Topics in geometry. In memory of Joseph D'Atri

Description

This collection of articles serves to commemorate the legacy of Joseph D'Atri, who passed away on April 29, 1993, a few days after his 55th birthday. Joe D' Atri is credited with several fundamental discoveries in ge­ ometry. In the beginning of his mathematical career, Joe was interested in the generalization of symmetrical spaces in the E. Cart an sense. Symmetric spaces, differentiated from other homogeneous manifolds by their geomet­ rical richness, allows the development of a deep analysis. Geometers have been constantly interested and challenged by the problem of extending the class of symmetric spaces so as to preserve their geometrical and analytical abundance. The name of D'Atri is tied to one of the most successful gen­ eralizations: Riemann manifolds in which (local) geodesic symmetries are volume-preserving (up to sign). In time, it turned out that the majority of interesting generalizations of symmetrical spaces are D'Atri spaces: natu­ ral reductive homogeneous spaces, Riemann manifolds whose geodesics are orbits of one-parameter subgroups, etc. The central place in D'Atri's research is occupied by homogeneous bounded domains in en, which are not symmetric. Such domains were discovered by Piatetskii-Shapiro in 1959, and given Joe's strong interest in the generalization of symmetric spaces, it was very natural for him to direct his research along this path Front Matter....Pages i-xxi Non-Linear Elliptic Equations on Riemannian Manifolds with the Sobolev Critical Exponent....Pages 1-100 Symmetric Cones....Pages 101-121 Pseudo-Hermitian Symmetric Spaces of Tube Type....Pages 123-154 Homogeneous Riemannian Manifolds Whose Geodesies Are Orbits....Pages 155-174 On the D -Module and Formal-Variable Approaches to Vertex Algebras....Pages 175-202 The Lowest Eigenvalue for Congruence Groups....Pages 203-212 Signatures of Roots and a New Characterization of Causal Symmetric Spaces....Pages 213-229 Admissible Limit Sets of Discrete Groups On Symmetric Spaces of Rank One....Pages 231-240 D’Atri Spaces....Pages 241-284 Multiple Point Blowup Phenomenon in Scalar Curvature Equations on Spheres of Dimension Greater Than Three....Pages 285-294 The Harish-Chandra Realization for Non-Symmetric Domains in ℂ n ....Pages 295-314 How Many Lorentz Surfaces Are There?....Pages 315-330 On a Theorem of Milnor and Thom....Pages 331-348 Riemannian Exponential Maps and Decompositions of Reductive Lie Groups....Pages 349-354 Weakly Symmetric Spaces....Pages 355-368