Nontrivial Vector Research Articles

Energy Properness and Sasakian-Einstein Metrics

In this paper, we show that the existence of Sasakian-Einstein metrics is closely related to the properness of corresponding energy functionals. Under the condition that admitting no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of Sasakian-Einstein metric implies a Moser-Trudinger type inequality. At the end of this paper, we also obtain a Miyaoka-Yau type inequality in Sasakian geometry.

Division rings whose vector spaces are pseudofinite

AbstractVector spaces over fields are pseudofinite, and this remains true for vector spaces over division rings that are finite-dimensional over their center. We also construct a division ring such that the nontrivial vector spaces over it are not pseudofinite, using Richard Thompson's group F. The idea behind the construction comes from a first-order axiomatization of the class of division rings all whose nontrivial vector spaces are pseudofinite.

Characterizing spheres by conformal vector fields

In this short note we consider an n-dimensional compact Riemannian manifold (M, g) of constant scalar curvature S = n(n − 1)c and show that the presence of a nontrivial conformal vector field ξ on M forces S to be positive. Then we show that an appropriate control on the energy of ξ makes M to be isometric to the n-sphere Sn(c).

Holomorphic versus algebraic equivalence for deformations of real-algebraic Cauchy–Riemann manifolds

Since Poincare’s celebrated paper [19] published in 1907, there has been a growing literature concerned with the equivalence problem for real submanifolds in complex space (see e.g., [4, 6, 7, 11, 13, 14, 22] for some recent works as well as the references therein). One interesting phenomenon, observed by Webster for biholomorphisms of Levi nondegenerate hypersurfaces [23], is that the biholomorphic equivalence of some types of real-algebraic submanifolds of a complex space implies their algebraic equivalence. In this paper, we show that this very phenomenon holds for algebraic deformations of germs of minimal holomorphically nondegenerate realalgebraic CR submanifolds in complex space. Let us recall that a germ of a real-algebraic CR submanifold (M,p) ⊂ (Cn, p) is minimal if there exists no proper CR submanifold N ⊂ M through p of the same CR dimension as M . It is holomorphically nondegenerate if there exists no nontrivial holomorphic vector field tangent to M near p (see [21]). An algebraic deformation of (M,p) is a real-algebraic family of germs at p of real-algebraic CR submanifolds (Ms, p)s∈Rk in Cn, defined for s ∈ Rk near 0, such that M0 = M . We say that two such deformations (Ms, p)s∈Rk and (Nt, p′)t∈Rk are biholomorphically equivalent if there exists a germ of a realanalytic diffeomorphism φ : (Rk, 0) → (Rk, 0) and a holomorphic submersion B : (Cz × Cu, (p, 0)) → (Cn, p′) such that z → B(z, s) is a biholomorphism sending (Ms, p) to (Nφ(s), p′) for all s ∈ Rk close to 0. We shall say that such

Supersymmetric solutions of gauged five-dimensional supergravity with general matter couplings

We perform the characterization program for the supersymmetric configurations and solutions of the supergravity theory coupled to an arbitrary number of vectors, tensors and hypermultiplets and with general non-Abelian gaugings. By using the conditions yielded by the characterization program, new exact supersymmetric solutions are found in the SO(4, 1)/SO(4) model for the hyperscalars and with SU(2) × U(1) as the gauge group. The solutions contain also non-trivial vector and massive tensor fields, the latter being charged under the U(1) sector of the gauge group and with self-dual spatial components. These solutions are black holes with AdS2 × S3 near horizon geometry in the gauged version of the theory and for the ungauged case we found naked singularities. We also analyze supersymmetric solutions with only the scalars ϕx of the vector/tensor multiplets and the metric as the non-trivial fields. We find that only in the null class the scalars ϕx can be non-constant and for the case of constant ϕx we refine the classification in terms of the contributions to the scalar potential.

Blowing up Kähler manifolds with constant scalar curvature, II

In this paper we prove the existence of Kahler metrics of constant scalar curvature on the blow up at finitely many points of a compact manifold that already carries a constant scalar curvature Kahler metric. In the case where the manifold has nontrivial holomorphic vector fields with zeros, we give necessary conditions on the number and locations of the blow up points for the blow up to carry constant scalar curvature Kahler metrics.

