We determine all orbits of two prehomogeneous vector spaces rationally over an arbitrary perfect field in this paper.

In [4], the author introduced two parabolic flows; the almost Hermitian flow and the almost Hermitian curvature flow on a compact almost complex manifold. In this paper, we will develop some regularity results for solutions to these flows. We will derive higher order derivative estimates in the presence of a curvature bound. We also exhibit a long-time existence obstruction for solutions to the almost Hermitian curvature flow by showing smoothing estimates for the curvature and torsion.

We revisit the problem of constructing a Stratonovich-Weyl correspondence for each generic representation of a the real diamond group which was already considered in [B. Cahen, Riv. Mat. Univ. Parma 4 (2013), 197-213]. Here we use the complex Weyl calculus on the Fock space. This allows us, in particular, to obtain some closed formulas for the Stratonovich-Weyl symbols of the representation operators.

Let $M^2_{\kappa}$ be the complete, simply connected, Riemannian 2-manifold of constant curvature $\kappa \leq 0$. Let $E$ be a closed, simply connected subspace of $M^2_{\kappa}$ with the property that every pair of points in $E$ is connected by a rectifiable path in $E$. We show that under the induced path metric, $E$ is a complete CAT($\kappa$) space. We also show that the natural notions of angle coming from the intrinsic and extrinsic metrics coincide for all simple geodesic triangles.

A. D'Agnolo and M. Kashiwara proved that their enhanced solution functor induces a fully faithful embedding of the triangulated category of holonomic $\cal D$-modules into the one of $\mathbb{R}$-constructible enhanced ind-sheaves. In this paper, we introduce a notion of $\mathbb{C}$-constructible enhanced ind-sheaves and show that the triangulated category of them is equivalent to its essential image. Moreover we show that there exists a t-structure on it whose heart is equivalent to the abelian category of holonomic $\cal D$-modules.

We classify real hypersurfaces of a non-flat complex space form of complex dimension 2 satisfying the condition that the Ricci tensor S is the transversal Killing tensor field on the direction of the structure vector field $\xi$ and $S\xi = \beta\xi$ where $\beta$ is a function.