arXiv research

Eigenvalues of Dirac operator converge on spin manifolds with Riemannian flows.

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 decom…

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 t…

Study shows rigidity for spin^c manifolds with foliated boundaries.

Classifies manifolds with specific spinors and constructs parallel spinors.

In this paper we complete the classification of spin manifolds admitting parallel spinors, in terms of the Riemannian holonomy groups. More precisely, we show that on a given n-dimensional Riemannian manifold, spin structures with parallel spinors are in one to one correspondence with lifts to Spin_n of the Riemannian …

In spin geometry, traceless cyclic homogeneous Riemannian manifolds equipped with a homogeneous spin structure can be viewed as the simplest manifolds after Riemannian symmetric spin spaces. In this paper, we give some characterizations and properties of cyclic and traceless cyclic homogeneous Riemannian manifolds and …

We prove that Riemannian $Spin(7)$ holonomy manifolds carry octonionic-Kähler structure.

Spin-harmonic structures on low-dimensional manifolds.

We describe, by their holonomy groups, all complete simply connected irreducible non-locally symmetric pseudo-Riemannian SpinC manifolds which admit parallel spinors. So we generalize the Riemannian SpinC case and the pseudo-Riemannian Spin one.

We derive various pinching results for small Dirac eigenvalues using the classification of $\text{spin}^c$ and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for $\text{spin}^c$ manifolds which involves a general study on convergence of Riemannian manifolds with a pr…

Classifies Spin(7) structures on compact 8-manifolds with abelian fundamental group.

Odd-dimensional Riemannian manifolds admit pure spin-c Killing spinors if and only if they are α-Sasakian.

New formula found for a unique invariant 8-form on Riemannian manifolds with Spin(9) structure.

Study on Rarita-Schwinger operator on spin manifolds, finding non-trivial kernels.

Analytic definition of spin structure in 4-manifolds and 3-manifolds.

The paper proves a 'long neck principle' for Riemannian spin manifolds with positive scalar curvature.

The paper studies curvature identities and solitons on Spin(7)-manifolds.

Study shows certain spin manifolds can't meet DEC condition.

We study spin structures on compact simply-connected homogeneous pseudo-Riemannian manifolds (M = G/H, g) of a compact semisimple Lie group G. We classify flag manifolds F = G/H of a compact simple Lie group which are spin. This yields also the classification of all flag manifolds carrying an invariant metaplectic stru…

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…

We characterize certain CR structures of arbitrary codimension (different from 3, 4 and 5) on Riemannian Spin$^c$ manifolds by the existence of a Spin$^c$ structure carrying a strictly partially pure spinor field. Furthermore, we study the geometry of Riemannian Spin$^c$ manifolds carrying a strictly partially pure spi…

We present a definition of Riemannian manifold in noncommutative geometry. Using products of unbounded Kasparov modules, we show one can obtain such Riemannian manifolds from noncommutative spin^c manifolds; and conversely, in the presence of a spin^c structure. We also show how to obtain an analogue of Kasparov's fund…

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 t…

We study conformal $Spin$-subgeometry of submanifolds in a semi-Riemannian $Spin$-manifold, focusing on conformal $Spin$-manifolds $(M,[h])$ and their Poincaré-Einstein metrics $(X,g_+)$. Our approach is based on the spectral theory of Dirac operator in the ambient $Spin$-manifold, and associated spinor valued meromorp…

Develops a theorem for a 6D manifold with boundary.

Study of higher spin Killing spinors on 3D manifolds, proving rigidity and providing explicit expressions.

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 p…

Generalizes Kastler-Kalau-Walze theorem to even-dimensional manifolds.

Study on harmonic flow of Spin(7)-structures in 8D manifolds.

The Rosenberg index vanishes if a manifold admits a wide Riemannian band or cube-like domain.

Short proof given for a complex formula in spin manifold geometry.

Functional for Spin(7) forms defined on compact manifolds.

In this article, we prove new rigidity results for compact Riemannian spin manifolds with boundary whose scalar curvature is bounded from below by a non-positive constant. In particular, we obtain generalizations of a result of Hang-Wang \cite{hangwang1} based on a conjecture of Schroeder and Strake \cite{schroeder}.

New spin frame transformations affect Dirac equations without preserving metric structures.

On a closed 4-dimensional Riemannian manifold, we give a lower bound for the square of the first eigenvalue of the Yamabe operator in terms of the total Branson's Q-curvature. As a consequence, if the manifold is spin, we relate the first eigenvalue of the Dirac operator to the total Branson's Q-curvature. On a closed …

Revises Schwarzschild manifold rigidity proof for spin manifolds.

The paper proves a mass theorem for non-spin manifolds with low regularity curvature.

We show that on every Spin(7) manifold there always exists a unique linear connection with totally skew-symmetric torsion preserving a nontrivial spinor and the Spin(7) structure. We express its torsion and the Riemannian scalar curvature in terms of the fundamental 4-form. We present an explicit formula for the Rieman…

Study on spaces of metrics with intermediate curvature bounds.

New flow deforms Riemannian metrics smoothly.

Totally umbilical hypersurfaces in Spin^c manifolds with special spinors have constant mean curvature.

In this paper, we extend the Hijazi type inequality, involving the Energy-Momentum tensor, to the eigenvalues of the Dirac operator on complete Riemannian Spin$^c$ manifolds without boundary and of finite volume. Under some additional assumptions, using the refined Kato inequality, we prove the Hijazi type inequality f…

In this paper we explain how to define "lower dimensional'' volumes of any compact Riemannian manifold as the integrals of local Riemannian invariants. For instance we give sense to the area and the length of such a manifold in any dimension. Our reasoning is motivated by an idea of Connes and involves in an essential …

Twistor theory applied to special holonomy manifolds.

We study Spin(9)-structures on 16-dimensional Riemannian manifolds and characterize the geometric types admitting a connection with totally skew-symmetric torsion.

Study geometry on $G_2$-manifolds, focusing on $Spin(7)$ and $G_2$.

The paper proves formulas and theorems for J-Witten deformation on specific manifolds.