Hyperbolic conservation laws on the sphere. A geometry-compatible finite volume scheme

We consider entropy solutions to the initial value problem associated with scalar nonlinear hyperbolic conservation laws posed on the two-dimensional sphere. We propose a finite volume scheme which relies on a web-like mesh made of segments of longitude and latitude lines. The structure of the mesh allows for a discrete version of a natural geometric compatibility condition, which arose earlier in the well-posedness theory established by Ben-Artzi and LeFloch. We study here several classes of flux vectors which define the conservation law under consideration. They are based on prescribing a suitable vector field in the Euclidean three-dimensional space and then suitably projecting it on the sphere’s tangent plane; even when the flux vector in the ambient space is constant, the corresponding flux vector is a non-trivial vector field on the sphere. In particular, we construct here “equatorial periodic solutions”, analogous to one-dimensional periodic solutions to one-dimensional conservation laws, as well as a wide variety of stationary (steady state) solutions. We also construct “confined solutions”, which are time-dependent solutions supported in an arbitrarily specified subdomain of the sphere. Finally, representative numerical examples and test cases are presented.

Aharonov–Bohm effect with many vortices

The Aharonov–Bohm (A–B) effect is the prime example of a zero-field-strength configuration where a nontrivial vector potential acquires physical significance, a typical quantum mechanical effect. We consider an extension of the traditional A–B problem, by studying a two-dimensional medium filled with many point-like vortices. Systems like this might be present within a type II superconducting layer in the presence of a strong magnetic field perpendicular to the layer, and have been studied in different limits. We construct an explicit solution for the wave function of a scalar particle moving within one such layer when the vortices occupy the sites of a square lattice and have all the same strength, equal to half of the flux quantum. From this construction, we infer some general characteristics of the spectrum, including the conclusion that such a flux array produces a repulsive barrier to an incident low-energy charged particle, so that the penetration probability decays exponentially with distance from the edge.

Regular and quasiregular isometric flows on Riemannian manifolds

We study the nontrivial Killing vector fields of constant length and the corresponding flows on smooth Riemannian manifolds. We describe the properties of the set of all points of finite (infinite) period for general isometric flows on Riemannian manifolds. It is shown that this flow is generated by an effective almost free isometric action of the group S1 if there are no points of infinite or zero period. In the last case, the set of periods is at most countable and generates naturally an invariant stratification with closed totally geodesic strata; the union of all regular orbits is an open connected dense subset of full measure.

Vector bundles on contractible smooth schemes

We discuss algebraic vector bundles on smooth k-schemes X contractible from the standpoint of A1-homotopy theory; when k=C, the smooth manifolds X(C) are contractible as topological spaces. The integral algebraic K-theory and integral motivic cohomology of such schemes are those of Speck. One may hope that, furthermore, and in analogy with the classification of topological vector bundles on manifolds, algebraic vector bundles on such schemes are all isomorphic to trivial bundles; this is almost certainly true when the scheme is affine. However, in the nonaffine case, this is false: we show that (essentially) every smooth A1-contractible strictly quasi-affine scheme that admits a U-torsor whose total space is affine, for U a unipotent group, possesses a nontrivial vector bundle. Indeed, we produce explicit arbitrary-dimensional families of nonisomorphic A1-contractible schemes, with each scheme in the family equipped with “as many” (i.e., arbitrary-dimensional moduli of) nonisomorphic vector bundles, of every sufficiently large rank n, as one desires; neither the schemes nor the vector bundles on them are distinguishable by algebraic K-theory. We also discuss the triviality of vector bundles for certain smooth complex affine varieties whose underlying complex manifolds are contractible but that are not necessarily A1-contractible

Killing vector fields of constant length on Riemannian manifolds

We study the nontrivial Killing vector fields of constant length and the corresponding flows on complete smooth Riemannian manifolds. Various examples are constructed of the Killing vector fields of constant length generated by the isometric effective almost free but not free actions of S1 on the Riemannian manifolds close in some sense to symmetric spaces. The latter manifolds include “almost round” odd-dimensional spheres and unit vector bundles over Riemannian manifolds. We obtain some curvature constraints on the Riemannian manifolds admitting nontrivial Killing fields of constant length.

