Showing papers in "Asian Journal of Mathematics in 2001"

Parallel spinors and connections with skew-symmetric torsion in string theory

Abstract: We describe all almost contact metric, almost hermitian and G2-structures admitting a connection with totally skew-symmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the ∇-parallel spinors. In particular, we obtain solutions of the type II string equations in dimension n = 5,6 and 7.

Deformations of Symplectic Lie Algebroids, Deformations of Holomorphic Symplectic Structures, and Index Theorems

Abstract: Recently Kontsevich solved the classification problem for deformation quantizations of all Poisson structures on a manifold In this paper we study those Poisson structures for which the explicit methods of Fedosov can be applied, namely the Poisson structures coming from symplectic Lie algebroids, as well as holomorphic symplectic structures For deformations of these structures we prove the classification theorems and a general a general index theorem

Weak $Spin(9)$-Structures on 16-dimensional Riemannian Manifolds

Abstract: The aim of the present paper is the investigation of $Spin(9)$-structures on 16-dimensional manifolds from the point of view of topology as well as holonomy theory. First we construct several examples. Then we study the necessary topological conditions resulting from the existence of a $Spin(9)$-reduction of the frame bundle of a 16-dimensional compact manifold (Stiefel-Whitney and Pontrjagin classes). We compute the homotopy groups $\pi_i (X^{84})$ of the space $X^{84}= SO(16) / Spin(9)$ for $i \le 14$. Next we introduce different geometric types of $Spin(9)$-structures and derive the corresponding differential equation for the unique self-dual 8-form $\Omega^8$ assigned to any type of $Spin(9)$-structure. Finally we construct the twistor space of a 16-dimensional manifold with $Spin(9)$-structure and study the integrability conditions for its universal almost complex structure as well as the structure of the holomorphic normal bundle.

Finite jet determination of holomorphic mappings at the boundary

Abstract: Let M,M' be smooth real hypersurfaces in N-dimensional space and assume that M is k-nondegenerate at a point p in M. We prove that holomorphic mappings that extend smoothly to M, sending a neighborhood of p in M diffeomorphically into M' are completely determined by their 2k-jet at p. As an application of this result, we also give sufficient conditions on a smooth real hypersurface which guarantee that the space of infinitesimal CR automorphisms is finite dimensional.

Convex polyhedra in lorentzian space-forms ⁄

Abstract: Aleksandrov (Ale51) characterized the metrics induced on convex polyhedra in E 3 ;H 3 and S3. We give analogs for compact and complete polyhedra in Lorentzian space-forms. There are three types of convex polyhedra in the de Sitter space S 3 1 . One, which includes generalized hyperbolic polyhedra, was treated in (Sch98a). For the second, we characterize the induced metrics, and show that each is obtained on a unique polyhedron satisfying a natural condition at inflnity. For the last type | compact polyhedra bounding compact domains | we describe the induced metrics, and give an existence and uniqueness result for a smaller class of metrics. The results on complete polyhedra are consequences of the study of the metrics induced on convex polyhedra in a natural extension of H 3 by S 3. We also characterize the metrics induced on compact, convex polyhedra in the Minkowski space E3 . Those description are partly similar to those obtained in the Riemannian cases, but they also involve new elements of a metric and combinatorial nature.

