A foliated metric rigidity theorem for higher rank irreducible symmetric spaces

LetF be a foliation of a compact manifold with a transverse invariant measure of finite total mass. We prove that ifF admits a leafwise metric such that every leaf is an irreducible symmetric space of noncompact type and higher rank, then any other leafwise metric of nonpositive curvature is also symmetric along any leaf in the support of the transverse measure. A rank one version of this result is also exposed.The second author is partially supported, by a Seed Grant from The Ohio State University.     

Equicontinuous foliated spaces

Riemannian foliations are characterized as those foliations whose holonomy pseudogroup consists of local isometries of a Riemannian manifold. Their main structural features are well understood since the work of Molina. In this paper we analyze the more general concept of equicontinuous pseudogroup of homeomorphisms, which gives rise to the notion of equicontinuous foliated space. We show that equicontinuous foliated spaces have structural properties similar to those known for Riemannian foliations: the universal covers of their leaves are in the same quasi-isometry class, leaf closures are homogeneous spaces, and the holonomy pseudogroup is indeed given by local isometries.     

Integral foliated simplicial volume of aspherical manifolds

Simplicial volumes measure the complexity of fundamental cycles of manifolds. In this article, we consider the relation between the simplicial volume and two of its variants — the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space.     

The Newton transformation and new integral formulae for foliated manifolds

In this article, we show that the Newton transformations of the shape operator can be applied successfully to foliated manifolds. Using these transformations, we generalize known integral formulae (due to Brito–Langevin–Rosenberg, Ranjan, Walczak, etc.) for foliations of codimension one. We obtain integral formulae involving rth mean curvature of the second fundamental form of a foliation, the Jacobi operator in the direction orthogonal to the foliation, and their products. We apply our formulae to totally umbilical foliations and foliations whose leaves have constant second order mean curvature.     

Integral formulae on hypersurfaces in Riemannian manifolds

The coefficients of the complete set of n fundamental forms of a hypersuface V_n−1 imbedded in an n-dimensional Riemannian space V_n, as recently introduced[(5)], are used to construct certain tensor fields over V_n−1 which display some remarkable features. In particular, the divergences of these tensor fields can be expressed very simply in terms of polynomials involving the curvature tensor of V_n, the coefficients of the n fundamental forms, and the r^th curvatures of V_n−1. As the result of an application of the generalized divergence theorem of Gauss to these relations a set of integral formulae on V_n−1 is obtained. The integrands of these integral formulae can be expressed very simply in terms of the n fundamental forms of V_n−1. By successive specialization it is indicated how known integral theorems([2], [3], [6], [7], [8]) can be derived as particular cases, which is possible partly as a result of the fact that the polynomial referred to above vanishes identically whenever V_n is a space of constant curvature. This research was supported by the National Research Council of Canada. Entrata in Redazione il 21 agosto 1970.     

Visible actions on symmetric spaces

A visible action on a complex manifold is a holomorphic action that admits a J-transversal totally real submanifold S. It is said to be strongly visible if there exists an orbit-preserving anti-holomorphic diffeomorphism σ such that σ|_S = id. In this paper we prove that for any Hermitian symmetric space D = G/K the action of any symmetric subgroup H is strongly visible. The proof is carried out by finding explicitly an orbit-preserving anti-holomorphic involution and a totally real submanifold S. Our geometric results provide a uniform proof of various multiplicity-free theorems of irreducible highest weight modules when restricted to reductive symmetric pairs, for both classical and exceptional cases, for both finite- and infinite-dimensional cases, and for both discrete and continuous spectra.     

Killing forms on symmetric spaces

Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew-symmetric. We show that a compact simply connected symmetric space carries a non-parallel Killing p-form (p?2) if and only if it isometric to a Riemannian product Sk×N, where Sk is a round sphere and k>p.     

Projectively symmetric spaces

Sunto In questo lavoro l'autore estende la nozione di simmetria al caso di una varietà reale C dotata di una connessione proiettiva normale, introducendo la classe degli spazi proiettivamente simmetrici. L'autore prova che ogni spazio proiettivamente simmetrico p-iperbolico è proiettivamente equivalente ad uno spazio Riemanniano simmetrico di Einstein con curvatura negativa. Infine l'autore prova che una varietà Riemanniana completa n-dimensionale (n>/2) localmente affinemente simmetrica, proiettivamente simmetrica in senso proprio e proiettivamente omogenea è proiettivamente equivalente alla sfera Sⁿ oppure allo spazio proiettivo realeRP ⁿ.     

Quantum symmetric spaces

Let G be a semisimple Lie group, g its Lie algebra. For any symmetric space M over G we construct a new (deformed) multiplication in the space A of smooth functions on M. This multiplication is invariant under the action of the Drinfeld-Jimbo quantum group U_hg and is commutative with respect to an involutive operator . Such a multiplication is unique. Let M be a kählerian symmetric space with the canonical Poisson structure. Then we construct a U_hg-invariant multiplication in A which depends on two parameters and is a quantization of that structure.     

Parabolic symmetric spaces

We study here systems of symmetries on |1|-graded parabolic geometries. We are interested in smooth systems of symmetries, and we discuss non-flat homogeneous |1|-graded geometries. We show the existence of an invariant admissible affine connection under quite weak condition on the system.     

Integral geometric formulae for projection functions

Principal kinematic and Crofton formulae are established for projection functions of convex bodies. In the case of centrally symmetric bodies, the results are expressed in terms of Radon transforms of the projection functions.Supported in part by a grant from the National Science Foundation (DMS-8908717).     

Complex crowns of Riemannian symmetric spaces and non-compactly causal symmetric spaces

In this paper we define a distinguished boundary for the complex crowns of non-compact Riemannian symmetric spaces . The basic result is that affine symmetric spaces of can appear as a component of this boundary if and only if they are non-compactly causal symmetric spaces.

     

Pseudo-Riemannian manifolds modelled on symmetric spaces

We construct irreducible pseudo-Riemannian manifolds (M, g) of arbitrary signature (p, q) with the same curvature tensor as a pseudo-Riemannian symmetric space which is a direct product of a two-dimensional Riemannian space form M ₂(c) and a pseudo-Euclidean space with the signature (p, q ? 2), or (p ? 2, q), respectively.