Homogeneous Symmetric Spaces Research Articles

Special Vinberg cones and the entropy of BPS extremal black holes

We consider the static, spherically symmetric and asymptotically flat BPS extremal black holes in ungauged N = 2 D = 4 supergravity theories, in which the scalar manifold of the vector multiplets is homogeneous. By a result of Shmakova on the BPS attractor equations, the entropy of this kind of black holes can be expressed only in terms of their electric and magnetic charges, provided that the inverse of a certain quadratic map (uniquely determined by the prepotential of the theory) is given. This inverse was previously known just for the cases in which the scalar manifold of the theory is a homogeneous symmetric space. In this paper we use Vinberg’s theory of homogeneous cones to determine an explicit expression for such an inverse, under the assumption that the scalar manifold is homogeneous, but not necessarily symmetric. As immediate consequence, we get a formula for the entropy of BPS black holes that holds in any model of N = 2 supergravity with homogeneous scalar manifold.

Some remarks about disjointly homogeneous symmetric spaces

Let $1\le p<\infty$. A symmetric space $X$ on $[0,1]$ is said to be $p$-disjointly homogeneous (resp. restricted $p$-disjointly homogeneous) if every sequence of normalized pairwise disjoint functions from $X$ (resp. characteristic functions) contains a subsequence equivalent in $X$ to the unit vector basis of $l_p$. Answering a question posed recently, we construct, for each $1\le p<\infty$, a restricted $p$-disjointly homogeneous symmetric space, which is not $p$-disjointly homogeneous. Moreover, we prove that the property of $p$-disjoint homogeneity is preserved under Banach isomorphisms.

Aspects of Differential Geometry II

Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. Book II deals with more advanced material than Book I and is aimed at the graduate level. Chapter 4 deals with additional topics in Riemannian geometry. Properties of real analytic curves given by a single ODE and of surfaces given by a pair of ODEs are studied, and the volume of geodesic balls is treated. An introduction to both holomorphic and Kahler geometry is given. In Chapter 5, the basic properties of de Rham cohomology are discussed, the Hodge Decomposition Theorem, Poincare duality, and the Kunneth formula are proved, and a brief introduction to the theory of characteristic classes is given. In Chapter 6, Lie groups and Lie algebras are dealt with. The exponential map, the classical groups, and geodesics in the context of a bi-invariant metric are discussed. The de Rham cohomology of compact Lie groups and the Peter--Weyl Theorem are treated. In Chapter 7, material concerning homogeneous spaces and symmetric spaces is presented. Book II concludes in Chapter 8 where the relationship between simplicial cohomology, singular cohomology, sheaf cohomology, and de Rham cohomology is established. We have given some different proofs than those that are classically given and there is some new material in these volumes. For example, the treatment of the total curvature and length of curves given by a single ODE is new as is the discussion of the total Gaussian curvature of a surface defined by a pair of ODEs.

CHARGE ORBITS OF SYMMETRIC SPECIAL GEOMETRIES AND ATTRACTORS

We study the critical points of the black hole scalar potential V BH in N = 2, d = 4 supergravity coupled to nV vector multiplets, in an asymptotically flat extremal black hole background described by a 2(nV+1)-dimensional dyonic charge vector and (complex) scalar fields which are coordinates of a special Kähler manifold. For the case of homogeneous symmetric spaces, we find three general classes of regular attractor solutions with nonvanishing Bekenstein–Hawking entropy. They correspond to three (inequivalent) classes of orbits of the charge vector, which is in a 2(nV+1)-dimensional representation RV of the U-duality group. Such orbits are nondegenerate, namely they have nonvanishing quartic invariant (for rank-3 spaces). Other than the ½-BPS one, there are two other distinct non-BPS classes of charge orbits, one of which has vanishing central charge. The three species of solutions to the N = 2 extremal black hole attractor equations give rise to different mass spectra of the scalar fluctuations, whose pattern can be inferred by using invariance properties of the critical points of V BH and some group theoretical considerations on homogeneous symmetric special Kähler geometry.

