Intersections of Quadrics with Spheres
by
Tsasa Lusala
Technische Universität Berlin, Germany

Poster

In Euclidean space Rⁿ⁺² we study the intersections of central quadrics with spheres where we consider the intersections as hypersurfaces in Sⁿ⁺¹(1). It is our aim to characterize such intersections within the class of all hypersurfaces in Sⁿ⁺¹(1) with Weingarten operator of maximal rank. The methods we use are similar to methods from affine hypersurface theory.

Summary:

A hypersurface immersion in a space form is called non-degenerate if the second fundamental form defines a semi-Riemannian metric. The symmetric difference tensor field C:=ѹ-Ѳ, where ѹ and Ѳ are the Levi-Civita connections of the first fundamental form (induced metric) and the second fundamental form, resp., has interesting geometric properties. In Blaschke's affine hypersurface theory, the analogous equation C=0 characterizes quadrics, while in the general relative geometry quadrics are characterized by the equation [C\tilde]=0, [2], [3], where [C\tilde] is the traceless part of the difference tensor field C. In this paper we study intersections of hyperquadrics and hyperspheres in Euclidean space Rⁿ⁺²; we consider such intersections as hypersurfaces in Sⁿ⁺¹(1) and assume the hypersurfaces to be non-degenerate. Do such intersections satisfy the equation [C\tilde]=0 as regular immersions in spheres? We observe that if a quadric and a sphere are centered at the same point, then the intersection has this property; otherwise, the necessary condition for such intersection to satisfy the condition [C\tilde]=0 is that the immersion has to be totally umbilical in Sⁿ⁺¹. The next question is to know whether a (hyper)surface of a sphere Sⁿ⁺¹ satisfying the condition [C\tilde]=0 is contained in an open part of some quadric Qⁿ⁺¹ ( =/= Sⁿ⁺¹) of Rⁿ⁺². We investigate this situation for n >= 2 and in detail for dimension 2; we use results from [1] where non-isoparametric regular immersions without umbilics in S³(1) are completely classified. We characterize surface immersionsin S³(1) satisfying [C\tilde]=0 in terms of Euclidean quadrics.

References

[1] T. Lusala: Non-isoparametric surfaces in S³(1) with two distinct non-zero principal curvature functions, to appear.

[2] K. Nomizu, U. Pinkall: Cubic form theorem for affine immersions, Results in Mathematics, 13 (1988), 338-362.

[3] U. Simon, A. Schwenk-Schellschmidt, H. Viesel: Introduction to the Affine Differential Geometry of Hypersurfaces, Lecture Notes, Science University of Tokyo (1991), ISBN 3 7983 1529 9.

Date received: May 2, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-37.