We describe the Cartan-Münzner polynomials of degree four on an $N$-dimensional Euclidean space by using the square norm of the moment map of certain group actions. In the case of OT-FKM type, the action is a spin action which is not necessarily transitive on isoprametric hypersurfaces.

For smooth manifolds equipped with various geometric structures, we construct complexes that replace the de Rham complex in providing an alternative fine resolution of the sheaf of locally constant functions. In case that the geometric structure is that of a parabolic geometry, our complexes coincide with the Bernstein-Gelfand-Gelfand complex associated with the trivial representation. However, at least in the cases we discuss, our constructions are relatively simple and avoid most of the machinery of parabolic geometry. Moreover, our method extends to contact and symplectic geometries (beyond the parabolic realm).

In this paper, we define a reducibility for a triply periodic minimal surface, and study hyperelliptic minimal surfaces and trigonal minimal surfaces in terms of the reducibility. Restrictions on the topological type of the above minimal surfaces are given.

In this present note, we introduce the notion of Monge characteristic systems for regular overdetermined systems associated with type-changing equations and clarify a relation between the rank of these characteristic systems and the type of the structure equations defined from the point of view of contact geometry of second order.

In this paper, we investigate the group of automorphisms of the induced orthogonal almost complex structures on isoparametric hypersurfaces in a seven-dimensional sphere, which is obtained by the multiplication of the octonions.

We show that the intersection of real flag manifolds in the complex flag manifold consisting of sequences of complex subspaces in a complex vector space is an antipodal set, which is a generalization of that in a Hermitian symmetric space of compact type.

This is the continuation of our previous paper “Contact Geometry of Second Order I”, where we have formulated the contact equivalence of systems of second order partial differential equations for a scalar function as the geometry of $PD$ manifolds of second order. In this paper, we will discuss the Two Step Reduction procedure in Contact Geometry of Second Order. In fact we establish the Second Reduction Theorem for $PD$ manifolds $(R; D^1, D^2)$ of second order admitting the first order covariant systems $\tilde N$. Utilizing the covariant system $\tilde N$, we construct the intermediate object $(W; C^:, N)$, called the $IG$ manifold of corank $r$, as a submanifold of the Involutive Grassmann bundle $I^r(J)$ over the contact manifold $(J,C)$, where $J = R/\text{Ch}\,(D^1)$. We will seek the condition when the equivalence of $(R; D^1, D^2)$ is reducible to that of $(W; C^*, N)$. Moreover, when $\text{Ch}\,(N)$ is non-trivial, the equivalence of $(W; C^*, N)$ is further reducible to that of $(Y; D_N^*, D_N)$, where $Y = W/\text{Ch}\,(N)$. This theorem gives a sufficient condition for the existence of higher dimensional characteristics of $(R; D^1, D^2)$. By analyzing the construction parts of the Two Step Reduction procedure, we will show several examples of Parabolic Geometries, which are, through the Second Reduction Theorem, associated with the geometry of $PD$ manifolds of second order.