scispace - formally typeset
Search or ask a question

Showing papers on "Distribution (differential geometry) published in 2003"


Journal ArticleDOI
TL;DR: In this paper, the authors considered the problem of finding the holomorphic tangent space of a real-analytic submanifold of a complex vector space of finite dimension.
Abstract: Let E be a complex vector space of finite dimension and let K ⊂ GL(E) be a compact connected subgroup. Then for fixed a ∈ E the orbit K := K(a) is a real-analytic submanifold of E that inherits various structures from E. For instance, choosing a K-invariant positive definite inner product (x|y) on E makes K a Riemannian manifold on which K acts transitively by isometries. On the other hand, K inherits from E a Cauchy–Riemann structure (CR-structure), that is given by the distribution of the maximal complex subspaces Hx K := Tx K ∩ iTx K of the real tangent spaces Tx K ⊂ E, x ∈ K , together with the complex structure on every Hx K (multiplication by i). The subspace Hx K is called the holomorphic tangent space to K at x (see [9] and [4] as general references for CR-manifolds). Of interest for the geometry of the orbit K = K(a) with respect to its CR-structure is the study of the CR-functions (or more generally CR-mappings) on K , i.e. of smooth functions f : K → C that satisfy the tangential Cauchy–Riemann differential equations in the sense that the restriction of the differential df to every holomorphic tangent space is complex linear. For instance, all holomorphic functions defined in an open neighbourhood of K ⊂ E give by restriction real-analytic CR-functions on K . Actually, we deal with the more general continuous CR-functions on K (which satisfy by definition the tangential CR-equations in the distribution sense, or equivalently, which are locally uniform limits of sequences of smooth CR-functions due to the approximation theorem of BaouendiTreves [6]). In this context it is of interest to determine the space of all

34 citations


Journal ArticleDOI
TL;DR: In this paper, the geometry of the space of isometry classes of Riemannian metrics with a 2-sided curvature bound on a fixed compact smooth manifold of dimension at least five is studied.
Abstract: This paper is devoted to large scale aspects of the geometry of the space of isometry classes of Riemannian metrics, with a 2-sided curvature bound, on a fixed compact smooth manifold of dimension at least five. Using a mix of tools from logic/computer science, and differential geometry and topology, we study the diameter functional and its critical points, as well as their distribution (density) within the space and the structure of their neighborhoods.

34 citations


11 Jun 2003
TL;DR: In this article, the authors considered the Schrodinger type dierential expression HV = r r + V, where r is a C 1 -bounded Hermitian connection on a vector bundle E of bounded geometry over a manifold with metric g and positive C 1-bounded measure dµ, and V = V1 + V2, where 0 V1 2 L 1 (EndE) and 0 V2 2 L 2 (End E) are linear self-adjoint bundle endomorphisms.
Abstract: We consider the Schrodinger type dierential expression HV = r r + V, where r is a C 1 -bounded Hermitian connection on a Hermitian vector bundle E of bounded geometry over a manifold of bounded geometry (M,g) with metric g and positive C 1 -bounded measure dµ, and V = V1 + V2, where 0 V1 2 L 1 (EndE) and 0 V2 2 L 1 (EndE) are linear self-adjoint bundle endomorphisms. We give a sucient condition for self-adjointness of the operator S in L 2 (E) defined by Su = HV u for all u 2 Dom(S) = {u 2 W 1,2 (E): R hV1u,uidµ 0 is a constant and 0 is a positive distribution on M.

16 citations


Journal ArticleDOI
Hans Triebel1
TL;DR: In this article, the authors deal with wavelet frames for a large class of distributions on euclidean n-space, including all compactly supported distributions, and characterize the global, local, and pointwise regularity of the distribution considered.
Abstract: This paper deals with wavelet frames for a large class of distributions on euclidean n-space, including all compactly supported distributions. These representations characterize the global, local, and pointwise regularity of the distribution considered.

16 citations


Journal ArticleDOI
TL;DR: In this paper, the Witten current defined in Ref. 13 is shown to be an analogue of Hida distribution, Wiener chaos being replaced by Chen iterated integrals.
Abstract: On p. 1096 of Ref. 31, we remark that the Witten current defined in Ref. 13 is an analogue of Hida distribution, Wiener chaos being replaced by Chen iterated integrals. We clarify this remark in the case of a Feynman path integral on a manifold.

12 citations


01 Jan 2003
TL;DR: In this paper, the authors generalize the notion of contact manifold by allowing the contact distribution to have codimension two and show that the complex structure on a three-dimensional complex contact manifold is determined solely by the underlying contact distribution.
Abstract: We generalise the notion of contact manifold by allowing the contact distribution to have codimension two. There are special features in dimension six. In particular, we show that the complex structure on a three-dimensional complex contact manifold is determined solely by the underlying contact distribution.

7 citations


Posted Content
TL;DR: In this article, it was shown that the social equilibrium set of an exchange economy, with consumption space as a subset of a Banach space, is a manifold, and that the set of social weights of equilibrium associated with a given distribution of the initial endowments is finite.
Abstract: In this paper we prove that the social equilibrium set, of an exchange economy, with consumption space as a subset of a Banach space is a Banach manifold, and this characterization does not depend on the number of commodities. In the way to obtain this characterization we will show that the set of social weights of equilibrium, associated with a given distribution of the initial endowments, is finite.

