Abstract:
In the finite dimensional case, a hypersurface $\Sigma$ in $(n+1)$-dimensional Euclidean space has $n$ principal directions: the eigenvectors of the Weingarten operator at a given point of $\Sigma$. The algorithm for finding the principal directions is well known for this case: one needs to find the roots of the characteristic polynomial $n$-th degree and to solve a system of linear equations. For the hypersurfaces of the infinite dimensional Hilbert space this algorithm fails. Moreover, it is possible that the Weingarten operator has no eigenvalues at all. In the present paper, we use another approach to the problem of finding principal directions of a hyperquadric of parabolic type. Given a local representation of an arbitrary nonzero vector, we explicitly find a point of the surface at which this vector has a principal direction.
Keywords:Hilbert space, Weingarten operator, principal direction of hyperquadric.
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