Topics in Geometry

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.

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.

Non-Linear Elliptic Equations on Riemannian Manifolds with the Sobolev Critical Exponent.- Symmetric Cones.- Pseudo-Hermitian Symmetric Spaces of Tube Type.- Homogeneous Riemannian Manifolds Whose Geodesics Are Orbits.- On the D-Module and Formal-Variable Approaches to Vertex Algebras.- The Lowest Eigenvalue for Congruence Groups.- Signatures of Roots and a New Characterization of Causal Symmetric Spaces.- Admissible Limit Sets of Discrete Groups on Symmetric Spaces of Rank One.- D'Atri Spaces.- Multiple Point Blowup Phenomenon in Scalar Curvature Equations on Spheres ofDimension Greater Than Three.- The Harish-Chandra Realization for Non-Symmetric Domains in ?n.- How many Lorentz Surfaces Are There?.- On a Theorem of Milnor and Thom.- Riemannian Exponential Maps and Decompositions of Reductive Lie Groups.- Weakly Symmetric Spaces.