7 citations


Journal ArticleDOI
TL;DR: In this article, the structural group of a (4m + 3)-dimensional manifold equipped with an almost contact 3-structure is reduced to Sp(m) £ I3.
Abstract: Hypersurfaces of a Riemannian manifold equipped with a hypercosymplectic 3-structure are studied. Integrability condi- tions for certain distributions on the hypersurface are investigated. Geometry of leaves of certain distribution are also studied. The quaternionic analog of almost complex structures is the almost quaternion (hypercomplex) which is defined by three local (global) al- most complex structures which satisfy the quaternionic relations as the imaginary quaternions satisfy ((3)). Quaternion Kahler manifolds and hyper-Kahler manifolds are special and interesting cases of Riemannian manifolds with almost quaternion and almost hypercomplex structure, respectively. Quaternion Kahler manifolds are Einstein, while hyper- Kahler manifolds are Ricci flat. An almost contact 3-structure was defined by Kuo ((5)) and it is closely related to both almost quaternion and almost hypercomplex structures. Hypersurfaces of manifolds with almost hypercomplex struc- ture inherit naturally three almost contact structures which constitute an almost contact 3-structure. An almost contact metric 3-structure manifold is always (4m + 3)-dimensional. The structural group of the tangent bundle of a (4m + 3)-dimensional manifold equipped with an almost contact 3-structure is reducible to Sp(m) £ I3. In particular, if each almost contact metric structure of an almost contact metric 3-structure is Sasakian, then this structure is called a Sasakian 3-structure. Riemannian manifolds with Sasakian 3-structure

6 citations


Proceedings ArticleDOI
TL;DR: In this article, the Nijenhuis tensor characteristic distributions on a non-integrable four-dimensional almost complex manifold are investigated for integrability, singularities and equivalence.
Abstract: In this paper the Nijenhuis tensor characteristic distributions on a non-integrable four-dimensional almost complex manifold is investigated for integrability, singularities and equivalence.

5 citations


Patent
26 Aug 2003
TL;DR: Auxiliary end heating devices on an elongated, heated hot-melt distribution manifold body assist in maintaining uniform temperatures throughout all portions and passageways of the manifold body.
Abstract: Auxiliary end heating devices on an elongated, heated hot melt distribution manifold body assist in maintaining uniform temperatures throughout all portions and passageways of the manifold body. The heating devices preferably take the form of thick film electrical resistive heaters of plate-like construction. The end heaters are on their own control circuit separate and apart from the circuit for other heaters for remaining portions of the manifold body. Special isolating slots are formed in lower corners of the manifold body between supporting standoffs and overhead melt passageways in the manifold body.

3 citations


Journal ArticleDOI
TL;DR: In this paper, Liu et al. developed a Morse Theory for the sub-Riemannian action functional on the space of horizontal curves, i.e., everywhere tangent to the distribution.
Abstract: We consider a Riemannian manifold $(\mathcal M,g)$ and a codimension one distribution $\Delta\subset T\mathcal M$ on $\mathcal M$ which is the orthogonal of a unit vector field $Y$ on $\mathcal M$. We do not make any nonintegrability assumption on $\Delta$. The aim of the paper is to develop a Morse Theory for the sub-Riemannian action functional $E$ on the space of horizontal curves, i.e. everywhere tangent to the distribution $\Delta$. We consider the case of horizontal curves joining a smooth submanifold $\mathcal P$ of $\mathcal M$ and a fixed point $q\in\mathcal M$. Under the assumption that $\mathcal P$ is transversal to $\Delta$, it is known (see [P. Piccione and D. V. Tausk, Variational aspects of the geodesic problem is sub-Riemannian geometry , J. Geom. Phys. 39 (2001), 183–206]) that the set of such curves has the structure of an infinite dimensional Hilbert manifold and that the critical points of $E$ are the so called {\it normal extremals} (see [W. Liu and H. J. Sussmann, Shortest paths for sub-Riemannian metrics on rank–$2$ distribution , Mem. Amer. Math. Soc. 564 (1995)]). We compute the second variation of $E$ at its critical points, we define the notions of $\mathcal P$-Jacobi field, of $\mathcal P$-focal point and of exponential map and we prove a Morse Index Theorem. Finally, we prove the Morse relations for the critical points of $E$ under the assumption of completeness for $(\mathcal M,g)$.


Journal ArticleDOI
TL;DR: In this article, the authors studied some links between autoparallel distributions and the factorization of a riemannian manifold and proved a splitting theorem for Lie groups with bi-invariant metrics.
Abstract: We study some links between autoparallel distributions and the factorization of a riemannian manifold. Finally, we prove a splitting theorem for Lie groups with biinvariant metrics.

Posted Content
TL;DR: In this paper, the authors determine the asymptotics of the joint eigenfunctions of the torus action on a toric Kahler variety and show that they behave like Gaussians centered at the corresponding classical torus, and that there is a universal Gaussian scaling limit of the distribution function near its center.
Abstract: We determine the asymptotics of the joint eigenfunctions of the torus action on a toric Kahler variety. Such varieties are models of completely integrable systems in complex geometry. We first determine the pointwise asymptotics of the eigenfunctions, which show that they behave like Gaussians centered at the corresponding classical torus. We then show that there is a universal Gaussian scaling limit of the distribution function near its center. We also determine the limit distribution for the tails of the eigenfunctions on large length scales. These are not universal but depend on the global geometry of the toric variety and in particular on the details of the exponential decay of the eigenfunctions away from the classically allowed set.