Trigonometry of complex Hermitian -type homogeneous symmetric spaces

This paper contains a thorough study of the trigonometry of the homogeneous symmetric spaces in the Cayley-Klein-Dickson family of spaces of 'complex Hermitian' type and rank-one. The complex Hermitian elliptic CP^N and hyperbolic CH^N spaces, their analogues with indefinite Hermitian metric and some non-compact symmetric spaces associated to SL(N+1,R) are the generic members in this family. The method encapsulates trigonometry for this whole family of spaces into a single basic trigonometric group equation, and has 'universality' and '(self)-duality' as its distinctive traits. All previously known results on the trigonometry of CP^N and CH^N follow as particular cases of our general equations. The physical Quantum Space of States of any quantum system belongs, as the complex Hermitian space member, to this parametrised family; hence its trigonometry appears as a rather particular case of the equations we obtain.

Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry

A new method to obtain trigonometry for the real spaces of constant curvature and metric of any (even degenerate) signature is presented. The method encapsulates trigonometry for all these spaces into a single basic trigonometric group equation. This brings to its logical end the idea of an absolute trigonometry, and provides equations which hold true for the nine two-dimensional spaces of constant curvature and any signature. This family of spaces includes both relativistic and non-relativistic homogeneous spacetimes; therefore a complete discussion of trigonometry in the six de Sitter, minkowskian, Newton--Hooke and galilean spacetimes follow as particular instances of the general approach. Any equation previously known for the three classical riemannian spaces also has a version for the remaining six spacetimes; in most cases these equations are new. Distinctive traits of the method are universality and self-duality: every equation is meaningful for the nine spaces at once, and displays explicitly invariance under a duality transformation relating the nine spaces. The derivation of the single basic trigonometric equation at group level, its translation to a set of equations (cosine, sine and dual cosine laws) and the natural apparition of angular and lateral excesses, area and coarea are explicitly discussed in detail. The exposition also aims to introduce the main ideas of this direct group theoretical way to trigonometry, and may well provide a path to systematically study trigonometry for any homogeneous symmetric space.

K-Loops, gyrogroups and symmetric spaces

In this paper we present a new class of K-Loops consisting of homogeneous symmetric spaces of the noncompact type. The most famous physical example of a representative of this class is found in special relativity: The set of all relativistically admissible velocities, R c 3:= {υ ∈ R3: ¦v¦< c} (c the speed of light), together with the relativistic velocity composition law forms a K-Loop, as was already shown by A. Ungar in 1988. The set of the corresponding boosts turns out to be the effective homogeneous symmetric space SO (1,3)0/SO(3), where SO(1,3)0 denotes the proper orthochronous Lorentz group and SO(3) the subgroup of proper rotations in three dimensions, which is also found to be the set of fixed points of an automorphism σ: SO(1,3)0 → SO(1,3)0, whose action on a matrix A in the defining representation is given by σ(A) = (AT)−1. Moreover, the triple (so(l,3),so(3),s) consisting of the Lie algebras of SO(1,3) and SO(3) and the induced automorphism s = (dσ)e is an orthogonal symmetric Lie algebra of the noncompact type. This property of the Lie algebra makes SO(l, 3)/SO(3) an (effective) homogeneous symmetric space of the noncompact type. The method of exact decomposition allows endowing each homogeneous symmetric space of the noncompact type with the structure of a K-Loop in a similar manner. Finally, we treat some alternative approaches which lead to K-Loops that are isomorphic to the one described above.

The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces

We consider the dynamic interpolation problem for nonlinear control systems modeled by second-order differential equations whose configuration space is a Riemannian manifoldM. In this problem we are given an ordered set of points inM and would like to generate a trajectory of the system through the application of suitable control functions, so that the resulting trajectory in configuration space interpolates the given set of points. We also impose smoothness constraints on the trajectory and typically ask that the trajectory be also optimal with respect to some physically interesting cost function. Here we are interested in the situation where the trajectory is twice continuously differentiable and the Lagrangian in the optimization problem is given by the norm squared acceleration along the trajectory. The special cases whereM is a connected and compact Lie group or a homogeneous symmetric space are studied in more detail.

EIGENFUNCTIONS OF THE BELTRAMI-LAPLACE OPERATOR ON HOMOGENEOUS PSEUDO-RIEMANNIAN SYMMETRIC SPACES OF RANK 1

The Beltrami-Laplace operator on a homogeneous pseudo-Riemannian symmetric space of rank 1 is ultrahyperbolic in general. The present article gives a description of the eigenfunctions of this operator in a space of generalized functions with a restriction on their growth at infinity.