Singularities in crystalline curvature flows

Abstract: This paper discusses the behaviour of polygonal convex curves in the plane moving under crystalline curvature flows, in which the speed of motion of each edge is determined by a function of its length. The behaviour depends on the rate of growth of the speed as the length of the edge approaches zero: For slow growth — including the homogeneous case where speed is inversely proportional to a power α ∈ (0, 1) of the length — there are always solutions for which the enclosed area approaches zero while the length remains positive. If α > 1, then all solutions are asymptotic to homothetically contracting solutions, and if α = 1 then there is a range of different kinds of singularity that occur. 1. Crystalline curvature flows Several authors have considered crystalline curvature flows of polygonal curves in the plane, since their introduction in [T]. We refer the reader to [TCH] and [AG] for a discussion of the geometric and physical motivation for such flows. For present purposes we consider only convex curves, although the flows can be defined much more generally. In this case the flows can be defined in the following way: Let γ be a closed convex N -sided polygon in the plane, and label the edges γ0, . . . , γN−1 in an anticlockwise order. Let θi ∈ S = R/2πZ be the angle of the exterior normal of γi, and let `i be the length of γi. Moving γ by a crystalline curvature flow consists of finding a continuous family of polygonal curves γ(t) starting from γ so that each edge keeps the same direction but moves in the outward normal direction with speed vi determined by its length: (1) vi(t) = gi(`i). Here gi is a smooth function defined on (0,∞) which is monotone increasing for each i. This paper mostly concerns contraction flows, for which gi 0 and fi is a positive real number for each i. 1991 Mathematics Subject Classification. 53C44, 52A10, 34C11. 1

Low index minimal hypersurfaces of spheres

Abstract: LetM be a compact oriented non-equatorial minimal hypersurface of the unit n-dimensional sphere. Suppose that for any non-zero vector in w ∈ R there exists an orthogonal matrix B such that B(M) = M and B(w) 6= w. Since all known examples of minimal hypersurfaces have antipodal symmetry, they satisfy this condition. We prove that: i) the stability index of M is greater than or equal to n + 2 with strict inequality, unlessM is a Clifford hypersurface; ii) the difference between the first two eigenvalues of the Jacobi operator is less than or equal to n − 1 with strict inequality, unless the norm of the second fundamental form is constant; and iii) ifM has antipodal symmetry and is not a Clifford hypersurface then the index is greater than n+ 3. Moreover if the unit normal vector is even, the index is greater than 2n+ 2.

A generalized index theorem for Morse-Sturm systems and applications to Semi-Riemannian geometry

Abstract: We prove an extension of the Index Theorem for Morse–Sturm systems of the form −V ′′ + RV = 0, where R is symmetric with respect to a (non positive) symmetric bilinear form, and thus the corresponding differential operator is not self-adjoint. The result is then applied to the case of a Jacobi equation along a geodesic in a Lorentzian manifold, obtaining an extension of the Morse Index Theorem for Lorentzian geodesics with variable initial endpoints. Given a Lorentzian manifold (M, g), we consider a geodesic γ in M starting orthogonally to a smooth submanifold P of M. Under suitable hypotheses, satisfied, for instance, if (M, g) is stationary, the theorem gives an equality between the index of the second variation of the action functional f at γ and the sum of the Maslov index of γ with the index of the metric g on P . Under generic circumstances, the Maslov index of γ is given by an algebraic count of the P-focal points along γ. Using the Maslov index, we obtain the global Morse relations for geodesics between two fixed points in a stationary Lorentzian manifold.

Knot concordance and torsion

Abstract: Algebra, 2nd ed. , Prentice Hall, New Jersey, 1999.[F] R. Fox, A quick trip through knot theory , 1962 Topology of 3–manifolds and relatedtopics (Proc. The Univ. of Georgia Institute, 1961) 120–167 Prentice-Hall, EnglewoodCliffs, N.J.[FM] R. Fox and J. Milnor, Singularities of 2 –spheres in 4 –space and cobordism of knots ,Osaka J. Math. 3 (1966) 257–267.[G] C. Gordon, Problems , in Knot Theory, ed. J.-C. Hausmann, Springer Lecture Notesno. 685, 1977.[Gi] P. Gilmer, Slice knots in S 3 , Quart. J. Math. Oxford Ser. (2) 34 (1983), no. 135,305–322.[J] B. Jiang, A simple proof that the concordance group of algebraically slice knots isinfinitely generated , Proc. Amer. Math. Soc. 83 (1981), 189–192.[K1] R. Kirby, Problems in low dimensional manifold theory , in Algebraic and GeometricTopology (Stanford, 1976), vol 32, part II of Proc. Sympos. Pure Math., 273–312.[K2] R. Kirby, Problems in low dimensional manifold theory , Geometric Topology, AMS/IPStudies in Advanced Mathematics, ed. W. Kazez, 1997.[L1] J. Levine,

