• Kyungpook Mathematical Journal
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• v.63 no.1
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• pp.105-122
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• 2023

We present explicit formulas for the spectra of higher spin operators on the subbundle of the bundle of spinor-valued trace free symmetric tensors that are annihilated by Clifford multiplication over the standard sphere in odd dimension. In the even dimensional case, we give the spectra of the square of such operators. The Dirac and Rarita-Schwinger operators are zero-form and one-form cases, respectively. We also give eigenvalue formulas for the conformally invariant differential operators of all odd orders on the subbundle of the bundle of spinor-valued forms that are annihilated by Clifford multiplication in both even and odd dimensions on the sphere.

• Journal of the Korean Mathematical Society
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• v.34 no.1
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• pp.167-179
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• 1997

In the study of a manifold M, the exterior algebra $\Lambda^* M$ plays an important role. In fact, the de Rham cohomology theory gives many informations of a manifold. Another important object in the study of a manifold is its Clifford algebra (Cl(M), generated by the tangent space.

• Bulletin of the Korean Mathematical Society
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• v.35 no.4
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• pp.669-681
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• 1998

In this paper, we prove that on the complete Kahler manifold, if ${\rho}(x){\geq}-\frac{1}{2}{\lambda}_0$ and either ${\rho}(x_0)>-\frac{1}{2}{lambda}_0$ at some point $x_0$ or Vol(M)=${\infty}$, then the Clifford $L^2$ cohomology group $L^2{\mathcal H}^{\ast}(M,S)$ is trivial, where $\rho(x)$ is the least eigenvalue of ${\mathcal R}_x + \bar{{\mathcal R}}(x)\;and\;{\lambda}_0$ is the infimum of the spectrum of the Laplacian acting on $L^2$-functions on M.