Showing papers in "Annals of Global Analysis and Geometry in 1996"
TL;DR: In this article, a family of hypercomplex automorphisms for the Stiefel manifold of complex 2-planes in ℂn for all n > 2 is described.
Abstract: This paper describes a family of hypercomplex structures {% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFqessaaa!4076!\[\mathcal{I}\]a(p)}a=1,2,3 depending on n real non-zero parameters p = (p1,...,pn) on the Stiefel manifold of complex 2-planes in ℂn for all n > 2. Generally, these hypercomplex structures are inhomogenous with the exception of the case when all the pi's are equal. We also determine the Lie algebra of infinitesimal hypercomplex automorphisms for each structure. Furthermore, we solve the equivalence problem for the hypercomplex structures in the case that the components of p are pairwise commensurable. Finally, some of these examples admit discrete hypercomplex quotients whose topology we also analyze.
24 citations
TL;DR: In this paper, a generalized version of the regular interval theorem of Morse theory is proven using techniques from the theory of Riemannian submersions and conformal deformations.
Abstract: A generalized version of the regular interval theorem of Morse theory is proven using techniques from the theory of Riemannian submersions and conformal deformations. This approach provides an interesting link between Riemannian submersions (for real valued functions) and Morse theory.
22 citations
TL;DR: In this article, the transverse Calabi invariants of the contact structure E(α), the contact flow Fα and the transversal symplectic geometry of a contact manifold (M, α) were introduced.
Abstract: We exhibit new invariants of the contact structure E(α), the contact flow Fα and the transverse symplectic geometry of a contact manifold (M, α). The invariant of contact structures generalizes to transversally oriented foliations. We focus on the particular cases of orientations of smooth manifolds and transverse orientations of foliations. We define the transverse Calabi invariants and determine their kernels.
8 citations
TL;DR: In this paper, it was shown that the degree of the algebraic leaves for singular meromorphic foliations on ℂℙ2 can be extended to large classes of foliations and complex projective surfaces.
Abstract: Here we show that many numerical computations and bounds on the degrees of the algebraic leaves for singular meromorphic foliations on ℂℙ2 may be extended to large classes of foliations and complex projective surfaces.
5 citations
TL;DR: In this article, it was shown that the only symmetric Lie groups acting in Euclidean spaces by isometries are extrinsically symmetric iff they are parallel, i.e. satisfy MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLw
Abstract: The orbits of Lie groups acting in Euclidean spaces by isometries are extrinsically symmetric iff they are parallel, i.e. satisfy % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSijHikaaa!3764!\[\mathbb{Z}\]h = 0. Submanifolds characterized by the integrability condition \-R · h = 0 of this system % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSijHikaaa!3764!\[\mathbb{Z}\]h = 0 are called semi-parallel; they are the second order envelopes of the symmetric orbits. Let the orbit set of an action of SO(n, R) in E% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% aIXaaabaGaaGOmaaaaaaa!3773!\[\frac{1}{2}\]n(n−1) contain a Plucker submanifold. It is proved that 1) the only symmetric orbits are Plucker orbits and for n = 2ν > 4 the unitary orbits, 2) each of their second order envelopes is trivial, i.e. is a single orbit or its open part.
3 citations
TL;DR: In this article, the smooth bounded manifold with cylindrical ends obtained by blowing up near the singularities is defined, where B is a manifold with conical singularities, and denotes by Ω(n, n) the smooth, bounded manifold obtained by the same process.
Abstract: Let B be a manifold with conical singularities, and denote by ℬ the smooth bounded manifold with cylindrical ends obtained by blowing up near the singularities.
3 citations
TL;DR: In this article, a family of non-nilpotent Riemannian solvable Lie groups whose isotropy group has a prescribed compact Lie algebra is described, and a Lie algebra has been defined for them.
Abstract: We describe a family of non-nilpotent Riemannian solvable Lie groups whose isotropy group has a prescribed compact Lie algebra
3 citations
TL;DR: In this paper, the authors considered the isospectral deformations on nilmanifolds from the dynamical point of view and showed that the associated Hamiltonian systems of geodesic flows are decomposed into a collection of reduced systems which are left invariant under the deformations.
