On principal directions of hyperquadric in Hilbert space

A hypersurface in an $(n+1)$-dimensional Euclidean space has $n$ principal directions at each point: the eigenvectors of the Weingarten operator. And for a hypersurface in the infinite-dimensional Hilbert space, the Weingarten operator possibly has no eigenvectors. In the present paper we show that a hyperquadric in the Hilbert space determined by a positive definite quadratic form has principal directions under some additional assumptions. For a given direction we write an explicit expression for the point of the hyperquadric where this direction is principal. Also we give examples of these hyperquadrics.