Related papers: $G_2$-Poisson equation on homogeneous spheres

We solve for the spectrum and eigenfunctions of Dirac operator on the sphere. The eigenvalues are nonzero whole numbers. The eigenfunctions are two-component spinors which may be classified by representations of the SU(2) group with…

Let $M$ be a complete Riemannian manifold and $G$ a Lie subgroup of the isometry group of $M$ acting freely and properly on $M.$ We study the Dirichlet Problem \begin{align*} \operatorname{div}\left( \frac{a\left( \left\Vert \nabla…

$G_2$-Monopoles are solutions to gauge theoretical equations on noncompact $7$-manifolds of $G_2$ holonomy. We shall study this equation on the $3$ Bryant-Salamon manifolds. We construct examples of $G_2$-monopoles on two of these…

We study the eigenvalue problem of the squared Pauli-Lubanski vector, W^{2}, in the Spinor-Vector representation space and derive from it that the -s(s+1)m^{2} subspace with s=3/2, i.e. spin 3/2 in the rest frame, is pinned down by the one…

We consider two different $\text{SU}(2)^2$-invariant cohomogeneity one manifolds, one non-compact $M=\mathbb{R}^4 \times S^3$ and one compact $M=S^4 \times S^3$, and study the existence of coclosed $\text{SU}(2)^2$-invariant…

We study the spectrum of the Hodge-Laplacian on $1$-forms for left-invariant metrics on the Lie group $\operatorname{SU}(2) \cong S^3$ and its quotient $\operatorname{SO}(3)\cong P^3(\mathbb{R})$. To the best of our knowledge, we provide…

We consider Poisson's equation on the $n$-dimensional sphere in the situation where the inhomogeneous term has zero integral. Using a number of classical and modern hypergeometric identities, we integrate this equation to produce the form…

We introduce a gauge invariant topological definition of monopole charge in pure SU(2) gluodynamics. The non-trivial topology is provided by hedgehog configurations of the non-Abelian field strength tensor on the two-sphere surrounding the…

In this paper, it is shown that (with no additional assumptions) on a compact 7-dimensional manifold which admits a $G_2$-structure soliton solutions to the Laplacian flow of R. Bryant can only be shrinking or steady. We also show that the…

We investigate octonion product deformations coming from the parallelizable torsion of the 7-sphere $S^7$, obtaining a family of geometries from solutions of the Lagrangian formalism movement equations. This can be achieved by analyzing the…

Let $G=SO_0(1,n)$ be the conformal group acting on the $(n-1)$ dimensional sphere $S$, and let $(\pi_\lambda)_{\lambda\in \mathbb C}$ be the spherical principal series. For generic values of $\boldsymbol \lambda…

Let g be a G-invariant Einstein metric on a compact homogeneous space M=G/K. We use a formula for the Lichnerowicz Laplacian of g at G-invariant TT-tensors to study the stability type of g as a critical point of the scalar curvature…

We consider Lie(G)-valued G-invariant connections on bundles over spaces G/H, RxG/H and R^2xG/H, where G/H is a compact nearly Kaehler six-dimensional homogeneous space, and the manifolds RxG/H and R^2xG/H carry G_2- and Spin(7)-structures,…

We solve the Killing spinor equations of supersymmetric IIB backgrounds which admit one supersymmetry and the Killing spinor has stability subgroup G_2 in Spin(9,1) x U(1). We find that such backgrounds admit a time-like Killing vector…

For any $G$-invariant metric on a compact homogeneous space $M=G/K$, we give a formula for the Lichnerowicz Laplacian restricted to the space of all $G$-invariant symmetric $2$-tensors in terms of the structural constants of $G/K$. As an…

We consider the $G$-invariant spectrum of the Laplacian on an orbit space $M/G$ where $M$ is a compact Riemannian manifold and $G$ acts by isometries. We generalize the Sunada-Pesce-Sutton technique to the $G$-invariant setting to produce…

In some other context, the question was raised how many nearly K\"ahler structures exist on the sphere $\S^6$ equipped with the standard Riemannian metric. In this short note, we prove that, up to isometry, there exists only one. This is a…

A Hadwiger-type theorem for the exceptional Lie groups $G_2$ and $Spin(7)$ is proved. The algebras of $G_2$ or $Spin(7)$ invariant, translation invariant continuous valuations are both of dimension 10. Geometrically meaningful bases are…

Let $G$ be a permutation group on a finite set $\Omega$. The $k$-closure $G^{(k)}$ of the group $G$ is the largest subgroup of $\operatorname{Sym}(\Omega)$ having the same orbits as $G$ on the $k$-th Cartesian power $\Omega^k$ of $\Omega$.…

Lie symmetries of a Novikov geometrically integrable equation are found and group-invariant solutions are obtained. Local conservation laws up to second order are established as well as their corresponding conserved quantities. Sufficient…