The Hessian of a noncommutative polynomial has numerous negative eigenvalues

In this paper, we establish bounds on the degree of a symmetric polynomial p = p(x) = p(x ₁,..., x _g ) (with real coefficients) in g noncommuting (nc) variables x ₁,..., x _g in terms of the “signature” of its Hessian

A sharp form of Nevanlinna's second fundamental theorem

Letf be meromorphic in the plane. We find a sharp upper bound for the error term

A family of cyclic cubic polynomials whose roots are systems of fundamental units

We give a family of cyclic cubic polynomials whose roots are systems of fundamental units of the splitting fields. These polynomials are constructed by a linear fractional transformation from Shanks’ polynomials with rational coefficients.     

Two nonisomorphicK-automorphisms all of whose powers beyond one are isomorphic

As in the earlier example of two nonisomorphicK-automorphisms with isomorphic square by building an appropriate skew product of aK-automorphism that is not Bernoulli with an algebraic fiber, we get all powers beyond one of the two nonisomorphic transformations to be isomorphic. Furthermore, all the isomorphism maps are finite codings of the constructed partition names.     

Classification of hypersurfaces with parallel M(o)bius second fundamental form in Sn+1

Let Mn(n≥2) be an immersed umbilic-free hypersurface in the(n+1)-dimensional unit sphere Sn+1. Then Mn is associated witha so-called M(o)bius metric g, and a M(o)bius second fundamental form Bwhich are invariants of Mn under the M(o)bius transformation groupof Sn+1.In this paper, we classify all umbilic-free hypersurfaces withparallel M(o)bius second fundamental form.     

On surfaces whose Grassmann image has curvature of at least one

We study the curvature of a Grassmann manifold along area elements tangent to a nondegenerate Grassmann image of a regular surface. According to Wong's theorem, it is included within the limits of [0; 2]. Limiting cases were considered earlier by Muto, Borisenko, and Nikolaevskii. There exists a conjecture according to which the values of the described curvatures cannot all be greater than 1 for surfaces of dimension at least 3. If they are all greater than or equal to 1, then the surface is a hypersurface. The conjecture is proven for certain bounds from below on the dimension of the surface.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 77–91, 1990.In conclusion, the author wishes to thank Professor A. A. Borisenko for his constant attention to the progress of the problem's solution.     

Pointwise characterizations of curvature and second fundamental form on Riemannian manifolds

Let M be a complete Riemannian manifold possibly with a boundary?M.For any C~1-vector field Z,by using gradient/functional inequalities of the(reflecting)diffusion process generated by L:=?+Z,pointwise characterizations are presented for the Bakry-Emery curvature of L and the second fundamental form of?M if it exists.These characterizations extend and strengthen the recent results derived by Naber for the uniform norm‖RicZ‖∞on manifolds without boundaries.A key point of the present study is to apply the asymptotic formulas for these two tensors found by the first author,such that the proofs are significantly simplified.     

Clifford systems, algebraically constant second fundamental form and isoparametric hypersurfaces

In this paper we prove that a submanifold with parallel mean curvature of a space of constant curvature, whose second fundamental form has the same algebraic type as the one of a symmetric submanifold, is locally symmetric. As an application, using properties of Clifford systems, we give a short and alternative proof of a result of Cartan asserting that a compact isoparametric hypersurface of the sphere with three distinct principal curvatures is a tube around the Veronese embedding of the real, complex, quaternionic or Cayley projective planes. Received: 22 April 1998     

On the topology of a compact submanifold of a sphere with bounded second fundamental form

Leth be the second fundamental form of a compact submanifold of a unit sphere. We show that if ‖h(u, u)‖²<1/3 holds for any unit tangent vectoru at any point on the submanifold then it is a homotopy sphere.     

共形平坦黎曼流形中具有平行第二基本形式的超曲面

In this paper, we prove the followingTheorem. Let Cⁿ⁺¹ ( n >5 ) be a conformally flat Riemannian anifola of dimension n + 1 . If Mn is a hypersurface immersed isometrically in Cⁿ⁺¹ over which the second fundamental form is covariant constant, then there are three posible cases only:I . locally Mⁿ= S^p×S^q×S^r, p+q+r=n;Ⅱ . locally Mⁿ=S^p×S^q, p + q = n , where S^k is k- dimensional Riemannian space of constant curvature;III. Mⁿ is umbilical and conformally flat. Moreover, if Mⁿ is connected and complete, then the result holds globally.     

The stressed state in a neighborhood of a plane crack whose border has points of negative curvature

Using the example of a plane crack bounded by the limaçon of Pascal we study the influence of the points of negative curvature on the stressed state of a body in a neighborhood of border of the crack. It is shown that under tensile loads the damage of the body begins in a neighborhood of such points. In the case of shear loads prescribed on the edges of the crack damage of the body begins in a neighborhood of points of the border farthest from the points of negative curvature.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 83–90.     

Submanifolds of the Cayley projective plane with bounded second fundamental form

We consider totally complex submanifolds of the Cayley projective plane with estimates on the length squared of the second fundamental form. We determine those bounds for which the second fundamental form is parallel and for which the submanifold is totally geodesic. The case of totally real submanifolds is also included.