Stationary Curvatures of Two-Dimensional Totally Geodesic Submanifolds in the Grassmannian of Bivectors

An infinitesimal criterion indicating when a two-dimensional submanifold of a Riemannian symmetric space is totally geodesic is given. As an application, the classification of two-dimensional totally geodesic submanifolds of the Grassmannian of bivectors is given in a new way, and it is proved that the sectional curvature takes stationary values on tangent spaces of such submanifolds. Bibliography: 9 titles.     

Almost Schur lemma for manifolds with boundary

In this paper, we prove the almost Schur theorem, introduced by De Lellis and Topping, for the Riemannian manifold M of nonnegative Ricci curvature with totally geodesic boundary. Examples are given to show that it is optimal when the dimension of M is at least 5. We also prove that the almost Schur theorem is true when M is a 4-dimensional manifold of nonnegative scalar curvature with totally geodesic boundary. Finally we obtain a generalization of the almost Schur theorem in all dimensions only by assuming the Ricci curvature is bounded below.     

Infinitesimal Rotary Deformations of Surfaces and Their Application to the Theory of Elastic Shells

We present a variational generalization of the problem of infinitesimal geodesic deformations of surfaces in the Euclidean space E ³. By virtue of rotary deformation, the image of every geodesic curve is an isoperimetric extremal of rotation (in the principal approximation). The results are associated in detail with rotary-conformal deformations. The application of these results to the mechanics of elastic shells is given.     

Some Computational Aspects of Geodesic Convex Sets in a Simple Polygon

In this article, we deal with some computational aspects of geodesic convex sets. Motzkin-type theorem, Radon-type theorem, and Helly-type theorem for geodesic convex sets are shown. In particular, given a finite collection of geodesic convex sets in a simple polygon and an “oracle,” which accepts as input three sets of the collection and which gives as its output an intersection point or reports its nonexistence; we present an algorithm for finding an intersection point of this collection.     

Measurable rigidity of the cohomological equation for linear cocycles over hyperbolic systems

We show that any measurable solution of the cohomological equation for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a Hölder solution. More generally, we show that every measurable invariant conformal structure for a Hölder linear cocycle over a hyperbolic system coincides almost everywhere with a continuous invariant conformal structure. We also use the main theorem to show that a linear cocycle is conformal if none of its iterates preserve a measurable family of proper subspaces of R^d. We use this to characterize closed negatively curved Riemannian manifolds of constant negative curvature by irreducibility of the action of the geodesic flow on the unstable bundle.     

New Characterization of Geodesic Convexity on Hadamard Manifolds with Applications

In this paper, some new results, concerned with the geodesic convex hull and geodesic convex combination, are given on Hadamard manifolds. An S-KKM theorem on a Hadamard manifold is also given in order to generalize the KKM theorem. As applications, a Fan–Browder-type fixed point theorem and a fixed point theorem for the a new mapping class are proved on Hadamard manifolds.     

A Zoll Counterexample to a Geodesic Length Conjecture

We construct a counterexample to a conjectured inequality L ≤ 2D, relating the diameter D and the least length L of a nontrivial closed geodesic, for a Riemannian metric on the 2-sphere. The construction relies on Guillemin’s theorem concerning the existence of Zoll surfaces integrating an arbitrary infinitesimal odd deformation of the round metric. Thus the round metric is not optimal for the ratio L/D. F.B. supported by the Swiss National Science Foundation. C.C. supported by NSF grants DMS 02-02536 and DMS 07-04145. M.K. supported by the Israel Science Foundation (grants 84/03 and 1294/06)     

A Note on Collars of Simple Closed Geodesics

For compact hyperbolic Riemann surfaces, the collar theorem gives a lower bound on the distance between a simple closed geodesic and all other simple closed geodesics that do not intersect the initial geodesic. Here it is shown that there are two possible configurations, and in each configuration there is a natural collar width associated to a simple closed geodesic. If one extends the natural collar of a simple closed geodesic α by ε >0, then the extended collar contains an infinity of simple closed geodesics that do not intersect α.Mathematics Subject Classiffications (2000). primary: 30F45; secondary: 32G07     

Normal and tangential geodesic deformations of the surfaces of revolution

We study special infinitesimal geodesic deformations of the surfaces of revolution in the Euclidean space E ³.     

Some Applications of Geodesic Loops and Nonlinear Geometric Algebra in the Theory of Gravity

Algebraic systems called local geodesic loops and their tangent Akivis algebras are considered. Their possible role in the theory of gravity is discussed. Quantum conditions for infinitesimal quantum events are proposed.     

The characterization of totally geodesic submanifolds ofS m andCP m

We prove a theorem on ruled surfaces that generalizes a theorem of Ferus on totally geodesic foliations. On the basis of this theorem we obtain criteria for totally geodesic submanifolds ofS ^m andCP ^m that generalize and complement certain results of Borisenko, Ferus, and Abe. We give an application to the geodesic differential forms defined by Dombrowski in the case of submanifolds ofS ^m andCP ^m.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 106–116.The author is grateful to V. A. Toponogov for posing this problem and for attention to the work and to A. A. Borisenko for helpful criticisms.     

