In this article, we summarize the results on symmetric conformal geometries. We review the results following from the general theory of symmetric parabolic geometries and prove several new results for symmetric conformal geometries. In particular, we show that each symmetric conformal geometry is either locally flat or covered by a pseudo-Riemannian symmetric space, where the covering is a conformal map. We construct examples of locally flat symmetric conformal geometries that are not pseudo-Riemannian...

A regular normal parabolic geometry of type $G/P$ on a manifold $M$ gives rise to sequences $D_i$ of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative ${\nabla }^{\omega }$ on the corresponding tractor bundle $V$, where $\omega$ is the normal Cartan connection. The first operator $D_0$ in the sequence is overdetermined and it is well known that ${\nabla }^{\omega }$ yields the prolongation of this operator in the homogeneous case $M=G/P$. Our first main result...

Conformally flat metric $\overline{g}$ is said to be Ricci superosculating with $g$ at the point $x_0$ if $g_{ij}\left(x_0\right)={\overline{g}}_{ij}\left(x_0\right)$, ${\Gamma }_{ij}^k\left(x_0\right)={\overline{\Gamma }}_{ij}^k\left(x_0\right)$, $R_{ij}^k\left(x_0\right)={\overline{R}}_{ij}^k\left(x_0\right)$, where $R_{ij}$ is the Ricci tensor. In this paper the following theorem is proved: If $\gamma$ is a smooth curve of the Riemannian manifold $M$(without self-crossing(, then there is a neighbourhood of $\gamma$ and a conformally flat metric $\overline{g}$ which is the Ricci superosculating with $g$ along the curve $\gamma$.

The well known conformal covariance of the Dirac operator acting on spinor fields does not extend to its powers in general. For odd powers of the Dirac operator we derive an algorithmic construction in terms of associated tractor bundles computing correction terms in order to achieve conformal covariance. These operators turn out to be formally (anti-) self-adjoint. Working out this algorithm we recover explicit formula for the conformal third and present a conformal fifth power of the Dirac operator....

Let $(M^n,g)$ be a closed Riemannian manifold and $g_E$ the Euclidean metric. We show that for $m>1$, $\left(M^n\times {𝐑}^m,(g+g_E)\right)$ is not conformal to a positive Einstein manifold. Moreover, $\left(M^n\times {𝐑}^m,(g+g_E)\right)$ is not conformal to a Riemannian manifold of positive Ricci curvature, through a radial, integrable, smooth function, $\varphi :{𝐑}^{𝐦}\to {𝐑}^{+}$, for $m>1$. These results are motivated by some recent questions on Yamabe constants.

We discuss some aspects of the composition structure of twisted Verma modules for the Lie algebra $\mathrm{𝔰𝔩}(3,ℂ)$, including the explicit structure of singular vectors for both $\mathrm{𝔰𝔩}(3,ℂ)$ and one of its Lie subalgebras $\mathrm{𝔰𝔩}(2,ℂ)$, and also of their generators. Our analysis is based on the use of partial Fourier tranform applied to the realization of twisted Verma modules as $D$-modules on the Schubert cells in the full flag manifold for $\mathrm{SL}(3,ℂ)$.