A brief note on the spectrum of the basic Dirac operator

关于磁Dirac算子谱的一个注记

A Note on the Spectrum of Magnetic Dirac Operators

本文研究完备黎曼流形上磁Dirac算子及其带势版本的谱，给出了谱为极大或离散的充分条件，并证明了其可具有稠密特征值。

Extrinsic eigenvalue estimates for Dirac operator

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.

A note on metric connections for chiral and Dirac spinors

It is known that the bundle of Dirac spinors is produced as a direct sum of two bundles - the bundle of chiral spinors and its Hermitian conjugate bundle. In this paper some aspects of metric connections for chiral and Dirac spinors are resumed and their relation is studied.

Some Notes on Dirac operators on the ${\rm S}^3$ and ${\rm S}^2$ spheres

We describe both the Hodge - de Rham and the spin manifold Dirac operator on the spheres ${\rm S}^3$ and ${\rm S}^2$, following the formalism introduced by Kähler, and exhibit a complete spectral resolution for them in terms of suitably globally defined eigenspinors.

关于李代数胚、奇异叶状结构和狄拉克结构的拉回与爆破的注记

A note on pullbacks and blowups of Lie algebroids, singular foliations, and Dirac structures

本文研究了李代数胚、奇异叶状结构和狄拉克结构在常秩或横截性映射下的拉回关系，特别分析了爆破映射的情况，并建立了李代数胚爆破与其奇异叶状结构的关系。

A note on twisted Dirac operators on closed surfaces

We derive an inequality that relates nodal set and eigenvalues of a class of twisted Dirac operators on closed surfaces and point out how this inequality naturally arises as an eigenvalue estimate for the $\rm Spin^c$ Dirac operator. This allows us to obtain eigenvalue estimates for the twisted Dirac operator appearing in the context of Dirac-harmonic maps and their extensions, from which we also obtain several Liouville type results.

A note on Dirac spinors in a non-flat space-time of general relativity

Some aspects of Dirac spinors are resumed and studied in order to interpret mathematically the P and T operations in a gravitational field.

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.

Transversal Dirac operators on distributions, foliations, and G-manifolds: Lecture notes

In these survey lectures, we investigate the geometric and analytic properties of transverse Dirac operators. In particular, we define a transverse Dirac operator associated to a distribution that is essentially self-adjoint (Prokhorenkov-R result). We describe the Habib-R Theorem showing that the invariance of the spectrum of a basic Dirac operator on a Riemannian foliation. The Bruening-Kamber-R theorems give Atiyah-Singer type formulas for the equivariant index of transversally elliptic operators on G-manifolds and the index of basic Dirac operators on Riemannian foliations. These notes con…

狄拉克框架下的离散力学系统：一项提案

Discrete Mechanical Systems in a Dirac Setting: a Proposal

本研究利用Dirac结构提出离散Dirac力学的新形式，建立连续Dirac系统概念并定义离散Dirac系统，证明其可涵盖离散拉格朗日与哈密顿系统，且能处理两类约束。

A brief note on how Einstein's general relativity has influenced the development of modern differential geometry

We briefly review a few aspects of the development of differential geometry which may be considered as being influenced by Einstein's general relativity. We focus on how Einstein's quest for a complete geometrization of matter and electromagnetism gave rise to an enormous amount of theoretical work both on physics and mathematics. In connection with this we also bring to light how recent investigation on theoretical physics has led to new results on some branches of modern differential geometry.

The multiplicities of the equivariant index of twisted Dirac operators

In this note, we give a geometric expression for the multiplicities of the equivariant index of a Dirac operator twisted by a line bundle.

Extrinsic Bounds for Eigenvalues of the Dirac Operator

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.

关于Berger球面上eta不变量的一个数值等式的注记

Note on a numerical equality regarding the eta invariant on Berger spheres

本文通过数值计算发现Berger球面上的Dirac APS eta不变量与圆球面上的Dirac共形反常一致，并用广义Bernoulli多项式验证了已知猜想，同时得到了Weingart生成函数。

Lectures on families of Dirac operators and applications

These are the notes for a minicourse taught at the 2022 ICTP summer school `Frontiers in Geometry and Topology'. The goal is to introduce families of Dirac operators and how they can be used to study interactions between geometry and topology. In particular, we discuss the index theorem for families of twisted Dirac operators, and its applications to metrics of positive scalar curvature and the three-dimensional Weinstein conjecture.

关于Clifford丛与某些Finsler型空间的注记

A note on Clifford bundles and certain Finsler type spaces

本文研究Clifford丛扩张与Finsler型结构的关系，证明在平坦度量下，由（伪）黎曼度量和1-形式构造的Finsler结构、Clifford丛扩张及Dirac型算子之间存在三角映射。