Geodesic
Principal Directions of Hyperquadric of Parabolic Type in Hilbert Space
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Kazanskii Gosudarstvennyi Universitet. Uchenye Zapiski. Seriya Fiziko-Matematichaskie Nauki, Tome 151 (2009) no. 4, pp. 197-202

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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
Mots-clés : principal direction of hyperquadric. 
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     title = {Principal {Directions} of {Hyperquadric} of {Parabolic} {Type} in {Hilbert} {Space}},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
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     publisher = {mathdoc},
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     url = {http://geodesic.mathdoc.fr/item/UZKU_2009_151_4_a17/}
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V. E. Fomin. Principal Directions of Hyperquadric of Parabolic Type in Hilbert Space. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Kazanskii Gosudarstvennyi Universitet. Uchenye Zapiski. Seriya Fiziko-Matematichaskie Nauki, Tome 151 (2009) no. 4, pp. 197-202. http://geodesic.mathdoc.fr/item/UZKU_2009_151_4_a17/