Bieberbach theorems for solvable Lie groups

Abstract: for all a: € G. With the linear connection on G defined by the left invariant vector fields, it is known that Aff(G) is the group of connection-preserving diffeomorphisms of G, see [KT] Proposition 2.1. TERMINOLOGY. In this paper we will use the term lattice of a Lie group G, to denote a discrete cocompact subgroup of G. Further, we will say that an automorphism a of a Lie group G is unipotent if and only if its differential da (in the automorphism group of the corresponding Lie algebra (&) is unipotent. Analogously we will speak of an element acting unipotently on a Lie group G. For G = IR, the following three theorems have been proven by Bieberbach. See [W] or [C]: THEOREM I'. Let ir C M x 0(n) be a lattice. Then T = TT D M is a lattice of W, and F has finite index in TT. THEOREM 2. Let 7r,7r' C M xi 0(n) be lattices. Then every isomorphism 0 : TT -)■ TT' is a conjugation by an element ofR x GL(n. R). THEOREM 3'. Under each torus Z\\E; there are only finitely many flat manifolds which are covered by the torus.

Real hypersurfaces of Kahler manifolds.

Abstract: Building on work by S. M. Webster \ref[J. Differential Geom. 13 (1978), no. 1, 25--41; MR0520599 (80e:32015)] the author studies the geometry of the second fundamental form of a real hypersurface in a Kahler manifold. As an application he proves that a compact strictly pseudoconvex hypersurface $M\subsetC^n$ is isometric to a sphere provided that $M$ has constant horizontal mean curvature and the CR structure $T_{1,0}(M)$ is parallel in $T^{1,0}(C^n)$.

A focus on focal surfaces

Abstract: Congruences of lines in P3, i.e. two-dimensional families of lines, and their focal surfaces, have been a popular object of study in classical algebraic geometry. They have been considered recently by several authors as Arrondo, Goldstein, Sols, Verra. Aim of the paper under review is to study from the modern point of view the notions appearing in this context, so to prove in a rigorous way some classical results. More precisely, a congruence is a surface X G(1, 3), its focal locus F, in general a surface, is the branch locus of the natural projection qX from the incidence correspondence IX to P3. It results that a general line of X contains two foci counted with multiplicity, a line L whose points are all foci is called a focal line. In the paper under review the authors compute several invariants of the focal surface of a smooth congruence, e.g. its degree, class and sectional genus, and the degree of its nodal and cuspidal curves (assuming that these are the only 1-dimensional components of its singular locus). Then they study in detail some important examples of congruences, i.e. the bisecant lines of a smooth curve C in P3, the bitangents and the inflectional tangents of a smooth surface . In the first case they find that in general the focal surface F is all formed by focal lines, which are the stationary bisecants. In the other examples they find, among other results, that is a component of the focal surface with high multiplicity, and that at least one component of F is formed by focal lines.Motivated by the examples they state a series of conjectures about congruences whose focal surface is not irreducible or not reduced, and about singularities of congruences of bitangents or inflectionary tangents to not necessarily smooth surfaces of P3.

The structure equations of a complex Finsler manifold

Abstract: For a strongly pseudo-convex complex Finsler manifold M, a bundle U of adapted unitary frames is canonically defined. A non-linear Hermitian connection on U, invariant under local biholomorphic isometries, is given and it proved to be unique. By means of such connection, an absolute parallelism on U is determined and a new set of structure functions which generate all the isometric invariants of a Finsler metric is obtained. A pseudo-convex complex Finsler manifolds M, which admits a totally geodesic complex curve with a given constant holomorphic sectional curvature through any point and any direction, is called E-manifold. Main examples of E-manifolds are the smoothly bounded, strictly convex domains in C^n, endowed with the Kobayashi metric. A complete characterization of E-manifolds, using the previously defined structure functions, is given and a smaller set of generating functions for the isometric invariants of E-manifolds is determined.