Abstract: A one-parameter familyg(t) (0 ≤t ≤T) of Riemannian metric on a compact manifold is called an isospectral deformation of a metricg(0) if the Laplace-Beltrami operators associated to the metricsg(t) have the same spectra. Examples of non-trivial isospectral deformations were constructed on solvmanifolds for the first time by C.S. Gordon and E. Wilson on the basis of Kirillov theory. This paper considers the isospectral deformations on nilmanifolds from the dynamical point of view. First, we see for certain isospectral deformations that the associated Hamiltonian systems of geodesic flows are decomposed into a collection of reduced systems which are left invariant as Hamiltonian systems under the deformations. This fact is formulated by the “classical Lax equations”. Next, by using a quantization procedure, we attempt to obtain Lax equations for the “reduced Laplacians” from the “classical Lax equations”. As a result, we show that certain isospectral deformations by Gordon-Wilson are represented by the Lax equations.
1 citations
TL;DR: In this article, the problem of computing the number of meromorphic functions with prescribed poles on Rieiemann surfaces was formulated in terms of the cohomology of a blown-up twistor space, and then calculated the holomorphic Euler characteristic of this twistedor space.
Abstract: In Twistor Conformal Field Theory the Riemann surfaces and holomorphic functions of two-dimensional conformal field theory are replaced by “flat” twistor spaces (arising from conformally-flat four-manifolds) and elements of the holomorphic first cohomology. The analogue of a Laurent Series is the expansion of a cohomology element in “elementary states” and we calculate the dimension of the space of these states for twistor spaces of compact hyperbolic manifolds. Our method follows the strategy used in the classical problem of calculating the number of meromorphic functions with prescribed poles on Rieiemann surface. We express the problem globally (in terms of the cohomology of a blown-up twistor space), calculate the holomorphic Euler characteristic of this blown-up space, and then use some vanishing theorems to isolate the first cohomology term.
1 citations
TL;DR: In this paper, the integral of the (n − 2)-th mean curvature of the generalized convex set, the mixed volume of the convex hull of the set, and a reference set are involved in the inequality.
Abstract: We develop some geometric inequality for a kind of generalized convex set. The integral of (n − 2)-th mean curvature of the generalized convex set, the mixed volume of the convex hull of the set, and a reference convex set are involved in the inequality.
TL;DR: In this paper, the curvature of invariant metrics on the generalization of the classical homogeneous domain of Pyatetskii-Shapiro was studied and a continuous family of nonsymmetric homogeneous Kahler metrics with non-positive curvature was obtained.
Abstract: We study the curvature of invariant metrics on the generalization of the classical homogeneous domain of Pyatetskii-Shapiro, as given by D'Atri in [3]. We obtain all invariant Kahler metrics of either, nonpositive sectional curvature or nonpositive holomorphic sectional curvature, and determine the corresponding connected groups of isometries in each case. This yields a continuous family of nonsymmetric homogeneous Kahler metrics with nonpositive curvature.
TL;DR: In this paper, it was shown that on open manifolds of bounded geometry satisfying a certain spectral condition, the identity component of the identity of all bounded Sobolev diffeomorphisms is a submanifold of all the other components.
Abstract: We prove that on open manifolds of bounded geometry satisfying a certain spectral condition the component of the identity D
infw,0
supr
of form preserving diffeomorphisms is a submanifold of the identity component of all bounded Sobolev diffeomorphisms. D
infw,0
supr
inherits a natural Riemannian geometry and we can solve Euler equations in this context.
TL;DR: The generalized equation and the intrinsic generalized equation are considered in this paper and the solutions of the first one correspond to Riemannian submanifolds Mn(K) of constant sectional curvature of psedo-Riemannians manifolds.
Abstract: The generalized equation and the intrinsic generalized equation are considered. The solutions of the first one are shown to correspond to Riemannian submanifolds Mn(K) of constant sectional curvature of psedo-Riemannian manifolds % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaamaaDaaaleaaieGacaWFZbaabaGaaGOmaiaad6gacqGHsisl% caaIXaaaaOGaaiikamaanaaabaGaam4saaaacaGGPaaaaa!3D97!\[\overline M _s^{2n - 1} (\overline K )\] of index s, with % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabgc% Mi5oaanaaabaGaam4saaaaaaa!3965!\[K
e \overline K \], flat normal bundle and such that the normal principal curvatures are different from % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabgk% HiTmaanaaabaGaam4saaaaaaa!388B!\[K - \overline K \]. The solutions of the intrinsic generalized equation correspond to Riemannian metrics defined on open subsets of Rn which have constant sectional curvature. The relation between solutions of those equations is given. Moreover, it is proven that the submanifolds M under consideration are determined, up to a rigid motion of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaaaaa!36D0!\[\overline M \], by their first fundamental forms, as solutions of the intrinsic generalized equation. The geometric properties of the submanifolds M associated to the solutions of the intrinsic generalized equation, which are invariant under an (n − 1)-dimensional group of translations, are given. Among other results, it is shown that such submanifolds are foliated by (n − 1)-dimensional flat submanifolds which have constant mean curvature in M. Moreover, each leaf of the foliation is itself foliated by curves of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaaaaa!36D0!\[\overline M \] which have constant curvatures.
