arXiv subjects

Yasushi Homma

Publications and source records attributed to Yasushi Homma.

13 recordsLinked to original sources

arXiv2026

Eigenvalue value estimates and stability of positive quaternion-Kähler manifolds

In this article we study the stability problem for positive quaternion-Kähler manifolds. We give a description of infinitesimal Einstein deformations and destabilising directions in terms of Laplace eigenfunctions and a special class of symmetric 2-tensors. We also give improved eigenvalue estimates for the Hodge-Laplacian on 2-forms. On the parallel subbundle Sym^2 E of the 2-form bundle we prove a sharp lower bound for the first non-zero eigenvalue.

math.DG↗

arXiv2026

Higher spin Killing spinors on 3-dimensional manifolds

We define higher spin Killing spinors on Riemannian spin manifolds in arbitrary dimension and study them in detail in dimension three. We prove a rigidity result for 3-dimensional manifolds admitting higher spin Killing spinors and give expressions for higher spin Killing spinors on the 3-sphere and the 3-hyperbolic space explicitly. We also investigate the Killing spinor type equation on integral spin bundles.

math.DG↗

arXiv2020

The spinor and tensor fields with higher spin on spaces of constant curvature

In this article, we give all the Weitzenböck-type formulas among the geometric first order differential operators on the spinor fields with spin $j+1/2$ over Riemannian spin manifolds of constant curvature. Then we find an explicit factorization formula of the Laplace operator raised to the power $j+1$ and understand how the spinor fields with spin $j+1/2$ are related to the spinors with lower spin. As an application, we calculate the spectra of the operators on the standard sphere and clarify the relation among the spinors from the viewpoint of representation theory. Next we study the case of trace-free symmetric tensor fields with an application to Killing tensor fields. Lastly we discuss the spinor fields coupled with differential forms and give a kind of Hodge-de Rham decomposition on spaces of constant curvature.

math.DG↗

arXiv2020

Spectra of the Rarita-Schwinger operator on some symmetric spaces

We give a method to calculate spectra of the square of the Rarita-Schwinger operator on compact symmetric spaces. According to Weitzenböck formulas, the operator can be written by the Laplace operator, which is the Casimir operator on compact symmetric spaces. Then we can obtain the spectra by using the Freudenthal's formula and branching rules. As examples, we calculate the spectra on the sphere, the complex projective space, and the quaternionic projective space.

math.DG↗

arXiv2018

The kernel of the Rarita-Schwinger operator on Riemannian spin manifolds

We study the Rarita-Schwinger operator on compact Riemannian spin manifolds. In particular, we find examples of compact Einstein manifolds with positive scalar curvature where the Rarita-Schwinger operator has a non-trivial kernel. For positive quaternion Kähler manifolds and symmetric spaces with spin structure we give a complete classification of manifolds admitting Rarita-Schwinger fields. In the case of Calabi-Yau, hyperkähler, $G_2$ and Spin(7) manifolds we find an identification of the kernel of the Rarita-Schwinger operator with certain spaces of harmonic forms. We also give a classification of compact irreducible spin manifolds admitting parallel Rarita-Schwinger fields.

math.DG↗

arXiv2005

Estimating the eigenvalues on Quaternionic Kähler Manifolds

We study geometric first order differential operators on quaternionic Kähler manifolds. Their principal symbols are related to the enveloping algebra and Casimir elements for $\Sp(1)\Sp(n)$. This observation leads to anti-symmetry of the principal symbols and Bochner-Weitzenböck formulas for operators. As an application, we estimate the first eigenvalues of them.

math.DG↗

arXiv2003

Bochner-Weitzenböck formulas and curvature actions on Riemannian manifolds

Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give certain relations in the enveloping algebra, which induce not only identities for higher Casimir elements but also all Bochner-Weitzenböck formulas for gradients. As applications, we give some vanishing theorems.

math.DG↗

arXiv2002

The Bochner identities for the Kählerian gradients

We discuss algebraic properties for the symbols of geometric first order differential operators on almost Hermitian manifolds and Kähler manifolds. Through study on the universal enveloping algebra and higher Casimir elements, we know algebraic relations for the symbols like the Clifford algebra. From the relations, we have all the Bochner identities for the operators. As applications, we have vanishing theorems, the Bochner-Weitzenböck formula, and eigenvalue estimates for the operators on Kähler manifolds.

math.DG↗

arXiv2000

Spherical harmonic polynomials for higher bundles

We give a method of decomposing bundle-valued polynomials compatible with the action of the Lie group $Spin(n)$, where important tools are $Spin(n)$-equivariant operators and their spectral decompositions. In particular, the top irreducible component is realized as an intersection of kernels of these operators.

math.DG↗

arXiv2000

Clifford Homomorphisms and Higher Spin Dirac Operators

We present a generalization of the Clifford action for other representations spaces of $Spin(n)$, which is called the Clifford homomorphism. Their properties extend to the ones for the higher spin Dirac operators on spin manifolds. In particular, we have general Bochner identities for them, and an eigenvalue estimate of a Laplace type operator on any associated bundle.

math.DG↗

arXiv2000

The Higher Spin Dirac Operators on 3-Dimensional Manifolds

We study the higher spin Dirac operators on 3-dimensional manifolds and show that there exist two Laplace type operators for each associated bundle. Furthermore, we give lower bound estimations for the first eigenvalues of these Laplace type operators.

math.DG↗

arXiv1998

Submodels of Nonlinear Grassmann Sigma Models in Any Dimension and Conserved Currents, Exact Solutions

In the preceding paper(hep-th/9806084), we constructed submodels of nonlinear Grassmann sigma models in any dimension and, moreover, an infinite number of conserved currents and a wide class of exact solutions. In this paper, we first construct almost all conserved currents for the submodels and all ones for the one of ${\bf C}P^1$-model. We next review the Smirnov and Sobolev construction for the equations of ${\bf C}P^1$-submodel and extend the equations, the S-S construction and conserved currents to the higher order ones.

hep-th↗

arXiv1998

Nonlinear Grassmann Sigma Models in Any Dimension and An Infinite Number of Conserved Currents

We first consider nonlinear Grassmann sigma models in any dimension and next construct their submodels. For these models we construct an infinite number of nontrivial conserved currents. Our result is independent of time-space dimensions and, therfore, is a full generalization of that of authors (Alvarez, Ferreira and Guillen). Our result also suggests that our method may be applied to other nonlinear sigma models such as chiral models, $G/H$ sigma models in any dimension.

hep-th↗