Classification of indefinite hyper-Kähler symmetric spaces

Abstract: We classify indefinite simply connected hyper-Kahler symmetric spaces. Any such space without flat factor has commutative holonomy group and signature (4m, 4m). We establish a natural 1-1 correspondence between simply connected hyper-Kahler symmetric spaces of dimension 8m and orbits of the group GL(m,H) on the space (S4Cn)τ of homogeneous quartic polynomials S in n = 2m complex variables satisfying the reality condition S = τS, where τ is the real structure induced by the quaternionic structure of C2m = Hm. We define and classify also complex hyperKahler symmetric spaces. Such spaces without flat factor exist in any (complex) dimension divisible by 4.

Extension of Lipschitz maps into 3-manifolds

Abstract: We prove that the universal covering Y of a closed nonpositively curved 3-dimensional Riemannian manifold possesses the following Lipschitz extension property: there exists a constant c ≥ 1 such that every λ-Lipschitz map f : S → Y defined on s subset S of an arbitrary metric space X has a cλ-Lipschitz extension f¯ : X → Y.

A new construction of Einstein self-dual metrics

Abstract: We give a new construction of Ricci-flat self-dual metrics which is a natural extension of the Gibbons--Hawking ansatz. We also give characterisations of both these constructions, and explain how they come from harmonic morphisms.

Bialgebra and geometry of plane quartics

Abstract: Weak semisimplicity of the bialgebra of ternary quartic forms is proved. In fact there being no direct method of proceeding from the invariant to the geometric meaning ... // it be not obtained we should console ourselves with the reflexion that the uninterpreted forms are of little geometrical interest in the present state of knowledge; Besides if we regard the algebra as being merely helpful to geometry in the analytical formulations of results, it does not follow that everything in the algebra need be taken seriously from the geometrical point of view. (From No. 229 of \"The algebra of Invariants\" by J. H. Grace and A. Young, Cambridge, 1903) 1. First definitions and introduction. (1.0) Conventions. In the article, the ground field K is always of characteristic zero and algebraically closed. Let W be a finite-dimensional K-vector space, W* be the dual space of W. Sometimes, elements of W (denoted as a, 6, c.) will be called linear forms, elements of W* (denoted as p,q,r...) will be called vectors. Therefore, elements of P(W) are hyperplanes (or lines, if W is three-dimensional), elements of P^*) are points. Also, we will use geometric term pencil for two-dimensional subspaces of W (or W*). If {a, b) is a basis of a pencil F, then we will denote the pencil P by (Aa + //&) or by (Ka + Kb). Let W x W* -> K, (a,p) H+< a,p >G K, W* xW'+K, (p,a) H->eir be two natural pairings denoted with the same symbol <, > . We will say that two bases {ei,...,en} C W, {/i,...,/n} C W* are projectively dual , if < ei.fj >= 0 for i ^ j, < ei, fi >^ 0. Here, projective point of view means that we do not try to normalize elements of the bases and consider them up to proportionality. (1.1) Definition. Bialgebra. Bimultiplication law on W (defining a bialgebra structure on W) consists of two bilinear maps (both of which will be denoted by square brackets) WxW-^W\\ (a, 6) i-» [a, 6] e W*, W* x W* -+ W, (p,q) ^ [p,g]

Regularity of butterworth refinable functions

Abstract: Let ^N be the refinable function with Butterworth filter cos |(cos | + sm |)~ and let Sp(^iv) be the Fourier exponent of tytf of order p (0 < p < oo). It is proved that In2 ln2 and for 0 < p < oo ^^<.^s)-N^<^±rD. (iv>i) pln2 " ' In2 " In2 where ro E (0,1) is independent of p and N.

A classification of non-degenerate homogeneous equiaffine hypersurfaces in four complex dimensions

Abstract: We solve the classification problem as in the title. We present explicit defining equations and give a characterization in terms of easily computable affine invariants.