Toric vector bundles, branched covers of fans, and the resolution property

We associate to each toric vector bundle on a toric variety X ( Δ ) X(\Delta ) a “branched cover” of the fan Δ \Delta together with a piecewise-linear function on the branched cover. This construction generalizes the usual correspondence between toric Cartier divisors and piecewise-linear functions. We apply this combinatorial geometric technique to investigate the existence of resolutions of coherent sheaves by vector bundles, using singular nonquasiprojective toric threefolds as a testing ground. Our main new result is the construction of complete toric threefolds that have no nontrivial toric vector bundles of rank less than or equal to three. The combinatorial geometric sections of the paper, which develop a theory of cone complexes and their branched covers, can be read independently.

KILLING VECTOR FIELDS OF CONSTANT LENGTH ON LOCALLY SYMMETRIC RIEMANNIAN MANIFOLDS

In this paper nontrivial Killing vector fields of constant length and the corresponding ows on smooth complete Riemannian manifolds are investigated. It is proved that such a ow on symmetric space is free or induced by a free isometric action of the circle S1. Examples of unit Killing vector fields generated by almost free but not free actions of S1 on locally symmetric Riemannian spaces are found; among them are homogeneous (nonsimply connected) Riemannian manifolds of constant positive sectional curvature and locally Euclidean spaces. Some unsolved questions are formulated.

Gauss-Arnoldi quadrature for and rational Padé-type approximation for Markov-type functions

The efficiency of Gauss-Arnoldi quadrature for the calculation of the quantity is studied, where is a bounded operator in a Hilbert space and is a non-trivial vector in this space. A necessary and a sufficient conditions are found for the efficiency of the quadrature in the case of a normal operator. An example of a non-normal operator for which this quadrature is inefficient is presented.It is shown that Gauss-Arnoldi quadrature is related in certain cases to rational Padé-type approximation (with the poles at Ritz numbers) for functions of Markov type and, in particular, can be used for the localization of the poles of a rational perturbation. Error estimates are found, which can also be used when classical Padé approximation does not work or it may not be efficient.Theoretical results and conjectures are illustrated by numerical experiments.Bibliography: 44 titles.

Quantum and classical spins on the spatially distorted kagomé lattice: Applications to volborthiteCu3V2O7(OH)2∙2H2O

In volborthite, spin-$1∕2$ moments form a distorted kagom\'e lattice of corner sharing isosceles triangles with exchange constants $J$ on two bonds and ${J}^{\ensuremath{'}}$ on the third bond. We study the properties of such spin systems and show that despite the distortion, the lattice retains a great deal of frustration. Although subextensive, the classical ground state degeneracy remains very large, growing exponentially with the system perimeter. We consider degeneracy lifting by thermal and quantum fluctuations. To linear (spin wave) order, the degeneracy is found to stay intact. Two complementary approaches are therefore introduced, appropriate to low and high temperatures, which point to the same ordered pattern for ${J}^{\ensuremath{'}}gJ$. In the low-temperature limit, an effective chirality Hamiltonian is derived from nonlinear spin waves, which predicts a transition on increasing ${J}^{\ensuremath{'}}∕J$ from $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$-type order to a ferrimagnetic chirality stripe order with a doubled unit cell. This is confirmed by a large-$n$ approximation on the $\mathrm{O}(n)$ model on this lattice. While the saddle point solution produces a line degeneracy, $\mathrm{O}(1∕n)$ corrections select the nontrivial wave vector of the striped chirality state. The quantum limit of spin $1∕2$ on this lattice is studied via exact small system diagonalization and compares well with experimental results at intermediate temperatures. We suggest that the very-low-temperature spin frozen state seen in NMR experiments may be related to the disconnected nature of classical ground states on this lattice, which leads to a prediction for NMR line shapes.

Uniruled surfaces of general type

We give a systematic construction of uniruled surfaces in positive characteristic. Using this construction, we find surfaces of general type with non-trivial global vector fields, surfaces with arbitrarily non-reduced Picard schemes as well as surfaces with inseparable canonical maps. In particular, we show that some previously known pathologies are not sporadic but exist in abundance.

