On the Chebyshev nets in a hypersurface of a Weyl space

In [1], Zlatanov introduced the Chebyshev vector fields of the first and second kind and the geodesic vector fields for an n-dimensional net in the Weyl spaceW _n . After having defined, in [2], the Chebyshev and geodesic curvatures of the lines of an arbitrary net,the b-nets and the c-nets, Tsareva and Zlatanov studied, among other things, some properties of the Chebyshev nets. In this paper, we consider an n-dimensional net in the hypersurfaceW _n of the Weyl SpaceW _n+1 and study some properties of the Chebyshev vector fields of the first and second kind and the geodesic vector fields of this net. Finally, two theorems concerning the b-nets and c-nets inW _n are obtained.     

On the nets of equi-conjugate pairs of curves in a Weyl hypersurface

In this paper, equi-conjugate pairs of curves in a hypersurfaceW _n(g_ij,T_k) are defined and the Chebyshev nets of the first kind, the second kind and the geodesic nets which are formed by the tangent vector fields of these pairs of curves are investigated.     

Extended affine Weyl groups and Frobenius manifolds

We define certain extensions of affine Weyl groups (distinct from these considered by K. Saito [S1] in the theory of extended affine root systems), prove an analogue of Chevalley Theorem for their invariants, and construct a Frobenius structure on their orbit spaces. This produces solutions F(t₁, ..., t_n) of WDVV equations of associativity polynomial in t₁, ..., t_n-1, exp t_n.     

Ghost Symmetries and Multi-fold Darboux Transformations of Extended Toda Hierarchy

In this paper, the author constructs ghost symmetries of the extended Toda hierarchy with their spectral representations. After this, two kinds of Darboux transformations in different directions and their mixed Darboux transformations of this hierarchy are constructed. These symmetries and Darboux transformations might be useful in GromovWitten theory of CP¹.     

Extended Affine Weyl Groups: Presentation by Conjugation via Integral Collection

We give several necessary and sufficient conditions for the existence of the presentation by conjugation for a non-simply laced extended affine Weyl group. We invent a computational tool by which one can determine simply the existence of the presentation by conjugation for an extended affine Weyl group. As an application, we determine the existence of the presentation by conjugation for a large class of extended affine Weyl groups.     

On simplexes inscribed in a hypersurface

The sufficient conditions are obtained for the existence, on a hyper surface M Rⁿ, of k points whose convex hull forms a (k–1)-dimensional simplex, homothetic to a given simplex Rⁿ. In particular, it is shown that if M is a smooth hypersurface, homeomorphic to a sphere, such points will exist for any simplex Rⁿ. The proofs are based on simple topological considerations. There are six references.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 81–89, January, 1969.     

Defect of a nodal hypersurface

In this note we give formulas for the Hodge numbers of a nodal hypersurface in a smooth complex projective fourfold. Received: 10 May 2000 / Revised version: 7 November 2000     

On union curves in a Kaehler hypersurface

Summary This paper has been devoted to the study of the union curve and union curvature of a Kaehler hypersurface. The expression for the geodesic curvature of the union curve with respect to a congruence have been obtained. Entrata in Redazione il 25 ottobre 1970.     

The motive of a Weil hypersurface

We study a special class of hypersurfaces V_p in a projective space over an algebraically closed field of arbitrary characteristic. We compute the motive of V_p in terms of the motives of the curves.Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 555–560, November, 1972.The author wishes to express his deep gratitude to Yu. I. Manin under whose direction this paper was prepared.     

A function of direction associated with n congruences of an orthogonal ennuple in a Riemannian hypersurface

In a hypersurface V_n belonging to a Riemannian space V_n+1 we choose n congruences of an orthogonal ennuple of unit vectors of contravariant components λ _h¦ ⁱ (h=1,2,...,n) and denote, by ω_kk the normal curvature of the hypersurface in the direction of the unit vector of components λ _k¦ ⁱ . In the present paper, we have shown that the expression $$\frac{\partial }{{\partial s_k }}\omega _{kk} - 2\mathop \Sigma \limits_{h = 1}^n \omega _{kh} \gamma _{hkk}$$ is a function of direction, where the symbol ?/?s_k indicates the differentiation in the direction of the vector of components λ _k¦ ⁱ and that ω_kh (h≠k) and γ_?hk (?,h,k=1,2,...,n) are, respectively, the invariants of the geodesic torsion of the curve of the congruence with unit tangent vector of components λ _k¦ ⁱ and Ricci's coefficients of rotation of the orthogonal ennuple.     

On a theorem of Weyl

We give a short proof of the theorem of Weyl, as generalized by von Neumann, that for any hermitian operator there exists a Hilbert-Schmidt perturbation of arbitrarily small Hilbert-Schmidt norm such that the sum has purely point spectrum. The proof consists of a simple construction of the perturbation. *** DIRECT SUPPORT *** A6515003 00004     

A hypersurface in -jets

We give an example of a hypersurface in through whose stability group at is determined by -jets, but not by jets of any lesser order. We also examine some of the properties which the stability group of this infinite type hypersurface shares with the -sphere in .