Harmonic radial combinations and dual mixed volumes

Abstract: For star bodies, p-harmonic radial combinations were introduced and studied in several papers. In this paper we study the relations of the dual quermassintegrals of p-harmonic radial combinations of star bodies to their dual quermassintegrals and obtain the upper bound for the dual quermassintegrals of p-harmonic radial combinations of star bodies. For star bodies, p-harmonic radial combinations were introduced and studied in several papers. The aim of this article is to study them further, that is, we investigate the relations of the dual quermassintegrals of p-harmonic radial combinations of star bodies to their dual quermassintegrals and obtain the upper bound for the dual quermassintegrals of p-harmonic radial combinations of star bodies. 1. Preliminaries. By a convex body in E,n > 2, we mean a compact convex subset of E with nonempty interior. Let 5~ denote the unit sphere centered at the origin in E, and write On_i for the (n — l)-dimensional volume of 5 . Let B be the closed unit ball in E, write (jjn for the n-dimensional volume of B. Note that c^n = 7r/r(l + -), and On_i = nujn. n For each direction u G 5, we define the support function h(K,u) on 5 of the convex body K by h(K, u) — supj-u • x\\x G K} and the radial function p(K,u) on 5~ of the convex body K is p(K,u) = sup{A > 0\\Xu G K}. If p(K,u) is positive and continuous, call K a star body (about the origin), and write S for the set of star bodies (about the origin) of E. Sets A, B are called homothetic if A = XB + t with t G E and A > 0 or one of them is a singleton ( a one-point set). The polar body of a convex body K, denoted by if*, is another convex body defined by if* = {y\\x -y < 1 for all x G if}. The polar body has the well known property that h(K*,u) = l/p(K,u) and p(K*,u) = l/h(K,u), * Received November 9, 1999; accepted for publication September 4, 2000. ^Department of Mathematics, Sungkyunkwan University, Suwon 440-746, South Korea (ydchai@yurim.skku.ac.kr). This research was supported by the Brain Korea 21 Project and 98 SKKU Faculty Research Fund. * School of Electrical and Computer Engineering, Sungkyunkwan University, Suwon 440-746, South Korea. Current address: Department of Mathematics, Sungkyunkwan University, Suwon 440-746, South Korea (yslee@math.skku.ac.kr). This research is finantially supported by the BK21 Project. 493 494 Y. D. CHAI AND Y. S. LEE Let Kj be a star body in E with o € Kj, 1 < j < n. Then we define the dual mixed volumes V(Ki, • • •, Kn) by (1.1) V{Kw',Kn)^[ p(Kuu)-.-p{Kn,u)du, n JSn-l where du signifies the area element on S~. Let Vi(K1,K2) = V(K1,-..,Kl,K2,-..,K2). The dual quermassintegrals are the special dual mixed volumes defined by Wi(K) = Vi(K,B). Note that Wo(K) = V(K) is the volume of K, while Wn(K) = un does not depend onK. 2. Main Results. Fix a real p > 1. For K, L G 0 (not both zero), the p-harmonic radial combination A • K+p /J, • L G S is defined by (2.1) p(A • i^+p ii'L,u)= \\p(K, u)+ /i/9(L, w)-. We obtain easily from the definition the following. THEOREM 1 (Positive multisublinear). Let K,L G S, a real p > 1, and A,// > 0 (not both zero). Then, for any M2, • • •, Mn G S, V(\\.K+pfi'L,M2r..,Mn)<\\V(K,M2r--,Mn)+fiV(L,M2,...,Mn) where A + // = 1. Proof. In the definition of the p-harmonic radial combination, if we use the fact that f(x) = x~p (p > 1) is convex, then p(X K+p pL,u) < Xp(K, u) + pp{L, u), A -fp = 1. So using the definition of the dual mixed volume, we easily obtain V{\\'K+pp'L,M2,--,Mn) = / p(\\ K+p p L,u)p{M2,u) • p(Mn,u)du n Jsn-i < / fA/9(i i:,w)+/i/9(L,w)V(M2,u)---p(Mn,u)du nJSn-i\\ / = Ay(^,M2,...,Mn)+//T/(L,M2,.-.,Mn). D Now we consider the dual quermassintegrals of the p-harmonic radial combinations. In the following theorem we obtain the upper bound for the dual quermassintegrals of the j9-harmonic radial combinations of star bodies. THEOREM 2. Let K,L e S, a realp > 1, and A,/i > 0 (not both zero). Then, for the p-harmonic radial combination A • K+p p • L ^(A.^p/4.L)<(^)^i(^_i^>«)^^) (/^ip(L>tt)^du)* HARMONIC RADIAL COMBINATIONS AND DUAL MIXED VOLUMES 495 for i — 0,1, • • •, n and s > 1 and ^ + | = 1. Equality holds if and only if both K and L are balls such that \\PL = fx^K. Proof From the definition of the p-harmonic radial combination, we have 1 ,1 1 p(X-K+Pn-L,u)P p{K,u)P ^p(L,u)P _ Xp(L,u)P + np{K,uy p{K,u)Pp(L,u)P It follows from the inequality between arithmetic and geometric means that p(K,u)Pp(L,u)P p(X-K+p p-L,u) p = < vt, „,„, Xp{L,u)P + np{K,u)P p(K,u)Pp(L,u)P 2^Xp(K,u)Pp,p(L,u)P

