A differential-geometrical criterion for quadratic Veronese embeddings

We obtain a criterion for quadratic Veronese varieties. We prove that in the set of smooth $n$-dimensional submanifolds of the projective space $P^N$ of dimension $N=n(n+3)/2$ only the Veronese varieties have the following two properties: (i) the tangent projective spaces at any two points intersect in a point, (ii) the osculating projective space at every point coincides with the ambient space. This result is a generalization to arbitrary $n$ of the criterion for two-dimensional Veronese surfaces in $P^5$ proved by Griffiths and Harris. We also find a criterion for a pair of submanifolds of $P^N$ to be contained in the same Veronese variety. We obtain calculation formulae that enable one to use these criteria in practice.