mean curvature.

We extend the Friedrich inequality for the eigenvalues of the Dirac operator on Spinc manifolds with boundary under different boundary conditions. The limiting case is then studied and examples are given.

In this paper, we consider a Riemannian foliation whose normal bundle carries a parallel or harmonic basic form. We estimate the norm of the O'Neill tensor in terms of the curvature data of the whole manifold. Some examples are then given.

We study boundary value problems for the Dirac operator on Riemannian Spin$^c$ manifolds of bounded geometry and with noncompact boundary. This generalizes a part of the theory of boundary value problems by C. B\"ar and W. Ballmann for complete manifolds with closed boundary. As an application, we derive the lower bound of Hijazi-Montiel-Zhang, involving the mean curvature of the boundary, for the spectrum of the Dirac operator on the noncompact boundary of a Spin$^c$ manifold. The limiting case is then studied and examples are then given., Comment: Accepted in Proceedings of the London Mathematical Society

In this paper, we estimate the eigenvalues of the twisted Dirac operator on Kahler submanifolds of the complex projective space C P m and we discuss the sharpness of this estimate for the embedding C P d → C P m .

We develop a spinorial description of CR structures of arbitrary codimension. More precisely, we characterize almost CR structures of arbitrary codimension on (Riemannian) manifolds by the existence of a Spin $$^{c, r}$$ structure carrying a partially pure spinor field. We study various integrability conditions of the almost CR structure in our spinorial setup, including the classical integrability of a CR structure as well as those implied by Killing-type conditions on the partially pure spinor field. In the codimension one case, we develop a spinorial description of strictly pseudoconvex CR manifolds, metric contact manifolds, and Sasakian manifolds. Finally, we study hypersurfaces of Kahler manifolds via partially pure Spin $$^c$$ spinors.

On a compact Riemannian manifold N whose boundary is endowed with a Riemannian flow, we gave in El Chami et al. (Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary, 2015) a sharp lower bound for the first non-zero eigenvalue of the basic Laplacian acting on basic 1-forms. In this paper, we extend this result to the set of basic p-forms when $$p>1$$ . We then characterize the limiting case by showing that the manifold N is isometric to for some group $$\Gamma $$ where $$B'$$ denotes the unit closed ball. As a consequence, we describe the Riemannian product $${\mathbb {S}}^1\times {\mathbb {S}}^n$$ as the boundary of a manifold.

Given a Riemannian spin $$^c$$ manifold whose boundary is endowed with a Riemannian flow, we show that any solution of the basic Dirac equation satisfies an integral inequality depending on geometric quantities, such as the mean curvature and the O’Neill tensor. We then characterize the equality case of the inequality when the ambient manifold is a domain of a Kahler–Einstein manifold or a Riemannian product of a Kahler–Einstein manifold with $$\mathbb R$$ (or with the circle $$\mathbb S^1$$ ).

In this paper, we consider a compact Riemannian manifold whose boundary is endowed with a Riemannian flow. Under a suitable curvature assumption depending on the O'Neill tensor of the flow, we prove that any solution of the basic Dirac equation is the restriction of a parallel spinor field defined on the whole manifold. As a consequence, we show that the flow is a local product. In particular, in the case where solutions of the basic Dirac equation are given by basic Killing spinors, we characterize the geometry of the manifold and the flow., Comment: 22 pages

The complex projective space $\mathbb C P^2$ of complex dimension $2$ has a Spin$^c$ structure carrying K\"ahlerian Killing spinors. The restriction of one of these K\"ahlerian Killing spinors to a surface $M^2$ characterizes the isometric immersion of $M^2$ into $\mathbb C P^2$ if the immersion is either Lagrangian or complex., Comment: 18 pages

In this paper, we extend the Hijazi inequality, involving the energy–momentum tensor, to the eigenvalues of the Dirac operator on Spin c manifolds without boundary. The limiting case is then studied and an example is given.