Generic modules for extension algebras

Abstract: Let A be a tame hereditary algebra (finite-dimensional over an algebraically closed field), RTM (m > 1) the extension algebra of A. A generic jR-module M over an arbitrary ring R is by definition an indecomposable .R-module of infinite length, such that M considered as an End(M)module, is of finite length (its endolength). In this paper we investigate the generic modules of A (the repetitive algebra of A) and RTM. It is proved that RTM has at least 2m generic modules. Introduction. The notion of generic module was introduced in [1] by CrawleyBoevey. The concept seems to be quite natural and important. The generic modules even have a dominating position in the category of modules. In [2], it was shown that whether a finite-dimensional algebra over an algebraically closed field is tame or wild is determined completely by the behaviour of the generic modules for that algebra. In [3], Aronszajn and Fixman gave the concept of a divisible module for the Kronecker algebra and showed that for the Kronecker algebra there exists a unique indecomposable torsion-free divisile module. In [4], Ringel generalized the work of Aronszajn and Fixman and proved the same result for a tame hereditary algebra. Ringel's work, in fact, showed that for a tame hereditary algebra, there exists a unique generic module. In [6], we solved the existence and uniqueness of generic module for the tilted algebra determined by a tame hereditary algebra. Following [1], A generic i?-module M over an arbitrary ring R is by definition an indecomposable .R-module of infinite length, such that M considered as an End(M)module, is of finite length (its endolength). Of course, the generic modules with endomorphism ring a division ring just, form the vertices of the (Cohn) spectrum of R. By [1], the endomorphism ring of a generic module always is a local ring. Our purpose here is to investigate the generic module of the extension algebra RTM (defined below) for a tame hereditary algebra A. In section 1, we investigate the z/-orbits of generic modules for a repetitive algebra, we shall prove that Mod A has at least two zA-orbits of generic ^-modules (Theorem 1.2). In section 2, we shall prove our main result on generic modules of RTM: it^ has at least 2m generic modules (Theorem 2.4 and Corollary 2.5). Throughout this paper, we denote by k an algebraically closed field. An algebra means basic, connected and finite-dimensional A:-algebra. For an algebra A we denote by Mod A the category of all right A-modules, by mod A the full subcategory of Mod A consisting of all finitely generated right ^4-modules and by mod A the corresponding stable category. We shall use freely properties of the Auslander-Reiten sequences, irreducible maps, Auslander-Reiten translation r = DTr and r = TrD, and the Auslander-Reiten quiver TA of an algebra B, for which we refer to [5]. 1. z/-orbits of Generic Modules of Repetitive Algebras. Let A = kA be a tame hereditary algebra over an k. We denote by D(A) the derived category D(mod A) of bounded complexes over mod A. For the definition of derived category we refer to [7]. By DA we denote the minimal injective cogenerator of A, where D = Homfc(—, k) is the usual dual functor. Consider the repetitive algebra [7]: * Received January 21, 1999; accepted for publication May 23, 1999. t Department of Mathematics, Anhui University, Hefei, 230039, P. R. China (xndu@mars.ahu. edu.cn). Reserch was supported by the National Science Foundation of China.