# Spherical Tube Hypersurfaces

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

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

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

- DOI https://doi.org/10.1007/978-3-642-19783-3
- Copyright Information Springer-Verlag Berlin Heidelberg 2011
- Publisher Name Springer, Berlin, Heidelberg
- eBook Packages Mathematics and Statistics
- Print ISBN 978-3-642-19782-6
- Online ISBN 978-3-642-19783-3
- Series Print ISSN 0075-8434
- Series Online ISSN 1617-9692
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