© 2011

Spherical Tube Hypersurfaces

  • This is a research monograph which is quite unique in a number of ways

  • However, it is hard to state the main features of the book briefly for non-experts

  • As a result, I am afraid I cannot come up with simple selling points that would be understood by the general reader and even by mathematicians who are not experts in the area of several complex variables


Part of the Lecture Notes in Mathematics book series (LNM, volume 2020)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Alexander Isaev
    Pages 1-33
  3. Alexander Isaev
    Pages 35-40
  4. Alexander Isaev
    Pages 41-53
  5. Alexander Isaev
    Pages 55-82
  6. Alexander Isaev
    Pages 97-121
  7. Alexander Isaev
    Pages 123-184
  8. Alexander Isaev
    Pages 195-212
  9. Back Matter
    Pages 213-220

About this book


We examine Levi non-degenerate tube hypersurfaces in complex linear space which are "spherical," that is, locally CR-equivalent to the real hyperquadric. Spherical hypersurfaces are characterized by the condition of the vanishing of the CR-curvature form, so such hypersurfaces are flat from the CR-geometric viewpoint. On the other hand, such hypersurfaces are also of interest from the point of view of affine geometry. Thus our treatment of spherical tube hypersurfaces in this book is two-fold: CR-geometric and affine-geometric. As the book shows, spherical tube hypersurfaces possess remarkable properties. For example, every such hypersurface is real-analytic and extends to a closed real-analytic spherical tube hypersurface in complex space. One of our main goals is to provide an explicit affine classification of closed spherical tube hypersurfaces whenever possible. In this book we offer a comprehensive exposition of the theory of spherical tube hypersurfaces, starting with the idea proposed in the pioneering work by P. Yang (1982) and ending with the new approach put forward by G. Fels and W. Kaup (2009).


32-XX CR-geometry affine homogeneity spherical hypersurfaces tube hypersurfaces

Authors and affiliations

  1. 1.Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia

Bibliographic information


From the book reviews:

“The main goal and purpose of Isaev’s book is to explore the invariant theory of the special class of spherical tube hypersurfaces. … this book will be of interest and of value to everyone working on the equivalence problem for CR structures.” (Thomas Garrity, Bulletin of the American Mathematical Society, Vol. 51 (4), 2014)