Physical vacuum properties and internal space dimension

The paper addresses matrix spaces, whose properties and dynamics are determined by Dirac matrices in Riemannian spaces of different dimension and signature. Among all Dirac matrix systems there are such ones, which nontrivial scalar, vector or other tensors cannot be made up from. These Dirac matrix systems are associated with the vacuum state of the matrix space. The simplest vacuum system realization can be ensured using the orthonormal basis in the internal matrix space. This vacuum system realization is not however unique. The case of 7-dimensional Riemannian space of signature 7(-) is considered in detail. In this case two basically different vacuum system realizations are possible: (1) with using the orthonormal basis; (2) with using the oblique-angled basis, whose base vectors coincide with the simple roots of algebra E_{8}. Considerations are presented, from which it follows that the least-dimension space bearing on physics is the Riemannian 11-dimensional space of signature 1(-)& 10(+). The considerations consist in the condition of maximum vacuum energy density and vacuum fluctuation energy density.

Vector bundles of low geometric dimension over real projective spaces

We construct stably nontrivial vector bundles of dimension 5 and also stably nontrivial vector bundles of dimension 6 on real projective spaces of arbitrarily high dimensions. Consequently, we answer a problem posed by J. F. Adams.

Non-Abelian conversion and quantization of nonscalar second-class constraints

We propose a general method for deformation quantization of any second-class constrained system on a symplectic manifold. The constraints determining an arbitrary constraint surface are in general defined only locally and can be components of a section of a non-trivial vector bundle over the phase-space manifold. The covariance of the construction with respect to the change of the constraint basis is provided by introducing a connection in the ``constraint bundle'', which becomes a key ingredient of the conversion procedure for the non-scalar constraints. Unlike in the case of scalar second-class constraints, no Abelian conversion is possible in general. Within the BRST framework, a systematic procedure is worked out for converting non-scalar second-class constraints into non-Abelian first-class ones. The BRST-extended system is quantized, yielding an explicitly covariant quantization of the original system. An important feature of second-class systems with non-scalar constraints is that the appropriately generalized Dirac bracket satisfies the Jacobi identity only on the constraint surface. At the quantum level, this results in a weakly associative star-product on the phase space.

Continuity and Asymptotic Behavior of Integral Kernels Related to Schrodinger Operators on Manifolds

The continuity of integral kernels related to Schrodinger operators (the kernel of the heat equa- tion, Green's function) plays an important role in the study of different properties of quantum systems. In the Euclidean case, it was shown in (1) that sufficiently general functions of the Schrodinger operator have continuous integral kernels for scalar potentials belonging to the Kato class. In (2), this result was generalized to operators with nontrivial vector potential of the mag- netic field; in this case, arbitrary domains in the Euclidean space were admitted as configuration spaces. Simultaneously, in view of several problems in mesoscopy and gravitation quantization (3, 4), it seems to be of interest to study the integral kernels related to the Schrodinger operators on Riemann manifolds. In this paper, we presents several results concerning the existence and continuity of integral kernels for different operators generated by the Schrodinger operator on a manifold. In addition, we give several estimates for Green's function for the cases in which its arguments are far from one another or, conversely, close to one another. Let X be a ν-dimensional manifold of bounded geometry. By d(x, y) we denote the geodesic distance between points x, y ∈ X ;b yB(x, r) we denote an open ball of radius r centered at x ; and by D we denote the set {(x, y) ∈ X × X, x = y }⊂ X × X. We consider a 1-form A on X with smooth coefficients; this form determines the connection in a trivial linear bundle over X ;b y ∆ A we denote the corresponding Bochner Laplacian (for A = 0 , we obtain ∆A =∆ , which is the Beltrami-Laplace operator). The method used to prove our results requires several (rather weak) restrictions on the scalar potentials under study; in what follows, we assume that this potential belongs to the class P(X) of real functions U on X with the properties U+ := max( U, 0) ∈ L p 0 loc (X )a ndU− := max(− U, 0) ∈ n i=1 L p i (X),