Geodesic
On principal directions of hyperquadric in Hilbert space
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Труды геометрического семинара, Tome 147 (2005) no. 1, pp. 173-180

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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. 
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     title = {On principal directions of hyperquadric in {Hilbert} space},
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V. E. Fomin. On principal directions of hyperquadric in Hilbert space. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Труды геометрического семинара, Tome 147 (2005) no. 1, pp. 173-180. http://geodesic.mathdoc.fr/item/UZKU_2005_147_1_a16/