Let G be a connected Lie subgroup of the real orthogonal group 0(n). For the action of G on R', we construct linear subspaces a that intersect all orbits. We determine for which G there exists such an a meeting all the G-orbits orthogo- nally; groups that act transitively on spheres are obvious examples. With few exceptions all possible G arise as the isotropy subgroups of Riemannian symmetric spaces. Introduction. Let G be a compact Lie group acting on a real vectorspace V, and let K ,.) be a G-invariant inner product on V. Having a linear cross-section a c V of

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Polar coordinates induced by actions of compact Lie groups

We construct ten-dimensional supergravity solutions corresponding to the near horizon limit of IIB fivebranes wrapping special lagrangian three-cycles of constant curvature. The case of branes wrapping a three-sphere provides a gravity dual of pure = 2 super--Yang-Mills theory in D = 3. The non-trivial part of the solutions are seven manifolds that admit two G2 structures each of which is covariantly constant with respect to a different connection with torsion. We derive a formula for the generalised calibration for this general class of solutions. We discuss analogous aspects of the geometry that arises when fivebranes wrap other supersymmetric cycles which lead to Spin(7) and SU(N) structures. In some cases there are two covariantly constant structures and in others one.

Fivebranes wrapped on SLAG three-cycles and related geometry

A static minimal energy configuration of a super p-brane in a supersymmetric $(n+1)$-dimensional spacetime is shown to be a ``generalized calibrated'' submanifold. Calibrations in ${E}^{(1,n)}$ and ${\mathrm{AdS}}_{n+1}$ are special cases. We present several M-brane examples.

Supersymmetry and generalized calibrations

f(x,y)= (y,y ax +c). The diffeomorphisms in this class have complicated dynamics: in particular, they have periodic points of infinitely many periods and positive topological entropy (see [FM, S]). We can distinguish between these classes by considering the growth of the degrees of iterates of the diffeomorphism. Define deg(f ) to be the maximum of the degrees of the coordinate functions. This quantity is not a conjugacy invariant of f, hence not a dynamical invariant. We can construct a conjugacy invariant, which we call the dynamical degree, as follows:

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Polynomial diffeomorphisms of ². II. Stable manifolds and recurrence

Domains with Non-Compact Automorphism Group: A Survey☆

Bounded domains with prescribed group of automorphisms

Introduction I. Spinors in dimension 8m 1.1. Spinors and pinors 1.2. Constructing calibrations from spinors 1.3. Triality and the octonians II. Dimension 16 and Es II.1. A model for the Clifford algebra Cl(16) II.2. The orbit structure of Spin(16) on 8+(16) II.3. Maximal isotropy subgroups of Spin(16) II.4. Calibration on R 16 constructed by squaring spinors III. Additional results in higher dimension III.1. A differential system for each geometry III.2. Pure spinors and calibrations III.3. Spinors in dimensions 50 mod 8 References

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Calibrations and spinors

Here we consider complex manifolds M which are holomorphically separable, i.e., the global holomorphic functions (9 (M) separate the points of M. We investigate T n (n-torus) actions on M, where T"={(ei~ ei~ 0eN"}, and n is the complex dimension of M. We assume that the action is effective, i.e., if 0ET ~ is nonzero, then there exists z e M with O.z+z. Our final assumption on the action is one of smoothness; the map T n x M ~ M given by (0, z ) , O . z is C 1 in both variables and holomorphic in z. The most obvious examples of such actions are given by Reinhardt domains in ~", with the usual action O.z=(ei~ . . . . , e i ~ The standard Reinhardt action may also be changed as follows: if A is an algebraic automorphism of T n, then O.az=(AO).z. An obvious question that arises is whether all T"-actions can arise in this way. Our main results are contained in the following four theorems.

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T n -actions on holomorphically separable complex manifolds

Calibrations on R6

In this paper we investigate a class of Lie group actions on$\mathbb{C}^N $, the so-calledpolar actions, that naturally generalize the standard$\mathbb{T}^N $ actions. For a domain invariant under such an action (i.e., a generalized Reinhardt domain) we characterize the invariant plurisubharmonic functions and determine the envelope of holomorphy in geometric terms. For a generalized Reinhardt domain containing the origin of$\mathbb{C}^N $ we also compute its automorphism group.

Jiri Dadok

Papers

Bounded domains with prescribed group of automorphisms

Calibrations and spinors

T n -actions on holomorphically separable complex manifolds

Calibrations on R6

Generalized reinhardt domains