We establish a lower bound for the eigenvalues of the Dirac operator defined on a compact K\"ahler-Einstein manifold of positive scalar curvature and endowed with particular ${\rm spin}^c$ structures. The limiting case is characterized by the existence of K\"ahlerian Killing ${\rm spin}^c$ spinors in a certain subbundle of the spinor bundle. Moreover, we show that the Clifford multiplication between an effective harmonic form and a K\"ahlerian Killing ${\rm spin}^c$ spinor field vanishes. This extends to the ${\rm spin}^c$ case the result of A. Moroianu stating that, on a compact K\"ahler-Einstein manifold of complex dimension $4\ell+3$ carrying a complex contact structure, the Clifford multiplication between an effective harmonic form and a K\"ahlerian Killing spinor is zero.

À paraître dans Archiv der Mathematik.; International audience; We prove lower bounds for the eigenvalues of the Spin c Dirac operator on submanifolds. These estimates are expressed in terms of extrinsic and intrinsic quanti-ties. We also give estimates involving the Energy-Momentum tensor as well as conformal bounds. The limiting cases of these estimates give rise to particular spinor fields, called generalized twisted Killing spinors, which are also studied.

In this paper, we extend the study of generalized Killing spinors on Riemannian Spin$^c$ manifolds started by Moroianu and Herzlich to complex Killing functions. We prove that such spinor fields are always real Spin$^c$ Killing spinors or imaginary generalized Spin$^c$ Killing spinors, providing that the dimension of the manifold is greater or equal to 4. Moreover, we classify Riemannian Spin$^c$ manifolds carrying imaginary and imaginary generalized Killing spinors., Comment: 15 pages

We extend to the eigenvalues of the hypersurface Spinc Dirac operator well known lower and upper bounds. Examples of limiting cases are then given. Furthermore, we prove a correspondence between the existence of a Spinc Killing spinor on homogeneous 3-dimensional manifolds E ⁎ ( κ , τ ) with 4-dimensional isometry group and isometric immersions of E ⁎ ( κ , τ ) into the complex space form M 4 ( c ) of constant holomorphic sectional curvature 4c, for some c ∈ R ⁎ . As applications, we show the non-existence of totally umbilic surfaces in E ⁎ ( κ , τ ) and we give necessary and sufficient geometric conditions to immerse a 3-dimensional Sasaki manifold into M 4 ( c ) .

Simply connected three-dimensional homogeneous manifolds \({\mathbb{E}(\kappa, \tau)}\), with four-dimensional isometry group, have a canonical Spinc structure carrying parallel or Killing spinors. The restriction to any hypersurface of these parallel or Killing spinors allows to characterize isometric immersions of surfaces into \({\mathbb{E}(\kappa, \tau)}\). As application, we get an elementary proof of a Lawson type correspondence for constant mean curvature surfaces in \({\mathbb{E}(\kappa, \tau)}\). Real hypersurfaces of the complex projective space and the complex hyperbolic space are also characterized via Spinc spinors.

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 Bär-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.

We extend the Hijazi type inequality, involving the energy-momentum tensor, to the eigenvalues of the Dirac operator on complete Riemannian Spincmanifolds without boundary and of finite volume. Under some additional assumptions, using the refined Kato inequality, we prove the Hijazi type inequality for elements of the essential spectrum. The limiting cases are also studied.

On $Spin^c$ manifolds, we study the Energy-Momentum tensor associated with a spinor field. First, we give a spinorial Gauss type formula for oriented hypersurfaces of a $Spin^c$ manifold. Using the notion of generalized cylinders, we derive the variationnal formula for the Dirac operator under metric deformation and point out that the Energy-Momentum tensor appears naturally as the second fundamental form of an isometric immersion. Finally, we show that generalized $Spin^c$ Killing spinors for Codazzi Energy-Momentum tensor are restrictions of parallel spinors., To appear in IJGMMP (International Journal of Geometric Methods in Modern Physics), 22 pages