projective space (inhomogeneous representation) (hypersphere) These two examples are quadrics and are projectively equivalent. Simple examples, which are not quadrics can be obtained by the following constructions: (a) Glue one half of a hypersphere to a suitable hyperellipsoid in a smooth way. (b) In the first two examples replace the expression by . Remark: The real examples can not be converted into the complex case (projective space over ). In a complex projective space of dimension there are no ovoidal quadrics, because in that case any non degenerated quadric contains lines. But the following method guarantees many non quadric ovoids: For any non-finite projective space the existence of ovoids can be proven using transfinite induction. Finite examples Any ovoid in a finite projective space of dimension over a field of characteristic is a quadric. The last result can not be extended to even characteristic, because of the following non-quadric examples: For odd and the automorphism the pointset is an ovoid in the 3-dimensional projective space over (represented in inhomogeneous coordinates). Only when is the ovoid a quadric. is called the Tits-Suzuki-ovoid. Criteria for an ovoid to be a quadric An ovoidal quadric has many symmetries. In particular: Let be an ovoid in a projective space of dimension and a hyperplane. If the ovoid is symmetric to any point (i.e. there is an involutory perspectivity with center which leaves invariant), then is pappian and a quadric. An ovoid in a projective space is a quadric, if the group of projectivities, which leave invariant operates 3-transitively on , i.e. for two triples there exists a projectivity with . In the finite case one gets from Segre's theorem: Let be an ovoid in a finite 3-dimensional desarguesian projective space of odd order, then is pappian and is a quadric. Generalization: semi ovoid Removing condition (1) from the definition of an ovoid results in the definition of a semi-ovoid: A point set of a projective space is called a semi-ovoid if the following conditions hold: (SO1) For any point the tangents through point exactly cover a hyperplane. (SO2) contains no lines. A semi ovoid is a special semi-quadratic set which is a generalization of a quadratic set. The essential difference between a semi-quadratic set and a quadratic set is the fact, that there can be lines which have 3 points in common with the set and the lines are not contained in the set.