THE FOURIER-MELLIN-WHITTAKER TRANSFORM ON HOROSPHERES OF HOMOGENEOUS PSEUDO-RIEMANNIAN SYMMETRIC SPACES OF RANK 1

An integral transformation carrying the Laplace-Beltrami operator on a homogeneous symmetric pseudo-Riemannian space of rank 1 into an ordinary differential operator is constructed. Spaces dual with respect to this transformation are determined, along with bounded operators on them. Bibliography: 14 titles.

Fourier transform on the orispheres of homogeneous pseudo-Riemannian symmetric spaces of rank 1

Fourier transform on the orispheres of homogeneous pseudo-Riemannian symmetric spaces of rank 1

Bessel functions onPn

In this paper Bessel functions are defined in the homogeneous symmetric space GL(n,R)/O(n). Two definitions are given. One is an integral over the space itself, the other is a Euclidean integral. A relation between the integrals is proved. The use of this relation is shown explicitly in a low dimensional case. Some additional properties of these Bessel functions are then noted. I. Introduction. The Riemannian symmetric space Pnoi n X n symmetric matrices over the real numbers has appeared in many areas of mathematics. It has been studied in connection with multivariate statistics [J], [I], with analytic number theory [S], [M], and with the study of other higher rank symmetric spaces, such as the Siegel upper half space of n X n complex matrices with imaginary part in Pn. The above references have been cited merely as examples of the uses of Pn. One way to approach questions about Pn is to use harmonic analysis and special functions. Pn has many coordinate systems. In the geodesic polar coordinate system a general theory has been worked out by HarishChandra [HC] and Helgason [Hel]. Using some computations of BhanuMurti [BM], Audrey Terras has been able to make this theory explicit for Pn [T]. However, it seems that the so-called partial Iwasawa coordinate system is needed in the study of some questions that arise in number theory. This is the coordinate system that is used in this note. In fact, the functions and formulas proved here have already found application in the Fourier series expansion of Eisenstein series for GL(3, Z) due to recent work of Kaori Imai and Audrey Terras [I-T]. The purpose of this note will be to define some Bessel functions for Pn and prove some of their properties. To begin, however, we establish some notation and recall some basic facts about Pn. We will only mention those details of structure that directly concern us here.

ON THE EIGENFUNCTIONS OF THE BELTRAMI-LAPLACE OPERATOR ON HOMOGENEOUS SYMMETRIC SPACES OF RANK 1

This paper concerns itself with a differential equation which arises in the determination of the eigenfunctions of the Beltrami-Laplace operator on homogeneous symmetric spaces of rank 1. It is related to the Whittaker equation, and its coefficients are quadratic forms which do not depend on the variable of differentiation. All of the solutions of this equation in a space of generalized functions with restricted growth are described.Bibliography: 7 titles.

Reflexion spaces and homogeneous symmetric spaces

Let be S(x): yi—>x-y the left multiplication with x in M. This is an involutive map of M onto itself leaving x fixed, which may be interpreted as the reflexion in the point x. Let % be a finite dimensional Jordan algebra and / the set of invertible elements of §K. In general for x, yCi.1 their product xy is not in ƒ, so / does not inherit a multiplicative structure from 9I. However, x-y = 2x(xy'~)—xy~ is invertible ( [ l ] ) , and the multiplication x • y makes I a reflexion space. Every group is a reflexion space with the new product x-y = xy~x. Every set is a reflexion space with the trivial product x-y = y for all x and y. A reflexion space M where M is a connected paracompact Cmanifold and /x: MXM—+M is differentiate is called a differentiable reflexion space. The following construction gives examples. Let G be a connected Lie group, a an involutive automorphism of G and H a subgroup of G lying between the group of all fixed points of a and its identity component. Then G/His a homogeneous symmetric space and G(G/H, H) is a principal fibre bundle with base space G/H and structure group H. Let H operate on a connected manifold F on the left and let be GXHF the bundle associated with G(G/H% H) with typical fibre F (cf [2]). We denote the equivalence class of (g, x) EGXF'mGXHFby g®x. In case F i s apo in t , wehave G X ^ ^ = G / i J .