Infinitesimal Isometries Along Curves and Generalized Jacobi Equations

On a Riemannian manifold, a solution of the Killing equation is an infinitesimal isometry. Since the Killing equation is overdetermined, infinitesimal isometries do not exist in general. A completely determined prolongation of the Killing equation is a PDE on the bundle of 1-jets of vector fields. Restricted to a curve, this becomes an ODE that generalizes the Jacobi equation. A solution of this ODE is called an infinitesimal isometry along the curve, which we show to be an infinitesimal rigid variation of the curve. We define Killing transport to be the associated linear isometry between fibers of the bundle along the curve, and show that it is parallel translation for a connection on the bundle related to the Riemannian connection. Restricting to dimension two, we study the holonomy of this connection, prove the Gauss–Bonnet theorem by means of Killing transport, and determine the criteria for local existence of infinitesimal isometries.     

On the geodesic curvature of an integral curve of a two-dimensional pfaffian manifold inE 4

The concept of generalized geodesic curvature is introduced along with the invariant mean geodesic curvature vector for a two-dimensional Pfaffian manifold inE ₄, and the indicatrix of generalized geodesic curvature of integral curves of this manifold is constructed.It is proved that the ratio of the fourth quadratic form of the manifold to its first fundamental form is equal to the square of the modulus of the generalized geodesic curvature vector, and an analog of a theorem of Euler is deduced.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 22–26.     

Symmetric subvarieties in compactifications and the Radon transform on Riemannian symmetric spaces of noncompact type

We investigate the totally geodesic Radon transform which assigns a function to its integration over totally geodesic symmetric submanifolds in Riemannian symmetric spaces of noncompact type. Our main concern is focused on the injectivity and support theorem. Our approach is based on the projection slice theorem relating the totally geodesic Radon transform and the Fourier transforms on symmetric spaces. Our approach also uses the study of geometry concerned with the totally geodesic symmetric subvarieties in Riemannian symmetric spaces in terms of the cell structure of the Satake compactifications.     

On the Space of Kähler Potentials

We consider the geodesic equation for the generalized Kähler potential with only mixed second derivatives bounded. We show that given two such generalized Kähler potentials, there is a unique geodesic segment such that for each point on the geodesic, the generalized Kähler potential has uniformly bounded mixed second derivatives (in manifold directions). This generalizes a fundamental theorem of Chen (2000) on the space of Kähler potentials.© 2014 Wiley Periodicals, Inc.     

Periodic Cyclic Homology as Sheaf Cohomology Dedicated to Daniel Quillen on his 60th Birthday

We study a noncommutative version of the infinitesimal site of Grothendieck. A theorem of Grothendieck establishes that the cohomology of the structure sheaf on the infinitesimal topology of a scheme of characteristic zero is de Rham cohomology. We prove that, for the noncommutative infinitesimal topology of an associative algebra over a field of characteristic zero, the cohomology of the structure sheaf modulo commutators is periodic cyclic cohomology. We also compute the noncommutative infinitesimal cohomology of other sheaves. For example, we show that infinitesimal hypercohomology with coefficients in K-theory gives the fiber of the Jones–Goodwillie character which goes from K-theory to negative cyclic homology.     

The first nonzero eigenvalue of Neumann problem on Riemannian manifolds

In this article, the author obtained some comparison theorems of the first nonzero Neumann eigenvalue on domains in nonpositively curved Riemannian manifolds. The author first gives a generalized Szegö-Weinberger theorem (Theorem 1). Then the first nonzero Neumann eigenvalues for geodesic balls on nonpositively curved Riemannian manifolds are compared (Theorem 2). Based on these results, a “monotonicity principle” for the Neumann eigenvalues is derived. Then the author proves a stability theorem of maximality of the first nonzero Neumann eigenvalue of a geodesic ball among those of all domains with the same volume.     

一类流形的刚性定理

本文通过具有良好性质的子流形的存在性,证明了一类流形的一个刚性定理,并得到形如Ｂｏｎｎｅｔ－Ｍｙｅｒｓ定理的推论．我们还指出,在主要定理中全测地子流形的条件一般不能减弱为极小子流形．     

Inhomogeneous semigroups of commuting operators

The concepts of the homogeneously continuable semigroup of operators, and of infinitesimal and reproducing families of a semigroup, are introduced. The class of strongly continuous homogeneously continuable semigroups of commuting linear operators is discussed. This class contains in particular the class (C₀) of homogeneous semigroups. An analog of the Hill-Yosida theorem is proved for it.     

Pompeiu transforms on geodesic spheres in real analytic manifolds

We prove a support theorem for Pompeiu transforms integrating on geodesic spheres of fixed radiusr>0 on real analytic manifolds when the measures are real analytic and nowhere zero. To avoid pathologies, we assume thatr is less than the injectivity radius at the center of each sphere being integrated over. The proof of the main result is local and it involves the microlocal properties of the Pompeiu transform and a theorem of Hörmander, Kawai, and Kashiwara on microlocal singularities.