TL;DR: In this paper, the authors studied negatively curved Riemannian manifolds acting on a Lie group of isometries with principal orbits of codimension one and proved that the orbit space of such a manifold M is always homeomorphic to ℝ or Ω+ and this second case may occur only when either the singular orbit is a geodesic of M or when the space is simply connected.
Abstract: This paper is aimed at studying negatively curved Riemannian manifolds acted on by a Lie group of isometries with principal orbits of codimension one. The orbit space of such a manifold M is proved to be always homeomorphic to ℝ or ℝ+ and this second case may occur only when either the singular orbit is a geodesic of M or when the space is simply connected. Several corollaries are given.
TL;DR: In this article, it was shown that multiplicity-free Hamiltonian group actions whose moment maps are transversal to a Cartan subalgebra are in one-to-one correspondence with a certain collection of convex polytopes.
Abstract: Multiplicity-free Hamiltonian group actions are the symplectic analogs of multiplicity-free representations, that is, representations in which each irreducible appears at most once. The most well-known examples are toric varieties. The purpose of this paper is to show that under certain assumptions multiplicity-free actions whose moment maps are transversal to a Cartan subalgebra are in one-to-one correspondence with a certain collection of convex polytopes. This result generalizes a theorem of Delzant concerning torus actions.
TL;DR: In this paper, it was shown that as long as the values of the numerator remain close together relative to the distances between poles, the above conjecture remains true and ceases to be so otherwise.
Abstract: We prove a recent conjecture of Manton and Murray: if a polynomialp(z) of degreek — 1 is given, then anSU (2) monopole corresponding to a rational functionp(z)/q(z) with well-separated poles \1,...,\k is approximately made up from charge 1 monopoles located at points (1/2 In p(\i), \i). We show how the rate of approximation changes with the numeratorp(z) with the result that, as long as the values of the numerator remain close together relative to the distances between poles, the above statement remains true and ceases to be so otherwise.
TL;DR: In this paper, general rigidity theorems for both elliptic and higher elliptic genera under a natural condition on the first equivariant Pontrjagin classes were proved.
Abstract: We prove some general rigidity theorems for both elliptic and higher elliptic genera under a natural condition on the first equivariant Pontrjagin classes. We also obtain the vanishing of some higher elliptic genera.
TL;DR: In this article, splitting theorems for stable causal spacetimes admitting certain metric related reference frames are obtained in connection to timelike geodesic incompletenes.
Abstract: Splitting theorems for stable causal spacetimes admitting certain metric related reference frames are obtained in connection to timelike geodesic incompletenes.
TL;DR: The information metric arises in statistics as a natural inner product on a space of probability distributions and is potentially degenerate in the sense that it is positive semi-definite as discussed by the authors.
Abstract: The information metric arises in statistics as a natural inner product on a space of probability distributions. In general this inner product is positive semi-definite but is potentially degenerate.
TL;DR: In this paper, a review on normally hyperbolic operators of Huygens type is given, and the methods to determine Huygen operators were essentially influenced and developed by Paul Gunther.
Abstract: In this paper we give a review on normally hyperbolic operators of Huygens type The methods to determine Huygens operators we explain here were essentially influenced and developed by Paul Gunther
TL;DR: In this paper, the hyperkahler metrics associated to minimal singularities in the nilpotent variety of a semisimple Lie group were studied and the 4-dimensional ALE spaces were naturally realized within the context of coadjoint orbits and can be thought of as certain moduli spaces of SU(2) invariants instantons on ℝ4∖{O} with appropriate boundary conditions.
Abstract: We study the hyperkahler metrics associated to minimal singularities in the nilpotent variety of a semisimple Lie group. We show that Kronheimer's 4-dimensional ALE spaces are naturally realized within the context of coadjoint orbits and can be thought of as certain moduli spaces ofSU(2) invariants instantons on ℝ4∖{O} with appropriate boundary conditions.